This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
,l + 1> yield t h e o c c w e n c e
.
-
89
INFINITE RULES I N FINITE SYSTEMS
(1) - ( 4 ) a r e more a r l e s s obvious f r o m t h e d e f i n i t i o n o f I n f a c t , t h e y were mentioned because t h e y a r e h e l p f u l i n v i s u a l i z i n g
GE.
PROOF.
proof o f
( 5 ) . One way t o p r o v e ( 5 ) i s t o f i r s t a s s i g n
ordinals
the
to
d e r i v a t i o n s i n Fm and t h e n use t r a n s f i n i t e i n d u c t i o n o n t h e o r d i n a l
the p
to
has
a
show t h a t P ( U ) where P ( p ) I WF
through If p = 0
M
[if F occurs a t ( n , j ) i n GE and F
E WWnWj
F"
derivation i n
'
o f ordinal
2
t h e n e v e r y branch o f
p
eE
passing
c o n t a i n s an axiom].
t h e n P ( U ) h o l d s because ( a ) d e r i v a t i o n s o f o r d i n a l 0 a r e j u s t
t n e axioms o f
F and (b) because o f
p > 0
(1). I f
t h e n u s i n g t h e induc
t i o n h y p o t h e s i s and e i t h e r ( 3 ) o r (4) one can conclude
6 , CANONICAL
P(u).
FOR THE THEOREMS OF F"
DERIVATIONS
-
,
We s h a l l now proceed t o show how g i v e n an e f f e c t i v e d i s j u n c t o r @ on t h e formal Carnap t h e o r y F t h e spread tion o f E i n
m
FGR
6,
can be transformed i n t o a d e r i v a
( p r o v i d e d E had a d e r i v a t i o n i n
F").
-
Loosely
speak-
i n g t h e d e r i v a t i o n i s o b t a i n e d by a p p l y i n g t h e d i s j u n c t o r @ t o t h e
w.f.e. thus
o c c u r r i n g a t t h e nodes o f let Z
GE.
We need a f u n c t i o n f o r t h a t purpose:
be t h e ( p a r t i a l ) r e c u r s i v e f u n c t i o n such t h a t : C ( 0 ) = 0,
C ( < i > )= i, c(n*<j;) = Z{M) Thus
I(<Eo
,...,EP > )
= ((..
. (Eo
j
El) 63
@
. ...)
63 E p ) .
We a l s o need a f u n c t i o n which t e s t s whether we have reached o r n o t . Thus
an
axiom
l e t A b e t h e p a r t i a l r e c u r s i v e f u n c t i o n such t h a t : A ( 0 ) = 1, A
. . ..
( < i > )=.
A(
,
{ a 1 (i)
is) = A
(M)
* {U}
(i).
.
.,E > ) = 0 i f f a t l e a s t <Eo,. .,'E p> = 6 , ( ~ ) , P ) = 0 i s a n axiom o f F . Consequently i f A (GE(n) one o f t h e E o , ,E P then f o r a l l successors m o f n , A (GE ( m ) ) = 0.
Hence f o r
A(<Eo,.
90 6.1
E.
DEFINITION.
B A S I S STEP.
Fo4
each E E W we d e i i n e .the 6unct.Lon DE
a6 QoLLowb:
DE(0) = E .
S u p p h e that D E ( n )
RECURSION S T E P .
Case 1 .
G. K . LOPEZ-ESCOBAR
DE(n) = 0.
has been d e 6 i n e d .
i
Then 604
let
E N,
DE(n*) = 0
Case 2 .
DE(n) # 0.
Then compuie
A ( GE( n ) ) = 0 .
Subcase 2 . 1 .
GE(n) Then d e 6 i n e 604 & i E N
D E ( n * < i > ) = 0.
Subcase 2 . 2 .
A( 6,(n))
= 1. Then
DE(n*) =
d e 6 i n e do& a l l i E N
c ( GE(n*)).
I t i s an immediate consequence o f the d e f i n i t i o n t h a t DE w i l l be w e l l -
founded ( i . e .
I n : DE(M) # 0 )
t a i n s an axiom. Furthermore, i f
i s a t r e e ) i f f every branch o f E € W
f u n c t i o n i r r e s p e c t i v e o f whether D~
then
DE
is
a
total
6,
con-
recursive
i s w e l l founded o r not. Thus i f D E i s
w e l l founded, then a p p l y i n g the r e c u r s i o n lemma ( f i x e d p o i n t theorem)
of
r e c u r s i v e f u n c t i o n theory (see Rogers 1965) i t can be shown t h a t every w.f.e. o c c u r r i n g i n a node o f
D~
i s derivable i n
F&.
Rather than g i v e t h e ac-
t u a l proof we f i n d i t much more worthwhile t o e x p l a i n t h e r o l e o f t h e cond i t i o n s ( i ' ) - ( v ' ) of Section 3.Thus suppose t h a t we a r e a t a node n o f D E , t h a t F 6
U
and t h a t t h e occurrence
w - r u l e . Then i n writing
D~
n e i t h e r the parentheses nor
a):
Suppose t h a t we have a r e c u r s i v e l i s t i n g
...F. ..F
all
. ..
i s a c t i v e a t n because o f t h e
we have t h e f o l l o w i n g k i n d o f s i t u a t i o n (where we a r e
[O], F...F k, D e r F - ( I { z l ( k ) } ;
of
.. ., t h a t ...F...F [ k ] ) .
[l],
derivations i n
FER
of
i s , we have a z such t h a t f o r
In order t o be a b l e t o apply
91
I N F I N I T E RULES I N F I N I T E SYSTEMS
the r e c u r s i o n lemma we must be a b l e t o e f f e c t i v e l y determine a natural numb e r ? such t h a t F i r s t obtain
(1) (2)
...F . . . ) .
DerF"({^zj;
? can be obtained as
follows :
G E U such t h a t
G -I
...F...F
...F . . . F , [k]
G [k],
+F
k.
for a l l
Then using z and the e f f e c t i v e n e s s o f (2) (as formulated i n (v')) we determine a (3)
({zl}; G ) .
DerF-
Combining i t w i t h (1) we can determine a
(4)
Der
m
( { z2 } ;
...F .. .F) .
Then by repeated a p p l i c a t i o n s o f (i')
z2
- (iii')
such t h a t
( t h e exact number o f a p p l i c a -
t i o n s can be read o f f from n ) we can then o b t a i n a DerF" (
?I ;
.. .F. ..) .
The g i s t o f t h e above remarks i s t h a t that i s
if
in
z^
such t h a t
every branch of
6,
F then t h e r e i s a r e c u r s i v e d e r i v a t i o n o f E i n
t h e r e occurs an axiom of
f";
can
such t h a t
z1
FER t-
E. Thus, i n o r d e r t o prove Theorem 1 ( o f Section 3 ) i t E then i n every branch o f GE there occurs an axiom of F. But the l a t t e r i s t r u e f o r any ( f o r m a l ) Carnap theory (see Lemma 5.1). Thus the proof o f Theorem 1 i s now complete. s u f f i c e s t o show t h a t i f F" -I
7, A N
UNFAITHFUL A X I O M A T I Z A T I O NOF I N T U I T I O N I S M,
The o n l y (non - c o n t r i v e d ) case t h a t we know o f an u n r e s t r i c t e d w - r u l e which cannot be replaced by t h e r e c u r s i v e l y r e s t r i c t e d one i s i n i n t u i t i o n i s t i c number theory. Take, f o r example, f i r s t - o r d e r i n t u i t i o n i s t i c theory HA f o r m a l i z e d as i n Kleene 1952 and l e t sions o f HA w-rule.
HAm, HAER
number
be t h e exten-
obtained by adding t h e o - r u l e and t h e r e c u r s i v e l y r e s t r i c t e d
L e t m be t h e s t r u c t u r e
(w,+,-,O,').
shows t h a t f o r a l l sentences A o f
Then
an
easy
induction
HA:
(.1) i6 3 I= A ( L e . A h &tue i n m ) &en (.2) i6 "l I== 7 A, &en HA" I- I A .
HA" -I
A,
Furthermore, the methods used i n t h e l a s t chapter o f Kleene 1952 show f o r a l l sentences A o f HA: (.3)
i6
H A I R -I
A
&then
A .i~ fie.a-P.izabte.
Since there a r e sentences which a r e t r u e b u t not r e a l i z a b l e (e.g.
that
92
E . G. K . LOPEZ-ESCOBAR
vx [ 3 y T ( x , x , y ) v v y 7 T ( x , x , y ) ] ) i t follows t h a t HAm i s n o t a conservative extension of HA;,. The derivations of (.1) - ( . 3 ) a r e carried o u t in classical mathematics and t h u s n o t t o o relevant t o intuitionism and since H A i s supposed t o be a formalization o f ( a p a r t o f ) intuitionism, the counterexample i s also somewhat contrived. O n the whole i t i s much more natural t o use a n intui t i o n i s t i c metatheory when discussing i n t u i t i o n i s t i c s systems. I n t u i t i o n i s t i c systems a r e very often closed under Church's Rule (CR) and t h u s t h e f o l lowing argument o f Kreisel gives a very simple solution to the problem of replacing the OJ - r u l e by the recursively r e s t r i c t e d O J - r u l e (when the metatheory i s closed under CR). For simplicity l e t us consider H A , HA" and H A E R and l e t the meta theory M be closed under Church's rule. Then Kreisel's formulation o f the
-
-
-
problem i s t o show t h a t : (*)
{ A : II i - ( ( 3 c ) P r f H A ~ ( a o r ~ ' ) 5 j {A : M + ( 3 a ) P r f H A m ( a , r ~ l ) j GR
where P r f H A - and P r f H A "
GR
are (canonical) representations i n M of the
proof predicates of HA" a n d H A E R respectively. The proof of ( * ) i s a s fol 1ows : Suppose t h a t M l-(3a) P r f H A m ( a r r A 1 ) . Then since M i s closed under C R , M I- (3a),,PrfHAm(a, r ~ l ) . B u t then, because of the form of the predicates (
PrfHAm
3 a ) P r f H A m (a;'A7 GR
and )]
PrfHAm
. Thus
GR
M I-
,
M -I (
[(ja)GRPrfHA-(~,rA1)
3 a ) P r f H A m (a,'A').
-
GR
I n s p i t e of the above solution there i s s t i l l some room for improvement. I t seems t o us t h a t i f one i s to be completely f a i t h f u l t o the principles embodied in H A then one should t r y t o prove, i n t h e intLLit.ionhi6.tic me&t h e m y M, the following sentence:
Although we do n o t d o u b t t h a t i t i s rewarding t o be f a i t h f u l ,
in
this
particular instance we find t h a t i f we follow the i n t u i t i o n i s t i c metatheory (and thus look f o r solutions of the kind ( * ) - (***)) we do not r e a l l y de t e n i n e which chmaracteristics of the usual formalizationmof H A produce the r e s u l t t h a t H A i s n o t a conservative extension of HA,,
-
.
93
I N F I N I T E RULES I N F I N I T E SYSTEMS
The p o i n t o f view we p r e f e r t o adopt ( f o r t h e purpose o f t h i s paper) i s that
HA
whay
HA,
i s j u s t a formal system amongst many and then t o t r y t o f i n d
s e r v a t i v e extension o f
out
h a s t h e property t h a t HAm i s n o t a con-
as a f o r m a l system,
HA^^.
The d e f i n i t i v e answer s t i l l eludes u s . However we have reason t o
be-
HA
l i e v e t h a t i t has something t o do w i t h t h e f a c t t h a t a formal proof i n proof o r c o n s t r u c t i o n . That i s
i s an attempt t o represent the i n t u i t i v e formal proof i n HA
a
o f a sentence A i s c l o s e l y r e l a t e d t o t h e ( e f f e c t i v e )
c o n s t r u c t i o n t h a t v a l i d a t e s A . HAiR
s t i l l maintains a resemblance o f e f -
.
fectiveness, HAm does n o t (unless the netatheory i s i t s e l f c o n s t r u c t i v e ) I n order t o support t h i s b e l i e f we now introduce another f o r m a l i z a t i o n
for
f i r s t - o r d e r i n t u i t i o n i s t i c number theory which has t h e same c l a s s o f theorems as HA, b u t whose proofs try t o conceal as best as p o s s i b l e t h e
con
-
s t r u c t i v e content o f t h e theorem. I t i s f o r t h e l a t t e r reason t h a t we s h a l l c a l l i t U H A ( U n f a i t h f u l H e y t i n g ' s a r i t h m e t i c ) . I t has t h e property UHA"
that
i s a conservative extension o f UHA&. The formulae o f U H A a r e j u s t t h e formulae o f H A , however p r o o f s
in
U H A w i l l n o t i n v o l v e t h e formulae themselves b u t r a t h e r f i n i t e , non-empty, sequences o f formulae which we s h a l l c a l l c -bequents quents)
.
(read: confused se
AXIOMS OF U H A are those c-sequents i n which t h e r e occurs axiom o f HA.
STRUCTURAL RULES OF UHA a r e t h e r u l e s o f the form:
Fmm: Pho v,ided:
. ..pAn Ano . Aol
To conclude:
n
*
0 a p m W n od
{O,.
. .,n}
DECIDING RULES OF U H A a r e given by t h e schema:
Fhom:
Aot
To conclude:
*
Aor
.-r A n - l l
--
lAn-l
B I
B B
ANCIENT RULES OF U H A are given by t h e f o l l o w i n g schemata:
Fhom:
Ao,.
..,AM - l t B o
To conclude:
Rovided:
and
Aot
-..lAn-lrB1
AOt-*.tAn-l I C
Bg' *1 c .
i s a rule o f HA,
and correspondingly f o r t h e one premiss r u l e s o f
HA.
at
-
l e a s t one
94
E . G . K. LOPEZ-ESCOBAR
D e r i v a t i o n s o f c -sequent5 i n U H A a r e t h e n d e f i n e d i n t h e usual
tree
form. We say t h a t a f o r m u l a A i s p r o v a b l e i n U H A and express i t i n
sym-
U H A I-
bols: in
A, j u s t i n case t h e one - t e r m e d c-sequent
( A ) i s derivable
UHA.
7.1
Fox any 60hmLLea
LEMMA.
t h e ~ 0 U v w i n g c o n d i t i o n b atre
A 06 H A
e@L&Jdent:
(1) H A I(2)
A.
T h a t (1)
PROOF.
-
UHA k A.
(2) i s t r i v i a l .
To prove t h a t ( 2 )
P ( n ) stand f o r the following:
(1)
Hq.
06
A s t r a i g h t f o r w a r d i n d u c t i o n o n n shows t h a t P ( n ) h o l d s f o r
Suppose n e x t t h a t Then
(A).
let
II 0 a detLiwation i n U H A 0 6 length 5 n t h e n a t l e a s t t h e ,5amLLeae i n .the end C-beqUent a 6 n 0 pmuable i n
W I l [id
one
-
HA+
n
i s a d e r i v a t i o n i n UHA
o f t h e one termed
all
n.
c -sequent
A.
Thus as f a r as t h e c l a s s o f theorems i s concerned l i A and UHA a r e e q u i v a l e n t . The o n l y d i f f e r e n c e between t h e p r o o f i n H A
and
UHA i s
that
in
U H A t h e a c t u a l p r o v a b l e f o r m u l a can be h i d d e n w i t h extraneous formulae. I n
t h e f i n i t e case t h i s c h a r a c t e r i s t i c o f
U H A causes no problem because
it
i s r e l a t i v e l y s i m p l e t o search t h r o u g h f i n i t e s e t s . However i t does make a n enormous d i f f e r e n c e when we add t h e w - r u l e .
Fmm:
Ao,
.. . , A n - l ,
To c a n d u d e : For l e t
UHA“,
Aot
BxL
- - - rAn-l,
k < w
doh
WxB
U H A i R be U H A t w - r u l e ,
UHA
+ (recursively restricted
w - r u l e ) r e s p e c t i v e l y . By a p p r o p i a t e changes i n t h e p r o o f o f Lemma 7.1
we
o b t a i n t h e f o l l o w i n g lemma: 7.2
LEMMA.
F v h any nentence A
06
H A t h e ~ o U v w i n gc o n d i t i o n 6
atre
&pLiUdent:
(1) HA” I(2)
A,
U H A ” ~A.
On t h e o t h e r hand t h e spread c o n s t r u c t e d i n s e c t i o n 5 (see Lemma
g i v e s us t h e f o l l o w i n g :
7.3
LEMMA.
Foh a &? bwtencen A
06
HA
5.1 ( 5 ) )
INFINITE RULES I14 FINITE SYSTEMS
UHA"
id and o d y id U H A i R
I- A
UHA"
Thus we o b t a i n t h a t HA",
and UHA&
95
I-A.
are a l l e q u i v a l e n t
as
far
as t h e classes o f theorems are concerned.
W e doubt t h a t
m
UHA"
( o r i t s e q u i v a l e n t UH?$.R)
i s o f much use as f a r
as p r o o f - t h e o r y i s concerned, although there might be a p p l i c a t i o n s case o f the theory o f species where i t i s sometimes c l a s s i c a l w - complete f i r s t - order subtheory ( c f . like
U H A and UHA"
to
i n the have
a
Smorynski 1973). Systems
are made f o r t h e theory o f proofs and probably should
be considered i n conjunction w i t h the &-we11
- Mints - Simpson
Kreisel
customary
founded proofs discussed
in
1974.
REFERENCES, Grzegorczyk, A., 1958.
The ~
A. Mostowski, and C. Ryll-Nardzewski
n
i andc 2ke~ w - comjdcte LvLithe-tic,
Logic, v o l . 23, pp. Kent, C. 1967.
Thc Journal of Symbolic
188-206.
F.
RenMcRed w - hiLee doh c#&hctic,Notices
o f the American Mathemati-
c a l Society, v o l . 14, pp. 665. Kleene, S . C. 1952.
I n t r o d u c t i o n t o M e t m a t h c m a t i c a , 0. van Nostrand Publ. CO. New York, x + 5 5 0 pp.
K r e i s e l , G., 1974.
G. E. Mints, and S . G. Simpson
The U b e
06
Abb,Ouct
Laflguage LIZ Eleme~%&Vcy ! ~ k 2 i Z l c 7 , t / t 5 7 1 ~ & 5 : S o mPede
agogic E x a m p l e s , A r t i c l e i n Logic Colloquium, Lecture Notes i n Math-
ematics, vol . 453, Springer Verlag Publ
L6pez 1967.
- Escobar,
.
Co.
E. G. K. Rematlks OIZ a:? in~i~u.tmglanguage c 3 L t h coiz6ZtucZi.ve 60hmlLeas, Journal o f Symbolic Logic, vol . 32, pp. 305 318.
-
The
Nelson, G. C. 1971.
A ~u&tlw,thc~d.thic.t~dw - hiLec, Colloquium Mathematicum, v o l . 23,
1
- 3.
pp.
Ohya, T. 1970.
On
heCUlAiV2 h e S . & U d O l Z
06
pao,@
i l ? cl bLjA.?hi
COn64%L&Vk?
d i ~ L t & y long expesniovm, C o n c n t . Math. Univ. S t . Paul, v o l . pp. 4 3 - 52.
in18,
96
E . G.
K. LOPEZ-ESCOBAR
Rogers, H. 1967.
Theory of Recursive Functions and Effective computability, + 482 pp.
McGraw - Hi1 1 Publ. Company, x i x Routledge, N. A. 1950.
-
Gheek M a t h e m d c n . A r t i c l e i n Eureka, The Archimedeans' Jour
nal, October, pp. 3 - 4 . Shoenfield, J. R. 1959.
On a hen&Loted
w-hLLee, B u l l e t i n de 1'Academie Polonaise des
ences, Serie des S c i . Math., Astr.,
-
Sci
e t Phys., v o l . 7, pp. 405- 437.
Smorynski, C. 1973.
Chapter V
Of
Metamathematical Investigation of Intuitionis-
tic Arithmetic and Analysis, L e c t u r e Notes i n M a t h e m a t i c s , v o l . 344, Springer-Verlag Publ. Co. Smullyan, R. M. 1961.
Theory of Formal Systems, Annals o f Plathematics studies, 47, 142 pp., Princeton U n i v e r s i t y .
vol.
Takahashi, 1.1. 1970.
A theahcm
Takeuti, G. and 1963.
OIZ
M e 6 e c d ahdm ahitluneLLc lhlith thtlze w - u L e , J o u r n a l
of
the Mathematical Society o f Japan, v o l . 22, pp. 15 - 24.
A. Kino
On phedicaten ULith co11,5,ttu~ueLyi n d i n i t d y l o n g expke66iom, Journal o f t h e Mathematical Society o f Japan, vol. 15, pp. 176 - 190.
Department o f Mathematics U n i v e r s i t y o f Maryland C o l l e g e Park, Maryland, U . S . A .
ADDED IN P R I N T (Jan. 1977), ( 1 ) The use o f "Formal" i n been
2,
i s misleading. A b e t t e r choice w o u l d
have
"Recursively definable".
( 2 ) I would l i k e t o thank Professor A. T r o e l s t r a f o r p o i n t i n g o u t t h a t
D.
97
I N F I N I T E RULES I N F I N I T E SYSTEMS
P r a w i t z i n t h e a r t i c l e " C o m m e n t . s on Gentzen-type pfioceduteh and t h e c & ~ n i c d flation 06 .t)LLLth" (Proof Theory Symposium, L e c t u r e Notes i n Mathematics, v o l . 500) a l s o c o n s i d e r s t h e e f f e c t s o f a d d i n g a
non-
iterative classical disjunction t o i n t u i t i o n i s t i c logic. ( 3 ) G. C . M i n t s has shown t h a t t h e r e i s no p a r t i a l r e c u r s i v e f u n c t i o n
mapping f o r m u l a e
A o f intuitionistic arithmetic to locally
r e c u r s i v e p r o o f f i g u r e s o f H A t w - r u l e such t h a t :
3p E Rec P f i o ~ ~ ~ + ~ ( p->, A T ) ( A ) i s d e f i n e d and well-founded.
'I
correct
Non-Classical Logics, Model Theory and Computability. A . I . Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
SOME REMARKS by
L.
H.
DISCUSSIVELOGIC
ON
LOPES DOS SANTOS
S . Jaskowski (Jaskowski 1948) introduced
s e t o f a l l formulas i c S5, where
h i s discussive l o g i c D~ as the
A such t h a t M A belongs t o the nodal p r o p o s i t i o n a l l o g -
M i s the usual p o s s i b i l i t y operator. Generalizing, t h e d i s -
cussive l o g i c J(K) associated w i t h any modal l o g i c K may be defined as the set
{ A : M A € K). An axiomatics f o r
J(S5) i s presented
i n
C o s t a 1975,
and i n T. Furmanowski 1975 i s proved t h a t J ( S 4 ) coincides w i t h J ( S 5 ) .
In
the f i r s t p a r t of t h i s paper, we present axiomatics f o r system),
S4, S5, J(B) and J ( S 5 )
B ( t h e Brouwerian s p e c i a l l y appropriate t o r e f l e c t t h e i r
mutual i n c l u s i o n and non- i n c l u s i o n r e l a t i o n s . The p r o p o s i t i o n a l l o g i c J(S5) i s n o t closed under a l l r u l e s o f i n f e r ence which a r e u n i v e r s a l l y v a l i d i n t h e c l a s s i c a l p r o p o s i t i o n a l l o g i c , even under a l l those which a r e v a l i d i n the p o s i t i v e p r o k o s i t i o n a l
not
logic;
p a r t i c u l a r l y , i t i s not closed under m a t e r i a l detachement ( c f . JaEkowski 1948). (A But i f we p u t A + D ( A LtidcwdiwtQg impfie6 U ) f o r M A 3 U and A A U and
B are dincusniweLy conjoined) f o r M A & 0, where
i m p l i c a t i o n and & means c l a s s i c a l conjunction, then
3
means
classical
a l l v a l i d l a w s and
r u l e s o f the p o s i t i v e p r o p o s i t i o n a l l o g i c come t o be v a l i d i n J ( S 5 ) ,when= and & are replaced r e s p e c t i v e l y by +and t a i n e d ) . I n f a c t , N.C.A.
da Costa and
A (andusual d i s j u n c t i o n i s main-
L. D u b i k a j t i s
(see
Costa
and
D u b i k a j t i s , these Proceedings) showed how t o t r a n s l a t e J ( S 5 ) i n a language whose p r i m i t i v e connectives,besides c l a s s i c a l negation and c l a s s i c a l j u n c t i o n , are j u s t discussive i m p l i c a t i o n and discussive conjunction;
dis
-
they
a l s o showed how t o axiomatize i t by means o f t h e discussive counterparts o f t h e w e l l - k n o w n axioms and r u l e s o f t h e p o s i t i v e p r o p o s i t i o n a l c a l c u l u s 99
100
L. H. LOPES DOS SANTOS
( i n c l u d i n g Peirce law), plus a f i n i t e s e t o f axiom schemas r e l a t i n g negation t o the o t h e r connectives. I n the second p a r t o f t h i s paper, we extend
this
axiomatics i n order t o o b t a i n a n a t u r a l one f o r t h e discussive l o g i c assoc i a t e d w i t h t h e p r e d i c a t e l o g i c S 5 ( w i t h the Barcan formulas). F i n a l l y we note t h a t t h i s discussive l o g i c i s i d e n t i c a l t o the discussive l o g i c
asso-
c i a t e d w i t h the p r e d i c a t e l o g i c s4.
PART I ,
1, LANGUAGE, The Logicsconsidered i n P a r t I are t o be thought as f o r m u l a t e d i n a l a n guage whose p r i m i t i v e symbols a r e denumerably many
p,, p 2 ,
...,
propositional
letters
the connectives i (negation) and V ( d i s j u n c t i o n ) , t h e necess i t y operator L , and parentheses. The f o l l o w i n g d e f i n i t i o n s a r e adopted: Df.1.
MA = df.
Df.2.
(A
3
I L I A .
C) = df. (1A
v
C).
Throughout t h i s paper, the l e t t e r s A , C and
D, w i t h o r w i t h o u t numerical
subscripts, w i l l be employed as s y n t a c t i c a l v a r i a b l e s f o r formulas; f o r the sake o f b r e v i t y , the most e x t e r n a l parentheses o f a formula w i l l be
often
omitted.
2, THE
L O G I C S B, J(B) AND J ( S 5 ) .
The l o g i c B i s included i n J ( B ) , which i s included i n J ( S 5 ) . The f o l lowing theorems r e f l e c t a x i o m a t i c a l l y the s i t u a t i o n . THEOREM 1 .
B 0 axiornatizabLe by meam o d t h e &Uoeal(Ling axiom nchana and
hulu:
4A
(1)
LA,
(2)
L ( L A 3 A),
(3)
L ( L ( A 3 C)
(4)
L(A3 Lm),
(5)
Ada
(6)
0 a &utacl*oLogy,
=)
( L A D LC)),
LC 6hom L ( A = C ) indeh LLA @om L A ,
and
LA,
101
SOME REMARKS ON D l S C U S S l V E L O G I C
i n d m A dhom LA.
(7)
PROOF:
I t i s w e l l known t h a t B i s axiomatizable by t a k i n g as axioms
all
(LA =I LC),A=>,lA, and as r u l e s m a t e r i a l detachment and Godel's r u l e . Now l e t A E B; then A i s
t a u t o l o g i e s , a l l instances o f the schemas LA
3
A,L(A
3
C)
3
a t h e s i s o f t h i s w e l l -known axiomatics; we show e a s i l y , by i n d u c t i o n , t h a t LA
i s a t h e s i s o f the axiom system (1)-( 7 ) ; hence so i s A i t s e l f , b y
(7).
Conversely, schemas (1)- ( 4 ) and r u l e s (5) - ( 7 ) can be c l e a r l y derived in B ; t h e r e f o r e , a l l theses o f the axiom sys the w e l l -known axiomatics f o r tem (1)- (7) belong t o B
PROOF:
.
J (B) 0 axiomatizabLe by meanb ad (1)- (7) and:
THEOREM 2.
i n d e h A &om
(8)
-
L e t A E J(B);
LMA. then
MA E
B, by d e f i n i t i o n o f J(B),
f o r B i s closed under Godel's r u l e ; t h e r e f o r e , LM4
and
LMA E B,
i s a t h e s i s o f the a x i -
om system (1)- (8!, by Theorem l, and so i s A , by (8). On the o t h e r hand, l e t A be a t h e s i s o f t h e axiom system (1)- (8); i t can be e a s i l y shown, by i n d u c t i o n , t h a t f44 E B; t h e r e f o r e I t i s worth n o t i n g t h a t r u l e
-
A E J(B), (7)
by d e f i n i t i o n o f J(B).
i s superfluous i n the axiom
system
(1) (8) : THEOREM 3 . J (B) 0 axiamaAzabLe by meum ad (1)- ( 6 ) and ( 8 ) .
PROOF: By means o f (1)- (6) and (81, any formula A can be derived LA
from
as f o l l o w s :
(i)
LA ;
(ii) L(A
3 LF.24),
by ( 4 ) ;
(iii) LLMA , by (i), (ii) and (5); (iv) L ( L M (v)
LMA
(vi) A
3
W ) , by (2) ;
, by (iii), (iv) and (5);
, by (v) and
(8).
-
THEOREM 4 . J ( 5 5 ) 0 axiamatizabLe by meam v d (1) (6), ( 8 ) a n d (9)
L ( L A 3 LLA).
It i s w e l l known t h a t S5 can be axiomatized by t a k i n g as axioms a l l t a u t o l o g i e s , a l l instances o f the schemas LA 3 A, L(A 3 C) 3 ( L A 'LC),
PROOF:
102
L. H. LOPES DOS SANTOS
A 3 LMA, LA 3 LLA , and as r u l e s m a t e r i a l detachment and Gijdel ' s r u l e ; by i n d u c t i o n on the l e n g t h o f a d e r i v a t i o n o f A i n t h i s axiomatics, we show e a s i l y t h a t LA i s a t h e s i s of t h e axiomatics proposed f o r J(S5) i f A E S5.
Now l e t
C E J ( S 5 ) ; then MCES5, b y d e f i n i t i o n o f J ( S 5 ) , a n d L M C i s a t h e s i s o f
t h e axiom system proposed f o r J(S5);hence so i s C, by ( 8 ) . Conversely,if
C is
a t h e s i s o f t h e axiom system proposed f o r J ( S 5 ) , then i t i s easy t o show, b y induction, t h a t M C € S5; t h e r e f o r e C € J ( S 5 ) , ;b d e f i n i t i o n o f J ( S 5 ) . The l o g i c B i s indeed a pope,& p a r t o f J ( B ) , which i s a l s o a p h o p e h The non - i n c l u s i o n r e l a t i o n s can be thus established,taking
p a r t o f J(S5).
i n t o account Theorems 1, 2 and 4: THEOREM 5 .
B
0 n o t &Vbed undm the R d e ( 8 ) .
PROOF: With t h e h e l p o f simple Kripke diagrams, we may v e r i f y t h a t LM(Mpl 3 p,) € B and Mpl 3 p1 f? B
.
THEOREM 6 .
N o t & ivlsZances a6 ( 9 ) b d o n g t o
J(B).
PROOF: By means o f a simple Kripke diagram, we may v e r i f y t h a t J ( B ) , by d e f i n i t i o n o f J(B). ML(Lpl= LLp,) B B ; hence L(Lpl 3 LLpl)
3, THE
LOGICS
S4,
S5 AND J ( S 5 1 ,
The l o g i c s4 i s a subset o f S5, which i s a subset o f J(S5) .This chain o f i n c l u s i o n s may be a x i o m a t i c a l l y founded on Theorems 7 - 9 below. THEOREM 7 .
S 4 0 axiarnatizabLe by
mem o6 (2) and
(10) LLLA, id A 0 a M o L o g y ; (11) L ( L(A (12)
3
C)
3
L( LA
3
indeh C dhom L(A
LC));
3
C)
and
A.
PROOF: A l l theses o f t h i s axiomatics belong t o S4
, for
(2) and (10)-(12)
are c l e a r l y v a l i d i n s4. Proving t h e converse r e q u i r e s some lemmas.
The ~ o U o W i n g/uLee 0 vaeid i n t h e axiom bybteJ?i phopobed S4 i n Theahem 7 : i n 6 a A 6ham LA. LEMMA 7 . 1 .
PROOF:
T r i v i a l , by (2) and (12)
.
doh
103
SOMEREMARKS ON D I S C U S S I V E LOGIC
LEMMA 7 . 2 . The ~ o U o w i n ghu&e 0 v&d i n t h e axiom b y d t e m pkopoded doh 54 in Theohem 7: i n 6 e h L C 6hom L(A1= ( . . . = ( A n = C ) ) ...), L A j ( l < j < n ) .
PROOF: By i n d u c t i o n on n .
n = l ; from L ( A 1
CASE 1.
t h i s and L A , , CASE 2.
3
C ) , we d e r i v e
L(LA1
3
L C ) , by (11)-(12); f r o m
we d e r i v e L C , by (12).
n > 1; from LA1 and L ( A 1 =
(... = ( A n =
C ) ) ...), we d e r i v e L ( A 2
= ( ... 3 (An= C ) ) ...), by Case 1 above; from t h i s , i n d u c t i o n and LA (1 2 j 2 n ) , we d e r i v e L C .
hypothesis
j
COROLLARY. 16 C 0 a tau-totogicd cottbequence
d h v a b l e @om LA . (15 J Theohem 7 .
j
5 n)
06
A1,
..., A,,
i n t h e axiom hystem p o p o d e d
LEMMA 7 . 3 . L ( L A 3 L LA) 0 a v&d doh S4 i n Theahem 7 .
schema
06
then LC 0 60h
S4
in
t h e axiom dydtem p h o p o s e d
PROOF: L e t A be any formula, l e t C he t h e formula 1 A V A . By (10) a n d Lemma 7 . 1 , L (A = ( C 3 A ) ) i s a t h e s i s o f t h e axiom system r e f e r r e d above. From t h i s , by (11)-(12),we d e r i v e (i) L ( L A
3
L ( C =A)).
The formulas (ii) L ( L ( C = A )
(iii)
=
L(L( LC =LA)
L(LC = L A ) ) , 3
L(LLC =LLA))
a r e instances o f (ll), and
(iv) L ( L ( L L C
= LLA)
a(LLC
3
LLA))
i s an instance o f (2). From ( i ) - ( i v ) , by t h e C o r o l l a r y o f Lemma 7 . 2 , we der i v e L(LLC = ( L A
3LLA)).
3
L L A ) ) ; from t h i s , by (11)-(12),we g e t L ( L L L C S L ( L A
But L L L C i s an instance o f (10); therefore,L(LA
t h e s i s o f t h e axiom system
LLA) i s
a
r e f e r r e d above, by (12).
COROLLARY. The 60UoOuling hu&c 0 v a l i d i n t h e a x i o m a t i U p o p o s e d doh S 4
i n Theohem 7 : i n d e h L L A
dhom
LA.
Now we a r e a b l e t o complete t h e p r o o f o f Theorem 7 . I t i s w e l l
known
104
L . H. LOPES DOS SANTOS
t h a t S4 i s a x i o m a t i z a b l e by means o f a l l t a u t o l o g i e s , a l l i n s t a n c e s o f t h e schemas LA
=A
and L ( A
C)
3
L(LA
3
L C ) , m a t e r i a l detachement and Gtidel's
A € S4; i t i s n o t d i f f i c u l t t o show, by i n d u c t i o n , t h a t
r u l e . Now l e t
LA
i s a t h e s i s o f t h e axiom system proposed f o r S 4 i n t h e body o f T h e o r e m 7 , hence so i s A, by Lemma 7.1.
w i t h t h e h e l p o f Lemma 7.3;
THEOREM 8. S5 0 axiomdtizable by meam
Ln6m L A
(13)
06
(2).
(10)-(12)
and
MLA.
6hom
PROOF: S i m i l a r t o t h e p r o o f o f Theorem 7, c o n s i d e r i n g t h a t
a x i o m a t i z e d by means o f a l l t a u t o l o g i e s ,
=
L(A 3 C )
LEMMA 8 . 1
= L C)
L(LA
Lemmas 7.1-7.3
.
and
A
3
S5
can
be
a l l i n s t a n c e s o f t h e schemas LA=A,
LMA, m a t e r i a l detachement and Gtidel's r u l e .
a r e t o be c a l l e d f o r , as w e l l as:
L (A
3
LMA) 0 a v a l i d nchema a6 t h e axiam nyntem phvpaned 6vh
S5 Ln Thevhm 8.
PROOF:
I t i s e c s i l y v e r i f i a b l e t h a t a l l i n s t a n c e s o f t h e schemaML(A3LMA)
belong t o S 4; consequently, by Theorem 7, t h e y a r e a l l d e r i v a b l e by o f (2)
and
(10)-(12);
t h e r e f o r e , a l l i n s t a n c e s o f t h e schema
means
L(A
LMA)
a r e theses o f t h e axiom system proposed f o r S 5 i n Theorem 8, by (13). THEOREM 9 . J ( S 5 ) 0 axiamdtizable by meam ad (21, (14) PROOF:
Ln6m A
7.1-7.3
and
MA.
I t i s known t h a t J ( S 5 ) i s a x i o m a t i z a b l e by means o f (1)-(2)
(11)-(12)
(9),
6ham
(10)-(12)
and 8.1,
and
( 1 4 ) ( c f . Theorem 4 o f ' C o s t a
,
(4),
1975). Given Lemmas
i t i s easy t o show t h e e q u i v a l e n c e o f b o t h axiom systems.
The axiom system p r e s e n t e d f o r J ( S 5 ) i n T h e o r e m 9 i s r e a l l y an extens i o n o f t h e one p r e s e n t e d f o r S 5 i n Theorem 8, f o r any a p p l i c a t i o n o f r u l e (13) i s an a p p l i c a t i o n o f r u l e (14). The l o g i c S 5 i s indeed a p r o p e r p a r t
o f J ( S 5 ) , s i n c e we h a v e : 'THEOREM 10.
S 5 i~ not d o b e d u n d a h d e (14).
PROOF: Simple K r i p k e diagrams show us t h a t PI
3
e
M P ~ S5.
M(pl
Mpl)
€ S5 a n d
105
SOME REMARKS ON DISCUSSIVE LOGIC Although 5 4 i s a proper p a r t o f S5,J(S4) J(S5).
AS
i s n o t a proper
part
a m a t t e r o f f a c t , J(S4) i s identical t o J(S5). T h i s
t i t y , a l g e b r a i c a l l y e s t a b l i s h e d i n T. Furmanowski-1975, i s a n
of
iden-
immediate
consequence o f Theorem 9 and THEOREM 1 1 . nnd
J ( S 4 ) i b axiamatizabLe b y menflb 0 6
(2),
(10)-(12)
(14).
PROOF: L e t A € J ( S 4 ) ; t h e n MA € S 4 , by d e f i n i t i o n o f J ( S 4 ) , a n d M € S5, f o r s 5 i n c l u d e s 54; so A € ~ ( s 5 ) by , d e f i n i t i o n o f J ( S 5 ) ; therefore, A i s a t h e s i s o f t h e a x i o m a t i c s proposed f o r J ( S 4 ) , by Theorem 9.Conversely, l e t A be a t h e s i s o f t h i s a x i o m a t i c s ; by i n d u c t i o n , we show t h a t M A € 5 4 ; therefore
A € J ( S 4 ) , by d e f i n i t i o n o f J ( S 4 ) .
I f m i s a m o d a l it y
and K i s a modal l o g i c , t h e s e t {A: mA € K} i s
c a l l e d t h e m-fragment o f K. By d e f i n i t i o n , J ( K ) i s t h e M-fragment o f f o r any
K . I t i s n o t d i f f i c u l t t o v e r i f y t h a t B and
K
,
S5 a r e t h e L-frag-
ments r e s p e c t i v e l y o f J ( B ) and J ( S 5 ) . Nevertheless, t h e analogous assumpt i o n c o n c e r n i n g S 4 i s n o t t r u e : t o g e t h e r w i t h Theorem 11, i t would l e a d us t o t h e f a l s e c o n c l u s i o n t h a t S4 c o i n c i d e s
4, THE LOGICS
s4,
s5 AND
w i t h S5.
J(B)
The l o g i c s s4 and s 5 n e i t h e r a r e subsets o f J ( B ) n o r have i t as a subset.
p1 belongs t o J ( B ) and does n o t belong n o r o f S 4 . On t h e o t h e r hand, L(Lpl=- LLp,) belong t o S4 and i t does n o t belong t o J ( B ) ; so J ( B ) i n c l u d e s n e i t h e r S4 n o r S5.
On t h e one hand, t h e f o r m u l a Mpl= t o s 5 ; so
J ( B ) i s a subset n e i t h e r o f S 5
PART I 1
1, THE
AXIOM
SYSTEM J*,
We s h a l l c o n c e i v e t h e axiom system J* as expressed i n
a
language
whose p r i m i t i v e symbols are: a denumerable i n f i n i t e s e t o f i n d i v i d u a l v a r i ables; a nonvoid s e t o f n - a r y p r e d i c a t e l e t t e r s , f o r each n a t u r a l number n ,
106
L . H. LOPES DOS SANTOS
n 2 1; the c l a s s i c a l connectives i and V; t h e necessity operator L; t h e ex-
i s t e n t i a l q u a n t i f i e r 3 ; and parentheses. The s e t o f formulas i s r e c u r s i v e l y d e l i m i t e d as usual. The l e t t e r s x and q w i l l be employedas s y n t a c t i c a l vari a b l e s f o r i n d i v i d u a l v a r i a b l e s o f t h e object-language. standard r e s t r i c t i o n ,
to
the
the notation A ( x l y ) w i l l r e f e r t o the f o r m u l a
Subject
ob-
t a i n e d from A by r e p l a c i n g each f r e e occurrence o f x i n A by an o f q. F i n a l l y we i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n s :
D f . 1.
(A
D f . 2.
( A & C)
=df. l ( 1 A V l C ) ,
D f . 3.
(A
=df. ( ( A
D f . 6.
(1A V C),
=df.
C)
( A A C)
C) & (C
,
=df.
(MA
3
=
( M A & C).
df.
( A ->C)
3
1LlA
MA =
D f . 4. D f . 5.
C)
3
occurrence
df.
3
A)),
C),
J* h a s t h e f o l l o w i n g a x i o m schemas and r u l e s :
J*l.
L A , id A . i + a .tautology.
J*2.
L(LA
J*3.
L(L(A
3
J*4.
L(A
LMA).
J*5.
L n d e h C &tom L ( A
J*6.
1.ndehA dhom MA.
J*7.
L(A(x/y)
J*8.
1n6eh L ( 3 x A
3
3
A). C)
=
The n o t a t i o n J* I-
3
L(LA
3
LC)).
C ) and A.
3xA). C)
Ahom
L(A
3
C), id
x h
nv-t dhee i n C.
A means t h a t A i s a t h e s i s o f J*.
d i 6 C U A b i V t Logic amoiated w i t h t h e mod& phedicate logic 5 5 ( w i t h t h e B a a c a n d o m u l a ) .
THEOREM 1 2 . J* i d an adqyuate axiamalics doh t h e
PROOF: I n Theorem 6 o f Costa 1975 an axiomatics f o r t h e d i s c u s s i v e l o g i c associated w i t h t h e modal p r e d i c a t e l o g i c S5 i s p r e s e n t e d ; t h e e q u i v a -
107
SOME REMARKS ON DISCUSSIVE LOGIC
l e n c e between t h i s a x i o m a t i c s and J*, w i t h r e s p e c t t o t h e u s u a l d i f i n i t i o n s o f t h e q u a n t i f i e r s , can be e a s i l y shown.
2, THE AXiOM SYSTEM 3* The language o f t h e axiom system
J*
must comprise as p r i m i t i v e s y m b o l s
t h o s e o f J * w i t h t h e e x c e p t i o n o f t h e n e c e s s i t y o p e r a t o r , which i s t o b e r e p l a c e d by t h e b i n a r y d i s c u s s i v e c o n n e c t i v e s +and
A. In
;i* we
retain
D e f i n i t i o n s 1-3 and add: D f . 7.
LA =
( 1 A ->
D f . 8.
MA =
ILIA,
D f . 9.
OA =
l(A VIA)
df. df. df.
3*
The s e t o f theses o f
l(A V lA)),
i s c h a r a c t e r i z e d by t h e f o l l o w i n g axiom sche-
mas and r u l e s : Ax.Dl.
A 4( C + A ) .
Ax.02.
(A
Ax.03.
((A +C)
Ax.04.
(A A C) + A .
Ax.05.
( A A C) -> C .
Ax.06.
A ->(C
->(A
Ax.07.
A ->(A
V C).
Ax.08.
C ->(A
V C).
Ax.09.
(A
Ax.1.
A -.llA
4
(C
AD))
4( ( A 4 C )
->A)
+A.
A C)).
->
((C
Ax.2.
1 1 A ->A.
Ax.3.
l ( A V lA)->
C.
Ax.4.
l ( A V C) ->
l ( C V A).
Ax.5.
l ( A V C) ->
(1A A 1C).
Ax.6.
l ( l 1 A V C) ->
+U)
+ ( A +D)).
->U)
->((A
l ( A V C).
V C)
->D)).
L. H . LOPES W S SANTOS
108
->D) ->
Ax.7.
( l ( A V C)
Ax.8.
l((A V C) V
Ax.9.
l((A->
Ax.10.
l ( ( A A C) V
Ax.11.
1(1(A V C) V
Ax.12.
- I ( I ( A ->C)
Ax.13.
l ( l ( A A C) V
Ax.14.
l(A(x/y)
Ax.15.
i(3 xA
U ) -> l ( A U ) ->
C) V
=I
3
U)->
D).
((lA4C) V
D)).
V (C V
(A A 1(C V (A->l(C
U ) ->
V
a)). D)).
U) V l(1C
(l(1A V
U ) -> ( A - > l ( l C
V
U ) ->
D)).
a)).
(A A l ( 1 C V 0 ) ) .
3 x A ) ->
C ) ->
V
V
3xi(A
R1.
ln6eh C 6~om A ->C
R2.
Indm 3 x A ->C
l(A 3
3
A).
i d x 0 n o t 6hee i n C .
C)
and A. i d x A nut dhee i n C.
6hom A ->C,
The notation j*I- A means that A is a thesis if
3,
EQUIVALENCE BETWEEN J* AND
J*
J*,
Two axiom systems A and A ' are said to be equivalent with respect to a given set of definitions when: (i) this set is enough to assure general and univocal translability of the language of one system to the language of the other; (ii) any formula of A is a thesis of A if and only if it abbreviates in A ' , according to the definitions, a thesis of A ' ; (iii) any formula o f A' is a thesis of A' if and only if it abbreviates in A, according to t h e definitions, a thesis of A. We will prove that J* and j* are equivalent with respect to Definitions 5-7. Condition (i) above is obviously fulfilled by Definitions 5-7. If A is a formula of J * , let TI-A be the formula of 5* abbreviated by A according to Definition 7; if C i s a formula of J*,let T r C be the formula of J* abbreviated by C according to Definitions 5-6. Conditions (ii) and ( i i i ) may now be so rewritten: (ii') for any formula A of J*, J* I- A if and only if f* I- TFA; (iii) for any formula C of i*, *: I- C if and only if J*I- T r C . Proving their f u l f i l l m e n t requires s o m e lemmas. LEMMA 1 . The @%xdng
h d e A v&d
in
5.:
in6eh A ->U
d h a m A ->C
109
SOME REMARKS ON D I S C U S S I V E L O G I C
and C
3
D. i s w e l l k n o w n t h a t t h e r u l e i s d e r i v a b l e by means o f Axioms 01-
PRO0F:It
02
and R1.
LEMMA 2. j* IPROOF:
L(A(x/y) 3 3 x A ) .
B y A x i o m 4 and D e f i n i t i o n 1,
-
J* I- 1 ( A = A ) ->
(i)
by (i), Axiom 3
l ( A V 1A);
and Lemma 1 ,
( i i ) j* I- l ( A 3 A ) + O ( A ( x / y ) a 3 x A ) ; by Axiom 14, ( i i ) and Lemma 1 ,
z*
(iii)
+ O(A(x/y)=3xA).
I- i ( A ( x / y ) = 3 x A )
A p p l i c a t i o n s o f D e f i n i t i o n s 7 and 9 on ( i i i ) complete t h e p r o o f
LEMMA 3. The @ % ~ 4 ~ L hlLee n g b detLivabte i n :* : i n d e t L( 3 x A L ( A 3 C ) i d x h not &ee i n C. PROOF:
By D e f i n i t i o n s 7 and 9, L ( A
(i) by Axiom 3
C) i s
i ( A 3 C ) -> O(A 3 C ) ; and D e f i n i t i o n 9,
-
J* I-
(ii)
3
= C ) 6aom
O(AzC)->
O(3xA3C);
so we d e r i v e f r o m ( i ) . w i t h t h e h e l p o f (ii) and Lemma 1, l ( A 3 C ) -> O( 3 x A 3 C ) . (iii) L e t u s suppose t h a t x i s n o t f r e e i n C ; t h e n i t i s n o t f r e e i n
O(3xA
3
C);
t h e r e f o r e , we can d e r i v e 3 x l ( A 3 C ) -. O( 3 x A 3 C ) (iv) f r o m ( i i i ) , by R2; f r o m ( i v ) we get, by Axiom 15
->O( 3 x A
3
C ) , which i s
L( 3 x A
LEMMA 4 . Undemtanding L me vaeid i n j*
M a
.
PROOF:
3
C),
and Lemma 1, l ( 3 x A = C )
by D e f i n i t i o n s 7 and 9.
dedined symbot
06
:*,
t h e ochernab J * l - J * 6
C f . Costa and D u b i k a j t i s , t h e s e Proceedings, P a r t 11,
7, 9-11, 13, 20, 23, 26-29. LEMMA 5 .
16 J* I-
A,
then
PROOF: S t r a i g h t f o r w a r d by
j*
I-
Lemmas 2-4.
TFA.
Theorems
110
L.
, then
16 j*I- C
LEMMA 6 .
Understanding ->
PROOF:
H.
LOPES DOS SANTOS
J* I- TrC.
and A as d e f i n e d symbols o f J * , Axioms 01-A15
and R1-R2 a r e e a s i l y shown t o be v a l i d i n J*. 16 0 0 a .taLc*oLogicd comcquence
LEMMA 7 .
hiwabLe
dhom LA and
*;
LC i n
PROOF: L e t us suppose t h a t
06
A and C, t h e n LD 0 d e -
D i s a t a u t o l o g i c a l consequence o f A and
C ; by
Lemma 4 ,
(i)
j* I- L(A = ( C =D)).
(ii)
J* I- L(L(A
(iii)
j*
By ( i ) ,
-
+ L ( L ( C ID)=
=D))
L ( LA = L ( C ID),)),
I
L( LC DLU)):
Lemma 4,
( i i ) and
-
(iv)
I (C
I- L ( L A = L(C ID)).
J*
From LA, ( i v ) and Lemma 4, we d e r i v e (v)
L(C
= D);
f r o m (iii), ( v ) and Lemma 4 we g e t (vi)
L( LC
L D f r o m LC and ( v i ) , by Lemma
f i n a l l y we d e r i v e LEMMA 8.
J*
LD);
I
I- L((A->
j* I-
C) G ( M A
I C))
4.
and
L ( ( A A C ) E. ( M A & C ) ) .
PROOF: T r i v i a l , by Lemma 7 and Theorems 30-33 i n P a r t
I1 o f Costa a n d
D u b i k a j t i s , t h e s e Proceedings. LEMMA 9 .
*;
I- L ( ( T r T r A ->
J * I-
TrTrC)
L((T?TrA A T r T r C )
z r r T r ( A + C))
and
T r T r ( A A C)).
PROOF: C o n s i d e r i n g t h a t T r T r (A 4 C ) and T r T r ( A A C ) a r e r e s p e c t i v e l y
(MTrTrA
3
T z T r C ) and ( M T r T r A & T r T r C ) ,
t h e lemma i s an i m m e d i a t e con-
sequence o f Lemma 8. LEMMA 1 0 . The &UouLing h d e Lh waeid i n LfA
3
C).
5.:
L n 6 c h L ( 3 x A =_ 3 x C )
@om
LOGIC
SOME REMARKS ON D l S C U S S l V E
PROOF: From L(A
(i)
111
C ) we d e r i v e , by Lemma 7,
L(A = C )
and
L(C 3 A ) .
B u t Lemma 2 assures t h a t
s* I- L(C
(ii)
and j* I- L(A
3xC)
3
3
3xA);
so we d e r i v e (iii)L(A
3xC)
3
and
L(C
3
3xA)
and ( i i ) , byLemma 7. From ( i i i ) , by Lemma3, we g e t from (i) (iv)
L( 3 x A 3 3 x C )
and
L( 3xC
f r o m t h i s , by Lemma 7, we d e r i v e
LEMMA 1 1
.
3
L( 3xA
3xA); f
3xC).
The 6oLtotuLng hLLee 0 dehivable i n
3.:
i n d e h L ( MA
MC)
L(A E C)
dhom PROOF:
From
(i)
L(A
?
C ) we d e r i v e , by Lemma 7,
L(lA= iC)
and
L(iC = l A ) ;
by Lemma4, (ii)
J* I-
L(L(1A 2 1 C ) = L ( L l A Z L l C ) ) ,
(iii)j * I- L(L(1C 2 1 A ) 3 L ( L1C 3 L l A ) ) . By t h e same lemma, we d e r i v e
(iv)
L(L1A
from ( i ) - ( i i i ) ;
3
LlC)
J*
L(L1C X L l A )
f r o m ( i v ) , by a n o t h e r a p p l i c a t i o n o f the1,emma
L ( l L 1 A E l L l C ) , which i s LEMMA 1 2 .
and
I- A
L( MA-MC)
,
get
id and o n l y id j* I- T Z T r A.
PROOF: W i t h t h e h e l p o f Lemmas 7-11,
we show e a s i l y t h a t J * I - L ( A z T r T r A ) ,
by i n d u c t i o n on t h e l e n g t h o f A.(Remember t h a t T r T r l C , T r T r (C V 0 ) Tr"Tr3xC a r e r e s p e c t i v e l y
lTrTrC,
lemma f o l l o w s f r o m t h i s r e s u l t by
.
and
( T r T r C V T Z T r D ) and 3 X T r T r C ) . T h e Lemma 7.
LEMMA 1 3 . 16 D 0 a t a u t o L o g i c a t comequence hivable dhom LA and LC in J* PROOF:
we
by D e f i n i t i o n 8.
06
A and C, t h e n L D
de-
I t i s enough t o make o b v i o u s m o d i f i c a t i o n s i n t h e proofofLemma 7.
112
L . H. LOPES DOS SANTOS
LEMMA 14. J* PROOF:
I- L( L A
(1A ->l(A
V 1A))).
I t i s easy t o v e r i f y t h a t t h e c w s i d e r e d schema i s v a l i d i n
the
p r e d i c a t e l o g i c S5, which i s included !n t h e s e t o f theses o f J*.
LEMMA 1 5 . The ~ o ~ u ~ u 4i nd ge 0 u a e i d i n J* : indm L(3xA :3xC) 64om L(A
= C).
PROOF:
I t i s e n o u g h t o make o b v i o u s m o d i f i c a t i o n s i n t h e
proof o f
Lemma 10.
LEMMA 1 6 .
L(LTrTFA ETrTrA).
J* I-
PROOF: Considering t h a t T r TFLA i s (1Tr T? A ->lTr
T r A V 1 T r T r A)), t h e 1 emma
i s an immediate consequence o f Lemma 1 4 ,
LEMMA 1 7 . The 6oaRowing 4 d e L(A
= C).
s L C) 64om
0 detLivabLe i n J*: i n 6 e h L( L A
PROOF: From L(A E C), by Lemma 13, we g e t
(i)
L ( A = C)
and
L ( C = A);
from ( i ) and conveniently chosen instances o f J * 3 , we d e r i v e
=I
( i i ) L(LA
LC)
and
L(LC
3
LA),
by J * 5 . From ( i i ) a n d Lemma 13, we d e r i v e L ( L A
LEMMA 1 8 . J *
I- A
i6 and o n l y i6 J*
I-
=
LC).
T r TFA.
PROOF: By t h e same method employed i n t h e p r o o f o f
Lemma 12,
with
the
help o f Lemmas 13-17.
THEOREM 1 2 . J* I- A i6 and ondy id
-
3* I-
J* I- C i6 and o d y i6 J* IPROOF:
I f J* I- A, then
i* I- T
?*
I-
Tr C
.
~ A ,by Lemma 5; conversely,
then J * I- T r T r ” A , by Lemma 6, and i f and o n l y i f
TEA;
J* I - A ,
i f 3 * I-TFA,
by Lemma 18; hence J * I- A
T ~ A .We show s i m i l a r l y t h a t ;*I- C i f and
only if
J* I- T r C , w i t h t h e h e l p o f Lemmas 5, 6 and 12. We have thus seen t h a t j* i s e q u i v a l e n t t o t h e
discussive l o g i c J*
SOME REMARKS ON DlSCUSSlVE
associated w i t h t h e p r e d i c a t e l o g i c S5
.
113
LOGIC
I t i s worth noting t h a t
3*
i s
also equivalent t o the discussive l o g i c associated w i t h the predicate l o g i c S4.
T h i s d i s c u s s i v e l o g i c i s indeed i d e n t i c a l t o J*, a s we may
by t h e same methods employed i n P a r t I i n o r d e r t o show t h a t J ( S 4 ) cides
verify coin-
with J(S5).
REFERENCES C o s t a , N . C. A. da 1975,
Remmhn on Jao-kowski d i 6 c u s o i v e l o g i c , Reports o n Mathematical Logic, 4, 7-16.
Costa, N . C . A. da 197+,
and L. D u b i k a j t i s
On J G k o w s k i d i s c w n i u e Logic,
t h e s e Proceedings
.
Furmanowski, T. 1975,
Remmhn on di6cubniue poponLLLonul CaecLLeub, Studia Logica, 34, 39-43.
Jackows k i , S. 1948,
Rachunek zdaM &a ~ y ~ . t e m &dedukcyjnych u n p m x z n y c h , S t u d i a Soci e t a t i s S c i e n t i a r u m Torunensis, S e c t i o A, I , nP 5, 57-77. ( A n E n g l i s h t r a n s l a t i o n o f t h i s paper appeared i n S t u d i a Logica, 24 (1969), 143157.)
Centro de Logica. Epistemologia e Hist o r i a da C i s n c i a Uqiversidade Estadual de Campinas Campinas, G o Paulo, B r a z i l .
PART I 1 MODEL THEORY
Non-Classical Logics, Model Theory and Computability, A . I . Arruda, N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
SOME DIRECTIONSI N MODEL THEORY ( * I by
MIROSLAV BENUA
T h i s paper a t t e m p t s t o expose and a l e r t a s t u d e n t o f model t h e o r y t o new areas of r e s e a r c h . We, o f course, have t o make t h e u s u a l d i s c l a i m e r , t h a t because o f t h e tremendous g r o w t h o f model t h e o r y n o t h i n g c l o s e t o comp l e t e n e s s can be done i n an h o u r ' s l e c t u r e . To g i v e an example, we do discuss t h e i n t e r e s t i n g concept o f r e c u r s i v e model t h e o r y
proposed
not i n
Barwise and S c h l i p f 1976. (We m e n t i o n i t because o f a s t i l l unsolved problem which i s s i m p l e t o s t a t e and w a r r a n t s g r e a t e r d i s s e m i n a t i o n :
every
if
c o u n t a b l e model o f a t h e o r y i s r e c u r s i v e l y s a t u r a t e d , i s t h e t h e o r y
w-cate-
gorical ?) The c h o i c e h a s been made f i r s t l y by t h e f r e s h n e s s o f t h e m a t e r i a l and secondly by t h e d e s i r e t o b r i n g o u t as many c o n n e c t i o n s t o o t h e r f i e l d s ( s e t theory, p r o b a b i l i t y , computer s c i e n c e ) as p o s s i b l e . We avoided well-kown d i r e c t i o n s l i k e t h e outcomes o f M o r l e y - B a l d w i n l L a c h l a n c i t y work, f o r c i n g ,
therefore categori-
e t c . The vague problem i n t h e l a s t s e c t i o n i s aimed a t
t h e p h i l o s o p h i c a l l y minded members o f t h e audience; however, we
wish that
a p o s s i b l e outcome were p r e c i s e and t e c h n i c a l . To g i v e an i d e a o f t h e work i n model t h e o r y done i n t h e p a s t we
dis-
cuss b r i e f l y t h e m a j o r problems (and a g a i n we j u s t choose) model t h e o r y h a s been f a c i n g f o r some t i m e . More complete l i s t s may be f o u n d i n
Friedman
1975, and f o r model t h e o r y i n p a r t i c u l a r i n Robinson 1973.
(*) ro
P r e p a r a t i o n o f t h i s p a p e r was p a r t i a l l y s u p p o r t e d b y Fundagao d e Ampa-
5
P e s q u i s a do Estado d e Sao P a u l o (FAPESP)
e P r o j e t o s (FINEP), B r a z i 1 .
117
and
F i n a n c i a d o r a d e Estudos
118
MIROSLAV BENDA
THREE OUTSTANDING PROBLEMS Vaught's conjecture: The numbeh 0 6 c o u n t a b l e models 0 6 u t h e o h y in 5 w OR 2w. Foh a mohe g e n e h a t q u e s t i o n h e p l a c e " f h e o h y " by
a sentence
06
1
w1 w
.
T h i s c o n j e c t u r e has s t i m u l a t e d and i s s t i m u l a t i n g a l o t Our work on modeloids,on
of
research.
which we r e p o r t e d b e f o r e t h e c o n g r e s s , w a s p a r t i a l l y
i n f l u e n c e d by t h e d e s i r e t o p r o v e t h e c o n j e c t u r e . The f i r s t g e n e r a l
result
i s due t o M o r l e y (see M o r l e y 1970). The number o f c o u n t a b l e models o f a s e n tence
of
L
w1 w
i s 5 w 1 o r 2w
.
M o r l e y ' s r e s u l t and i t s e x t e n s i o n s
now w e l l understood thanks m a i n l y t o t h e work
are
o f Vaught (Vaught 1974),
B u r g e s s - M i l l e r (Burgess and M i l l e r 1975),and o t h e r s . C l o s e s t t o p r o v i n g t h e c o n j e c t u r e came G. Sacks whose work i s n o t y e t p u b l i s h e d b u t an account i t may be found i n H a r n i k and Makkai 1976. H a r n i k and M a k k a i , i n f a c t ,
of
proved
i n d e p e n d e n t l y a r e s u l t c l o s e t o Sacks' which i s easy t o s t a t e : a P C w l c l a s s whose c o u n t a b l e members have o n l y c o u n t a b l y many automorphisms has e i -
5 w or
ther
2 w c o u n t a b l e models. Because o f i n d i c e r n i b l e s ,
no f i r s t - o r -
d e r t h e o r y s a t i s f i e s t h e assumptions o f t h i s theorem. Sacks' r e s u l t hasmuch m i l d e r c o n d i t i o n on t h e automorphism s a t i s f i a b l e i n some E C A
classes.
t h i s r e s p e c t a r e s u l t o f M. Rubin and S.Shelah (see Rubin and Shelah
In
1975)
( v e r y ) weakly complements t h e above: I f a t h e o r y T has Skolem f u n c t i o n s and a l i n e a r c j r d e r i n g o f t h e u n i v e r s e t h e n i t has 2w c o u n t a b l e models. ( T h i s e x tends o u r e a r l i e r r e s u l t ' t h a t such a t h e o r y has 2 w models.) Most models o f t h e s e t h e o r i e s have 2 u automorphisms.
0 4 c a h d i n a l i t i e s 06 o p e c t h a c l o s e d undeh comple-
The spectrum problem: A ApeCthUm is t h e n e t
d i n i t e modePo mentation ?
06
a sentence
@.Ahe
T h i s problem was posed by H. Scholz i n 1957. M o s t o f t h e
research
w e n t i n t o establishing t h a t the spectra a r e very e f f e c t i v e sets o f n a t u r a l numbers. T h i s d i r e c t i o n was c u l m i n a t e d by Jones and Selman who
showed
t h a t t h e s p e c t r a a r e e x a c t l y t h e s e t s o f n a t u r a l numbers which a r e r e c o g n i z a b l e by n o n - d e t e r m i n i s t i c , T u r i n g machines t i m e bounded by 2", s t a n t , x l e n g t h o f i n p u t (see Jones and Selman 1972). T h i s l i n k s famous
P = NP problem t o t h e spectrum problem: I f P = NP t h e n
c a con-
the
now
spectr.
a r e c l o s e d under complementation.0n t h e m o d e l - t h e o r e t i c s i d e t h e problem was
SOME D I R E C T I O N S ON MODEL THEORY
recently investigated by Fagin (Fagin 1976 contains basic references)
119 who
among other things proved t h a t f o r any sentence I$ the spectrum of I$ or l+ i s c o f i n i t e . I n f a c t the s e t of sentences which a r e eventually true in a l most a l l models of cardinality n i s a complete and decidabletheory(studied e a r l i e r by Gaifman). Morley's conjecture: L e t T b e a t h e v h y . The Lahgeh t h e u n c v u n t a b l e c a h d i n a l , t h e mVhe m o d e l s 0 6 T v b t h a t c a h d i n a l i t y .
All the r e s u l t s we know of in t h i s direction are due t o Shelah and i t i s d i f f i c u l t t o give a simple account of them. R a t h e r we r e f e r the reader to his Tarski's Symposium lecture and t o Shelah 197+ ( s p e c i f i c a l l y t h e s e c tion on p. 283).
FINITE MODEL
THEORY,
Classical model theory has the ultimate r e s u l t about f i n i t e m o d e l s .
Two f i n i t e models are elementarily equivalent i f f they a r e isomorphic. Of course, t h i s i s an overstatement, as we have seen in the discussion o f the Scholz' problem. B u t the f a c t remains t h a t most methods and r e s u l t s of model theory apply almost exclusively t o i n f i n i t e models. The f i r s t concentrated e f f o r t on f i n i t e models was i n i t i a t e d oy Keisler: we s h a l l report
on some of his r e s u l t s which open unexpected connections. The basic idea i s t o study i n f i n i t e models which appear f i n i t e in some model of mathematics. The idea of using non-standard models t o get standard r e s u l t s was used by A.
Robinson and even e a r l i e r by C. Ryll-Nardzewski. B r i e f l y , l e t M be mathematics, by which we mean the collection of objects and relations in the public domain of mathematicians. This can be considered as a n ill-defined structure; model theory showed how we can get en extension of i t , M* , which c o u l d be characterized by saying t h a t the following holds about M*: whateveh c o u l d b e , i n .
S p e a k i n g p r e c i s e l y , M * i s saturated or a t l e a s t tol-saturated a n d t h e phrase " v b j e o t c v d d be" means t h a t the existence of the o b j e c t i s n o t forbidden by a f i r s t order formula. (These explanations are the main reason f o r the slow acceptance of non-standard methods). So M* h a s i n f i n i t e n a t u ral numbers because there i s no f i r s t - o r d e r formula which says t h a t e v e r y n a t u r a l number i s e i t h e r 0 or 1 or ... . M * i s richer than M in some
120
MIROSLAV BENDA
aspects b u t i n o t h e r s i t i s p o o r e r .
M* does n o t r e c o g n i z e
the
set
s t a n d a r d n a t u r a l numbers, f o r example; t h e o b j e c t s i t does r e c o g n i z e
of are
c a l l e d intmnd. A theorem a b o u t an i n t e r n a l o b j e c t can be t r a n s f o r m e d i n t o an i n f o r m a t i o n a b o u t p r o p e r t i e s o f s t a n d a r d o b j e c t s ( t h i n k o f M* as an u l -
M-
t r a p o w e r ) . K e i s l e r p u t s i t by s a y i n g " M * - ~ i n i t e mode& e x k i b d t h e
i n g behavioh od d i n i t e mode&". B u t t h e i n t e r n a l model t h e o r y o f M* i s t h e same as i n M so we
have
n o t g a i n e d much. And i t i s h e r e where we s h o u l d ask o u r s e l v e s t h e q u e s t i o n : "Why do we want t o s t u d y f i n i t e models". The answer t o t h i s i s c l e a r ,
fi-
n i t e models a r e everywhere. L e t us t a k e an example:
EXAMPLE, We have s u b j e c t s and want t o t e a c h them t o
distinguish
black
f r o m w h i t e . We can g e t o n l y s t a t i s t i c a l r e s u l t s so we s h o u l d s t u d y l a r g e r groups and we i n f a c t make t h e l e a p t o s t u d y an i n f i n i t e group A o f s u b j e c t s b u t one which i s f i n i t e i n M*. L e t
F ( a ) be t h e p r o b a b i l i t y t h a t a sub-
j e c t ~l e a r n s t h i s on t h e f i r s t t r i a l , and assume t h a t
F
we can t r a n s f e r t h e r e s u l t s back t o M . Q u e s t i o n s which about t h i s s i t u a t i o n a r e :
i s internal sothat m i g h t be
asked
( 1 ) what i s t h e p r o b a b i l i t y t h a t a s u b j e c t l e a r n s
t h e t a s k a f t e r two sessions (assume he does n o t f o r g e t i t ) . ( 2 )
what
i s
t h e average p r o b a b i l i t y o f l e a r n i n g t h e t a s k a f t e r two sessions. The answer t o ( 1 ) i s G(u) = F ( u )
+
(1 - F(a)).F(a),
and t o t h e second q u e s t i o n , ZG(a). 1 aEA IAl
o r i f we d e n o t e M * ) we h a v e
JG du
p(a)
=
&, t h e u n i f o r m measure on P(A)
(taken i n
.
O f course, t h e t a s k o f model t h e o r y i s n o t i n f i n d i n g
F(a) n o r
i n making t h e computations. The t a s k i s t o a b s t r a c t f r o m such examples
even and
seek r e s u l t s about t h e a b s t r a c t i o n s . K e i s l e r proposes t o s t u d y models where A i s i n t e r n a l t o measure on
P(A)
(see K e i s l e r 1976)
(A, p ,
Fi)i
I
M*, M * - f i n i t e b u t i n f i n i t e ; p i s a p r o b a b i l i t y Fi, i E I , i s an i n t e r n a l M * - f i n i t e
( t a k e n i n M*) and
sequence o f f u n c t i o n s o f v a r i o u s numbers o f arguments f r o m A i n t o
Io,1]*
121
SOME D I R E C T I O N S ON MODEL THEORY
( t h e r e a l s o f M*). We t a k e a h i n t f r o m t h e example and b u i l d o u t o f t h e f u n c t i o n s t e r m s u s i n g c o n t i n u o u s f u n c t i o n s f r o m R" i n t o
i n t e g r a t i o n w i t h r e s p e c t t o t h e measure (see K e i s l e r f o r d e t a i l s ) . a powerful language s t r o n g e r t h e n
Fi
and
R , s u p , inf
L
L w w b u t weaker t h a n
:if
w1 w
This i s
Fi admits
o n l y 0 and 1 as v a l u e s and we i n t e r p r e t 1 as "tfLue" t h e n we h a v e a n o r d i nary r e l a t i o n a l s t r u c t u r e
and any f i r s t - o r d e r s t a t e m e n t about i t
expressed by s a y i n g t h a t a c e r t a i n t e r m has v a l u e 1:
use
can
1 - x i n s t e a d o f n e g a t i o n , and sup i n s t e a d
instead o f conjunctions,
t h e e x i s t e n c i a l q u a n t i f i e r . Thus,the continuous f u n c t i o n s
be
multiplication of
a c t as connec-
t i v e s , s u p and inf as q u a n t i f i e r s . The i n t e g r a l i s a k i n d o f q u a l i t a t i v e q u a n t i f i e r not expressible i n L w w "about h a l f o f t h e elements s a t i s f y
.
I t a l l o w s us t o make statements
like
by w r i t i n g
$'I
+ / $ ~ I J
2
( 4 = Q ( x ) i s a term here = 1 i f Q i s t r u e and pression
0 otherwise); i f t h i s
i s - 1 we, as b e f o r e , t h i n k o f i t as t r u e and i t does mean
exthat
about h a l f , a c c o r d i n g t o 1-1, elements s a t i s f y Q . The s i m p l e s t most u s e f u l r e s u l t i n m o d e l t h e o r y Lllwenheim
r
theorem. To f o r m u l a t e t h e r e s u l t i n t h e p r e s e n t s e t - u p K e i s l e r
d e f i n e s as s u b s t r u c t u r e o f
85 A
i s t h e Skolem-
i s i n t e r n a l and
u,(a) =
( A , IJ
, Fi)
a structure
p8
i s t h e measure:
if
aEB
-
a 18 =
( A , u B , FJ
where
0 i f not.
So o n l y t h e exprcssons u s i n g i n t e g r a t i o n ( t h e q u a l i t a t i v e q u a n t i f i e r ) may have changed meaning. When t h e y do n o t we c a l l t h e s u b s t r u c t u r e elementary:
01 18 and
OZ
i s a n e l e m e n t a r y substructure o f any
al...an E A
(repeat A
a
i f f o r any term
t ( q .. . w n )
s e e t h e reasons above) t e v a l u a t e d i n
i s i n f i n i t e s i m a l l y close t o t evaluated i n
02 18 ( a t a l . . .a,).
SKOLEM-LOWENHEIM THEOREM ( K e i s l e r ) :
Let ti = \ A \ log
n 181 = n a n d
+
111 l o h a t
= 0 021 B
and
01 = ( A , I J , F ~ ) ~I 0 ab bedone. L e i n be n u c h n2 0 Thgn t h e h e in a 8 C_ k Auch t h a t
- z
IAl
.
0 a n eLementa/ry oubnthuctwre
06
m.
Tn duct aemobt
122
MIROSLAV BENDA
~ ( € 3 ) = 0. The i m p o r t o f t h e theorem t o s t a n d a r d models i n a
Note t h a t
s p e c i a l s i t u a t i o n i s t h i s : assume we have a c o n s t a n t term
oln = (n,
u n , d,...
no such t h a t i s
6,)
tarn. Then g i v e n tan=tmn f oIr B some B c n
such t h a t t Oln=
">no
then
t
and
E>
models
0 , there
is
and 1Blcn.E.
The p r o o f o f t h e theorem uses t h e weak l a w o f l a r g e numbers Stone-Weierstrass theorem 2
and t h e The theorem may be proved w i t h o u t t h e a s s u m p t i o n
.
n / I A l = 0 b u t t h e p r o o f i s e a s i e r w i t h i t . Under t h i s assumption a l m o s t a l l 6 : n + A a r e one-to-one. The computation o f t h i s i s s i m p l e and p r e sents a t i n y b i t o f t h e use o f non-standard p r o b a b i l i t y used t h r o u g h o u t K e i s l e r ' s work so we show i t i n f u l l : we t h i n k o f t h e f u n c t i o n d : n -->A as choosing n - t i m e s an element f r o m A r e p l a c i n g i t a f t e r each c h o i c e . The p r o b a b i l i t y o f choosing a one-to-one sequence i s 1 ). m
1.(1-
as
(
I
1
-
m
~
(1
- 2) ...(1 - 2 ) L (1 - L ) " . IAl
IAl
IAl
=) e-1 f o r i n f i n i t e m
B u t t h e l a w o f l a r g e numbers i s used i n t r y i n g t o show t h a t
( /t(x) d p p f o r almost a l l k-th choice
i s close t o B E A,
181 = n . 8 i s o b t a i n e d by n c h o i c e s f r o m A . W i t h x, = t ( k - t h c h o i c e ) . The ex-
we a s s o c i a t e a random v a r i a b l e
p e c t a t i o n o f each
E
(/.t(x)dp)'
xk
is
= (f.t(x)dp)"
a n d s i n c e we a r e r e p l a c i n g , x k ' s a r e independent. The l a w
of
large
numbers says t h a t 1 n
I k
xk =
E.
T h e sum o n t h e l e f t - h a n d s i d e i s a l m o s t t h e i n t e g r a l o f t i n O Z I B . f a c t t o make t h e i n d u c t i o n go t h r o u g h we need B e r n s t e i n ' s
In
i n e q u a l i t y and
123
SOME D I R E C T I O N S I N MODEL THEORY
e v e n t u a l l y use s a t u r a t i o n b u t t h i s g i v e s a sample o f t h e i d e a s used. Many o t h e r c l a s s i c a l theorems ( i n d i c e r n i b l e s , fiers
...
) have c o u n t e r p a r t s i n t h i s set-up.
elimination o f q u a n t i -
An i m p o r t a n t d i r e c t i o n i s
c o n v e r s i o n o f t h e non-standard model i n t o a s t a n d a r d p r o b a b i l i t y u s i n g t h e work o f
structure
P. Loeb (see Loeb 197+).
COST ALGEBRAS T h i s concept was i n s p i r e d by S i l v e r ' s work ( S i l v e r 197+) which i n t u r n arose f r o m a d e s i r e t o s i m p l i f y Jensen's 1972. S i l v e r p r e s e n t e d h i s work i n terms o f machines b u t some o f h i s remarks (see e s p e c i a l l y F o o t n o t e 5 p. 22) i n d i c a t e t h a t he was t h i n k i n g i n terms o f cheap Skolem c l o s u r e s
on as
we would p u t i t i n o u r t e r m i n o l o g y . The concept i s v e r y n a t u r a l , one c o u l d say i t a t t e m p t s t o b r i n g something o f t h e r e a l w o r l d i n t o mathematics. DEFINITION,
(a)
A C O A ~d g e b h a 0
A = (A,
paih
(A, C) w ~ U L ~ :
0 an d y e b m , i. e . each F,
Fn)n
i n a ~ u n C t i o non A
k
i n t o A (may be p a h t i d ) ; (b)
C = ( ( 8 , s), C ) 604
a1
u~he4e ( 8 , 5 ) 0 a fineah. o h d u and
each o p W o n
...ak 06
6 oh
6 an &?me& 06 B.
We t h i n k o f C ( 6 , al
... a k ) as
C
t h e d y e b h a A and doh e a c h Thuh
C ( 6 , al
t h e c o s t o f computing
... %) 6 6
The l i n e a r o r d e r on B g i v e s us a way t o compare t h e c o s t o f
UAbign.4
mgument
a t al
8.
... a k .
computing
v a r i o u s o p e r a t i o n s a t v a r i o u s arguments. EXAMPLE 1 . L e t A be t h e a l g e b r a o f T u r i n g m a c h i n e s ,
i.e.
we view
a T u r i n g m a c h i n e a s a ( p a r t i a l ) o p e r a t i o n on t h e n a t u r a l numbers.Letthe c o s t o f computing on an i n p u t be t h e number o f s t e p s r e q u i r e d t o complete t h e computation; l e t i t be w i f t h e machine does n o t h a l t . T h e l i n e a r o r d e r i n t h i s case i s w + 1. EXAMPLE 2 . L e t G be a group generated by a s e t o f g e n e r a t o r s . For
every
element o f G t h e r e a r e many words formed f r o m t h e g e n e r a t o r s w h i c h a r e e q u a l t o t h e element; l e t t h e h e i g h t o f an element a be t h e l e a s t number n
such
124
MIROSLAV
BENDA
t h a t i f i n a reduced word o n l y exponents k .-k w i t h 0 < k < n aupear t h e n t h e word i s n o t equal t o a.The g e n e r a t o r s g g e t assigned 1, and g 2 has 2 unl e s s i t i s equal t o a g e n e r a t o r .
The
cost
of
c o m p u t i n g x . g i s de-
f i n e d as t h e maximun o f t h e h e i g h t s o f t h e elements; t h e c o s t o f x-l .
computing
i s t h e h e i g h t o f x. Each c l a s s o f a l g e b r a s has i t s own concepts and techniques b u t t h e no-
t i o n o f a subalgebra i s p r e s e n t i n a l l a l g e b r a i c c o n s i d e r a t i o n s . T h i s
no-
t i o n g e n e r a l i z e s t o c o s t a l g e b r a s i n t h e f o l l o w i n g form.
DEFINITION, Let ( A , C ) b e S 5 A b c&ed
gument
id
.
al,
a C V h t dgeblla b € B (the a b - h u b d g e b h u ad A id 6ah any a p e h a t i o n
..., %
C(6, al,
€ S 06
6
ohdehl.
A
h&t
a6 A and any ah-
we have:
... a k ) < b $hen
I n Example 2,ann-subgroup
6
6(a1...% ) 6 S.
o f G i s a group i f f t h e i n d e x ( o r d e r ) o f G i s
< n T h i s shows t h a t a s e t S E A which i s n o t a subalgebra may be a b subalgebra f o r some b € 8 . I n t u i t i v e l y , f o r a p r a g m a t i s t who cannot spend more t h a n b a subalgebra and a b - s u b a l g e b r a a r e t h e same t h i n g .
A b - s u b a l g e b r a can be made, i n a n a t u r a l way,into a c o s t a l g e b r a w i t h a s h o r t e r c o s t s c a l e . The o p e r a t i o n s when reduced t o S w i l l be i n g e n e r a l p a r t i a l ; t h a t i s a reason f o r s t a r t i n g w i t h p a r t i a l o p e r a t i o n s a l r e a d y on t h e o r i g i n a l a l g e b r a . Also, i n o r d e r t o be a b l e t o speak o f r e a l s u b a l g e b r a s ,
B s h o u l d have a l a r g e s t element which i s n o t a c o s t o f any computation.Vari a n t s o f t h e d e f i n i t i o n i n v o l v i n g subsets o f t h e c o s t s c a l e e t c . c o u l d a l s o be explored. The i n t e n t o f J e n s e n ' s and S i l v e r ' s work i s preserved. I n f a c t t h e w h o l e o f S i l v e r ' s argument i s devoted t o a d e t a i l e d s t u d y o f a p a r t i c u l a r c o s t a l gebra which i s d e f i n e d on t h e o r d i n a l s which a l s o ' f o r m t h e c o s t s c a l e .
The
m a j o r t o o l i n t h e work a r e 6 - s u b a l g e b r a s which enable us t o have a c l o s e watch on what s e t s o f o r d i n a l s g e t c o n s t r u c t e d . I n S i l v e r ' s w o r d s
"
the
constructible universe slowly reveals i t s e l f " . More i n t e r e s t i n g v a r i a n t a r i s e s i f we l o o k a t Example 1 and compare i t w i t h t h e c u r r e n t i n t e r e s t i n computer s c i e n c e s . The n - s u b a l g e b r a s l i m i t t h e number o f s t e p s t h e T u r i n g machine may p e r f o r m by n and i n t h i s c a s e o p e r a t i o n s s i m p l y can n o t a p p l y t o l a r g e numbers
(> 2")
the
because t h e m a c h i n e
cannot even r e a d t h e i n p u t . We, t h e r e f o r e , c o n s i d e r t h e number o f s t e p s r e q u i r e d t o f i n i s h t h e computation b u t i n dependence o n t h e i n p u t .
The m o s t
125
SOME D I R E C T I O N S ON MODEL THEORY
n a t u r a l a b s t r a c t s e t t i n g would be a l g e b r a s on A* quences o f members o f
the set o f f i n i t e
se-
A.
I n o r d e r t o g i v e an example o f t h e use o f c o s t subalgebrasweshow t h a t t h e y were i m p l i c i t l y used i n an o l d p r o o f o f Ryll-Nardzewski 1952.
THEOREM ( Ryll -Nardzewski ) . Peanv a h i t h e t i c .LA n v t binLt&y axivmcLtizabte.
PROOF: Assume t h e c o n t r a r y . Then we have f i n i t e l y many axioms and
because
the
what we
t h e o r y has d e f i n a b l e Skolem f u n c t i o n s ( v i a t h e w e l l - o r d e r i n g )
have, e s s e n t i a l l y , i s f i n i t e l y many f u n c t i o n s and axioms w i t h o u t q u a n t i f i e r s which t e l l us t h e p r o p e r t i e s o f t h e f u n c t i o n s . The c o n s e q u e n c e
of
t h i s i s t h a t any s e t i n any model o f a r i t h m e t i c which i s c l o s e d u n d e r t h e f u n c t i o n s i s a model o f t h o s e f i n i t e l y many axioms. The n a t u r a l numbers t o g e t h e r w i t h t h e f u n c t i o n s g i v e r i s e t o a c o s t a l g e b r a somewhat
like
the
a l g e b r a i n Example 2. I f we t a k e f i n i t e l y many numbers, among them 0, e v e r y number can be expressed as a term i n t h e f u n c t i o n s and t h e f i n i t e l y "generators".
We d e f i n e t h e
cost
o f computing
6
many
a t n as t h e com-
p l e x i t y o f n (as a term) p l u s 1. A simple, b u t c r u c i a l , o b s e r v a t i o n i s t h a t there are f i n i t e n-subalgebras f o r every
n € N. Now e v e r y t h i n g s a i d up t o
t h i s p o i n t i s expressible i n arithmetic, i n p a r t i c u l a r t h e f o r m u l a
cp(n)
s a y i n g t h a t t h e r e i s a bounded n - a l g e b r a . T h i s f o r m u l a has t h e p r o p e r t y that O(0) A ( w n ) ( $ ( n ) - >
$(n+l))
h o l d s i n a l l models o f t h e f i n i t e l y many axioms. Consider t h e m o d e l
of
these axioms which i s generated i n a non-standard model o f t r u e number t h e o r y by an i n f i n i t e element w, u s i n g t h o s e f i n i t e l y many f u n c t i o n s . e v i d e n t t h a t i n t h i s model any therefore
l$(w)
It
i s
w-subalgebra i s i n f a c t t h e whole model,
holds there.
T h i s theorem c o u l d be f o r m u l a t e d f o r a b s t r a c t c o s t a l g e b r a s .
I t would
claim t h a t a cost algebra s a t i s f y i n g c e r t a i n conditions ( t h e cost d e f i n e d by c o m p l e x i t y o f terms and e x p r e s s i b l e i n f i r s t o r d e r e t c . ) h i l s an elementari l y e q u i v a l e n t c o s t a l g e b r a where subalgebras and n - s u b a l g e b r a s
are the
same f o r some n i n t h e c o s t s c a l e . More u s e f u l d i r e c t i o n though w o u l d b e t o keep t h e c o s t s c a l e f i x e d and e x p l o r e t h e c o s t f u n c t i o n s which a r e t i e d t o t h e c o m p l e x i t y o f terms. I n t h i s c o n n e c t i o n an axiom t o c o n s i d e r i s C(6t g("))
2 C(dog,")
126
M I R O S L A V BENDA
assuming the operations a r e closed under substitutions. A somewhat more concrete problem i s t o characterize the cost a l g e b r a constructed by Silver in more general terms; perhaps the f a c t t h a t algebras associated with Skolem functions on L , a r e J6nsson i s r e l e v a n t here (see the proofs of 7.4.9 and 7.4.10 in Chang and Keisler 1973).
SAMPLINGS, There a r e two kinds of tools: those of general use and those f o r s p e c i f i c operations. The method o f w h a t we c a l l samplings has been used so f a r in the l a t t e r capacity b u t we think t h a t i t s r i g h t place i s in t h e f i r s t category among ultraproducts, indicernibles e t c . The method consists ofextracting information about a structure from small samples of the s t r u c t u r e in the sense described below. The f i r s t r e s u l t of t h i s kind was obtained by Kueker; i t i s related below. The general problem in t h i s area i s b e s t expressed in Shelah 197+ : "Kueker i n defined some f i l t e r s .We c a n e a s i l y suggest more". We would l i k e t o add t h a t rather than f i l t e r s more samplings should be constructed; they more o r l e s s canonically d e f i n e the filters.
...
...
a he* S E P ( A ) such t h a t 60% euehy 6iand any d g e b m on A 06 countabLy many o p e a d h n b w e c a n dind which includes a and 0 closed undeh .the ope hat ion^.
DEFINITION. A OampLLng 05 A 0 nite
o e S
a EA
The definition may be stated equivalently f o r structures andelementary substructures instead of algebras and subalgebras. The LBwenheim-Skolem theorem ( f o r c o u n t a b l e l a n g u a g e s ) i s equivalent t o saying t h a t P ( A ) , the countable subsets of A , f o r m s W 1 a sampling o f A . More generally, i f w 5 K < I A l then P K ( A ) i s a sampling o f A .
EXAMPLE 1 .
EXAMPLE 2. Chang's conjecture f o r saying t h a t the s e t s of order 'type in Chang and Keisler 1973).
(K+,K) is forms a sampling of
(K++,K+) K+
equivalent t o (see 7.3.4
K++.
I t i s c l e a r t h a t there a r e no interesting samplings of countable
sets
127
SOME D I R E C T I O N S I N MODEL THEORY
because a c o u n t a b l e s e t w i t h a "successor" f u n c t i o n has no p r o p e r elementaof
r y s u b s t r u c t u r e . I n t u i t i v e l y , sampling i s a c o l l e c t i o n o f s m a l l subsets
A and t h e r e a r e so many o f them t h a t one g e t s an i d e a o f t h e s e t f r o m t h e p r o p e r t i e s o f t h e samples. To make t h i s c l e a r e r we a s s o c i a t e w i t h e v e r y sampling a f i l t e r o f l a r g e s e t s o f samples.
DEFINITION. L e X the
6iXtetl
(6
whae
I
(A,
S be a ( n o n - i x i v i d ) sampling on S genmrated by 6 d o o e d undm 6 , , ..., 6, ,... I
6, ,..., 6, ,. .. )
A . By
FS
we
denote
-id an d g e b m .
S = P,,,,(A),
I n t h e case when
06
FS
i s j u s t the f i l t e r generated
by
P w , ( A ) . I n general, i t i s countably c o m p l e t e
c l o s e d unbounded subsets o f
and normal. To express n o r m a l i t y and o t h e r t h i n g s i t i s c o n v e n i e n t t o i n t r o duce a q u a n t i f i e r : €
for
S
we w r i t e
I ... 6 ...1
(WS6)(
FS
€
(3p)
1(vS6)i write
...A . . . )
if
;
Ws and
IS mean i n t h e case o f
t h a t t h e s e t i n q u e s t i o n i s closed-unbounded and s t a t i o n a r y
S = Pu,(A)
respectively.
N o r m a l i t y t h e n means t h a t i t i s p o s s i b l e t o s w i t c h ( p a r t i a l l y ) W and WS:
( W x € A ) ( VS6) R ( f i )
R SAxS
where
. This
3
( WS6) ( x € 6) R
(x6 )
i s t h e p r o p e r t y on which most theorems ( b u t n o t a l l ,
see below) hinge. I t enables us t o p r o v e a r e s u l t on o m i t t i n g o f t y p e s w h i c h l o o s e l y speaking says t h a t i f a t y p e can be sampled by subtypes a l l o f which can be o m i t t e d t h e n t h e t y p e i t s e l f can be o m i t t e d (see Benda 1 9 7 t b f o r det a i l s : t h e t y p e i s c o u n t a b l e so i t has no sampling b u t we can r e l a t i v i z e the notion). Another r e s u l t proved u s i n g n o r m a l i t y i s e s s e n t i a l l y t h e f i r s t
result
i n t h i s f i e l d i f we do n o t c o u n t t h e p o w e r f u l samplings p r o v i d e d by
super-
compact c a r d i n a l s (see Kueker 1972):
THEOREM ( K u e k e r ) .
01 a bR/ructuhe on A
m
I= @
L e X @ be a oentence
.
Then:
06
Lwlw
, S
a oampfing od A and
( W S 6 ) m16 I=@
I n Barwise 1974 t h e s e r e s u l t s a r e extended. A use o f sampling i s i m plicit
a l s o i n t h e work o f Shelah on Whitehead c o n j e c t u r e
(Shelah 197+)
.
128
MIROSLAV BENDA
He d e f i n e s t h e r e what amounts t o a sampling o f a s t r o n g l i m i t
cardinal A
( s a t i s f y i n g some a d d i t i o n a l p r o p e r t i e s ) b u t he does n o t use t h e f i l t e r defunctions f i n e d above; i n s t e a d o f c o u n t a b l y many f u n c t i o n s he uses K ( < A ) w i t h t h e e f f e c t t h a t t h e f i l t e r becomes
K-complete.
The q u a n t i f i e r
asso-
ciated with i t satisfies:
if
( 3 S6)(A16
i s free)
then
A i s free
f o r an a l g e b r a A ( s a y a group) o f c a r d i n a l i t y
A.
I t w o u l d b e u s e f u l t o k n o w how t h e p r e s e r v a t i o n o f d i f f e r e n t
p r o p e r t i e s v a r i e s from sampling t o sampling. S p e c i f i c a l l y ,
i s the theorem
above f a l s e f o r some o f t h e c a n o n i c a l samplings we d i s c u s s e d b e f o r e ? These q u e s t i o n s may l a t e r appear as n a i v e as t h e q u e s t i o n whether u l t r a f i l t e r s o n w
have d i f f e r e n t p r o p e r t i e s . Samplings p l a y a r o l e i n s e t t h e o r y as w e l l . Assume t h a t we
have
a
model o f ZF w i t h a s e t A i n i t and we want t o e x t e n d i t e l e m e n t a r i l y so t h a t A s t a y s i n t h e e x t e n s i o n b u t t h e image o f A i s p r o p e r l y l a r g e r t h a n A ( s i t u a t i o n s l i k e t h i s occur f r e q u e n t l y i n l a r g e cardinal questions). are the properties if
6will
P ( . ) i s a p r o p e r t y such t h a t f o r e v e r y sampling S ( n o n - t r i v i a l )
we have
(Vs6)
P(6)
What
have i n t h e e x t e n s i o n ? A p a r t i a l answer i s t h i s : of A
t h e n P ( A ) i s t r u e i n t h e e x t e n s i o n . L e t us denote b y
QA t h e p r o p e r t i e s d e f i n e d i n t h e l a s t sentence. QUESTION:
Is
a t y p e ? When i s i t complete ?
Note t h a t i f t h e axiom o f determinateness h o l d s
QWl
i s a
complete
t y p e . (There m i g h t be some problems w i t h t h e axiom o f c h o i c e h e r e b u t t h e a l t e r n a t i v e d e f i n i t i o n o f a sampling i s e f f e c t i v e ) . ’ A s i m p l e f a c t w e n o t i c e d i s that
Qwl #
ordinal while
CAN
because
Owl c o n t a i n s t h e p r o p e r t y b e i n g a c o u n t a b l e
does n o t c o n t a i n t h e p r o p e r t y . Is ,Q
MODEL THEORY LIVE
=
Qwg?
UP TO ITS NAME?
J. S i l v e r once r e l a t e d . t o me a s t o r y a b o u t a man f r o m an i n d u s t r y who c a l l e d him w i s h i n g t o c o n s u l t him on model t h e o r y w i t h p r o s p e c t s o f i t i n m a r k e t i n g . Blunders l i k e t h i s a r e f a i r l y common,
using
people t h i n k ( r i g h t -
l y ? ) t h a t model t h e o r y r e f e r s t o t h e i m p o r t a n t and fundamental a c t i v i t y o f
SOME D I R E C T I O N S ON MODEL THEORY
129
getting a t f a c t s and predictions bymeans ofmodels.The model theory we study i s , of course, n o t suited, by i t s own d e f i n i t i o n , f o r producing quantitative r e s u l t s sought in applied a r e a s . B u t i t i s notalways the numbers t h a t weseek, sometimes we need a q u a l i t a t i v e judgment. This thought i s most e f f e c t i v e l y expressed in the following quotation from R. Thom 1969 ( p . 3 3 3 ) : "But a s s o o n a s we r u n i n t o d i f f i c u l t i e s , c o n t r a d i c t i o n s , feel ( l i k e i n E l e m e n t a r y P a r t i c l e t h e o r y now) o r when w e overwhelmed by t h e mass o f e m p i r i c a l d a t a t h e n t h e need a r i s e s f o r some c o n c e p t u a l g u i d a n c e i n o r d e r t o c l a s s i f y t h e d a t a and t o f i n d o u t t h e most s i g n i f i c a n t phenomena. I f s c i e n t i f i c p r o g r e s s i s t o be a c h i e v e d by o t h e r means t h a n p u r e c h a n c e and l u c k y g u e s s i t r e l i e s n e c e s s a r i l y o n a q u a l i t a t i v e understanding of t h e process studied". Many problems which need a q u a l i t a t i v e understanding could be readily suggested; we shall describe one which seems intimately connected withmodel theory and what we discussed above. The problem i s to imitate o n g e n e r a l structures the process by which natural s c i e n t i s t s obtain information about the.world we l i v e in. The value of better understanding of t h i s process i s cl ear. Let us be more s p e c i f i c . I n the f i r s t approximation we s e e t h a t the information i s obtained by taking f i n i t e samples of the s t r u c t u r e (i.e. the Universe). This i s a process we understand t o some degree i f we take countable samples and we also understand t h a t , in general, a f i n i t e sample has almost no information about the structure from which i t i s taken. If we take, f o r an example, an i n f i n i t e l i n e a r order , a f i n i t e sample of i t g i v e s absolutely no information a b o u t the order because f i n i t e l i n e a r orders a r e t h e same everywhere. This example i s unfair t o l i n e a r orders because in f a c t all structures exhibit t o a large degree such behavoir. (Ramsey's theorem being partly responsible f o r t h i s . ) B u t i f we pull a real out of a model of s e t theory we know t h a t i t may contain a l o t of information about the model.It may happen t h a t the real i s O d which codes u p information about a f a i r l y large part of the whole s t r u c t u r e . The key t o finding out a b o u t the model from the real i s t o i n t e r p r e t i t the r i g h t way. And t h i s i s e x a c t l y t h e methodology used in science. The f i n i t e sample i s interpreted, generalized, idealized into a pattern which i s subsequently checked on the universe by an experiment. Cat1 t h i s pkocess be usefully modelled on abstract s t r u c tures ? This would involve defining the possible interpretationsandspeci-
130
MI ROSLAV BENDA
fying t h e permissible experiments f o r v e r i f y i n g t h e conclusions o f t h e s e i n t e r p r e t a t i o n s . T h i s e s s e n t i a l l y asks f o r some measure of t h e amount of i n formation about a s t r u c t u r e which can be obtained from small p a r t s of i t . We a r e q u i t e f a m i l i a r with t h e r e v e r s e procedure, i n j e c t i n g i n f o r m a t i o n i n t o a s t r u c t u r e . A t i p i c a l example i s the proof of Morley's Upward C a t e g o r i c i t y r e s u l t using Keisler's two cardinal theorem. We have a s t r u c ture which i s not s a t u r a t e d and we i n j e c t t h i s information i n t o i t so t h a t , t o p u t i t antropomorphically, the s t r u c t r u r e "bnom" t h a t i t i s not s a t u r a t e d and "hemembehb" i t a f t e r passages t o equivalent s t r u c t u r e s . Can we turn t h i s technique a r o u n d ? Perhaps we should not b e t o o frugal i n t h e begining and allow countable samples because i t i s not f i n i t e n e s s which i s important; i t i s the i n t e r p r e t a t i o n s of the sample. And h e r e we come t o t h e beginings of model theory which s t a r t e d with the s t u d y o f i n t e r p r e t a t i o n s ; however, t h e s e a r e i n t e r p r e t a t i o n s of s y n t a c t i c a l o b j e c t s by semantical ones. The i n t e r p r e t a t i o n s under discussion a r e just t h e o p p o s i t e , they seek t o i l l u m i n a t e a semantical information by s y n t a c t i c a l means.This i s e x a c t l y t h e p r a c t i c e used i n science. The processes of t h e r e a l w o r l d l i k e the behavior of t h e e l e c t r o n o r the development of an e m b r i o a r e i n t e r preted by formal means l i k e H i l b e r t spaces and s t a b i l i t y theory r e s p e c t i v e l y ( t h e popularity of t h e c a t a s t r o p h e theory i s mainly due t o t h i s kind of i n t e r p r e t a t i o n ) . What model theory has s t u d i e d and s t u d i e s i s the r e l a t i o n of the t h e o r i e s encountered a b o v e , o r t h e o r i e s simply invented, t o a b s t r a c t s t r u c t u r e s which pass a s a s u b s t i t u t e f o r t h e r e a l world. Perhaps by turning some of i t s a t t e n t i o n t o t h e f i r s t s t e p ( r e a l world - theory and checking) w i l l model theory f u l l y l i v e u p t o i t s name.
REFERENCES
I
Barwise, K. 1974,
J.
M o b t o m k i ' n coUupbing 6unCtion, Fund. Math. 8 2 , 95-103.
Barwise, K. J . and J . S. Schlipf 1976,
An inthoduction .to h e c ~ i v d ybatwrated and hebplendent mod&, The Journal of Symbolic Logic, 41, nQ 2 , 531-536.
131
SOME D I R E C T I O N S ON MODEL THEORY
Benda, M. 197+a,
Modeloih,
197+b,
CompacAnan doh o m W n 5
i n preparation. 06
tgpeh, t o appear.
Burgess, J. and D. M i l l e r 1975,
Rematrkn on invahiavLt
duchiptiwe n e t theohg, Fund. Math. 90, n0 1,
53-75. Chang, C. C. and H. J. K e i s l e r 1973,
Model Theory
Amsterdam.
H.
Friedman, 1975,
, North-Holland,
One Iiundhed and .tm phubeemn i n mathemmatical l o g i c , T h e J o u r n a l o f Symbolic L o g i c , 40, n0 1, 113-129.
Fagin, R. 1976,
P h o b a b U e n on ljinite mod&,
The Jounal o f Symbolic Logic, 41,
n0 1, 50-58. Hanick, V. and M. Makkai 1976,
Vaugkt nelztcncen and Xhe c o u d n g theohem,
The J o u r n a l o f
Sym-
b o l i c Logic, 41, n0 1, 171-187. Jensen, B. R. 1972,
The dine n&um3.u~e06 com&ucLibLe kiehmchy,
Annals o f
Math.
Logic, 4, n0 3, 229-308. Jones, N . 0. and 1972,
A.
L. Selman
TwLing mackineh and t h e npectha ad @ & - o h d m 6ohmuean, P r o ceedings o f t h e 4 t h Symposium on Theory o f Computing, Denver, 185-1 96.
K e i s l e r , H. J. 1976,
Model Theory of Finite Structures, ( l e c t u r e n o t e s ) , S p r i n g o f 1976.
132
MIROSLAV
BENDA
Kueker, D. W . 1972,
Liiwenhh-Skobem and intehpalatian theahem6, B u l 1 . Amer. M a t h . SOC., 78, 211-215.
Loeb, P. A . 197+,
C o n v m i a n 6hom non,5tandahd t v
ntandahd
mCabUhe
~ p a c e n ,t o
appear. Morley, 1970,
M.
D.
The nwnbeh 06 countabbe mad&, nP 1, 14-18.
The J o u r n a l o f Symbolic Logic,35,
Robinson, A. 1973,
Meiamathematicd phobbemn,
The J o u r n a l
of
Symbolic L o g i c , 38,
nP 3, 500-516. Rubin, M. and S. Shelah 1915,
On fineahey ahdehed made&, N o t i c e s AMS, o c t . , A646.
Ryl 1 -Nardzewski
1952,
, C.
The h a k k 0 6 t h e axiam 06 i n d u d a n i n elementahy a h i t h n i e t i c Fund. Math.,
39,
,
239-263.
Shelah, S. 197+,
A compactnu6 theahem d V h n i n g d a h cahdin&,
i h e e algebhah,
W h i t e h e a d phabbem, and t h a n A u e h A u h , ( s e m i n a r n o t e s ) Louvai n. Silver, 197+,
J.
How t o eLikncLte t h e dine nthuotuhe dhom t h e wahk a6 J e a e n , t o appear,
Thom, R. 1969,
Topobogicd made& i r l bialogy,
Topology, 8 , nQ 3, 313-335.
Vaught, R. L. 1974,
lnvahiant oe& i n topology and l o g i c , Fund. Math. 82, 269-293.
133
SOME DIRECTIONS I N MODEL THEORY
Department o f M a t h e m a t i c s U n i v e r s i t y o f Washington S e a t t l e , Washington, U.S.A. and Departamento d e M a t e m i t i c a U n i v e r s i d a d e d e Sao P a u l o Sao Paulo, S P . ,
Brazil.
Non-Classical Logics, Model Theory and Computability, A . I . Arruda, N.C.A. da Costa and R. Chuaqui (eds .) 0 North-Holland Publishing Company, 1977
A SEMANTICALDEFINITIONOF by R O L A N D 0 8.
PROBABILITY
CHUAQUI
there are
two
subjects t h a t come t o mind. On t h e one hand, the axiomatic foundations
When one speaks about the Foundations o f P r o b a b i l i t y ,
of
the Calculus o f P r o b a b i l i t y , which i s a w e l l
- developped,independent
mathe-
matical d i s c i p l i n e ; on t h e o t h e r hand, t h e study o f p o s s i b l e i n t e r D r e t a t i o n s f o r p r o b a b i l i t y statements. As i s w e l l -known, there are several c o n f l i c t i n g i n t e r p r e t a t i o n s o f these statements h e l d by t h e d i f f e r e n t schools i n
the
subject. I n t h i s paper, based on the i n t e r p r e t a t i o n o f P r o b a b i l i t y r e l a t i n g i t t o t r u t h , I present a new d e f i n i t i o n o f a p r o b a b i l i t y measure i n seman-
t i c a l terms. Thus, most o f the content o f t h e paper deals w i t h
the
second
o f the foundational subjects mentioned above. Expressions o f the form ' I t i s probable t h a t . . . I same way as, although more guardedly than,
guide b e l i e f
'It i s true that...'.
in
the
Thus, both
expressions have gerundive f o r c e . (For an i l l u m i n a t i n g discussion see Lucas 1970, Ch. I,11). This k i n s h i p n a t u r a l l y leads t o attempt a d e f i n i t i o n P r o b a b i l i t y s i m i l a r t o T a r s k i ' s semantical d e f i n i t i o n o f t r u t h
of
(see Tarski
1935).
I b e l i e v e t h a t f o r many common language uses o f t h e word ' p r o b a b l e ' these I s h a l l
there i s no adequate measure o f p r o b a b i l i t y . However, w i t h
not be concerned. My aim i s t o a t t a i n a d e f i n i t i o n o f a probabilitymeasure, s a t i s f y i n g the axioms o f the Calculus, t h a t can account f o r al1,or a t l e a s t most, o f i t s a p p l i c a t i o n s i n c u r r e n t s c i e n t i f i c and s t a t i s t i c a l p r a c t i c e . My conception i s connected w i t h Carnap's d e f i n i t i o n (see Carnap 1950 o r
Carnap and Jeffreys
1971)
-
a c t u a l l y , t h e p i c t u r e on page 297 o f h i s
book i n s p i r e d some of my ideas
-
because I use model - t h e o r e t i c a l 135
1950
methods.
136
ROLAND0 6 . CHUAQUI
However, s i n c e I see P r o b a b i l i t y i n t h e same c a t e g o r y as t r u t h and n o t l o g i c a l t r u t h , t h e r e a r e i m p o r t a n t d i f f e r e n c e s i n t h e two conceptions. I n p a r t i c u l a r , Carnap
i n t e n d s h i s measure t o be d e f i n e d on t h e space o f a l l mod-
e l s , whereas I l i m i t t h e space t o t h o s e models t h a t p i c t u r e a p o s s i b l e o u t come. Which outcomes a r e p o s s i b l e i s determined by t h e laws o f t h e phenomenon i n v o l v e d . Thus, ' p o s s i b l e ' i s meant h e r e i n t h e sense o f p h y s i c a l pos
-
s i b i l i t y and n o t l o g i c a l p o s s i b i l i t y . out-
I n o r d e r t o o b t a i n a p r o b a b i l i t y measure i n t h e space o f p o s s i b l e comes,
I use an equal l i k e l i h o o d o r e q u i p r o b a b i l i t y r e l a t i o n . Being
based
on e q u i p r o b a b i l i t y , my d e f i n i t i o n i s a l s o r e l a t e d t o t h e c l a s s i c a l concept i o n . However, my equal l i k e l i h o o d r e l a t i o n w i l l be w e l l - d e f i n e d
and w i l l
n o t s u f f e r f r o m t h e p i t f a l l s o f t h e c l a s s i c a l one. The paper has an i n t r o d u c t o r y f i r s t s e c t i o n t h a t g i v e s a g e n e r a l
des-
c r i p t i o n o f t h e procedures i n v o l v e d , and a second s e c t i o n w i t h some m a t h e m a t i c a l p r e l i m i n a r i e s . I t c o n t i n u e s i n a t h i r d s e c t i o n w i t h t h e formal d e f i n i t i o n o f t h e s i m p l e p r o b a b i l i t y s t r u c t u r e s and t h e c o r r e s p o n d i n g p r o b a b i l i t y r e l a t i o n , n o t i o n s which a r i s e when t h e r e a r e no
equi-
sequences o f
outcomes. The n e x t s e c t i o n g i v e s necessary and s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f a p r o b a b i l i t y measure c o m p a t i b l e w i t h an e q u i v a l e n c e
relation
h a v i n g t h e same p r o p e r t i e s as t h e equal l i k e l i h o o d r e l a t i o n d e f i n e d p r e v i
-
o u s l y . T h i s s e c t i o n o f t h e paper c o n t a i n s t h e most i n t e r e s t i n g newmathenati c a l r e s u l t s o f t h e paper: necessary and s u f f i c i e n t c o n d i t i o n s f o r t h e
ex-
i s t e n c e o f a measure on a f i e l d o f s e t s , i n v a r i a n t under a group o f t r a n s formations. The f i f t h s e c t i o n d i s c u s s e s compound p r o b a b i l i t y which a r i s e when t h e r e a r e sequences o f outcomes, and ways
structures, of
defining
a
p r o b a b i l i t y measure f o r them. The paper concludes, i n s e c t i o n s i x , w i t h ex-
A reader n o t interested i n the technical material could read sections one a n d s i x t o
amples and methods f o r a p p l y i n g t h e p r o b a b i l i t y models g i v e n . understand t h e main i d e a s
.
A b r i e f o u t l i n e o f an e a r l i e r v e r s i o n o f t h e s e i d e a s appeared i n Chuaqui 1975, a l t h o u g h some o f them were a l r e a d y i n Chuaqui 1965.
1, GENERAL DESCRIPTION, We assume a language L w i t h some s e t o f n o n l o g i c a l c o n s t a n t s . F o r
the
moment we l e a v e t h e e x a c t d e s c r i p t i o n o f L undetermined. I t m i g h t b e a f i nitary f i r s t
- order
language, an i n f i n i t a r y language, o r a language o f some
137
S E M A N T I C A L D E F l N l T t O N OF P R O B A B I L I T Y
other type.
F o r t h e d e f i n i t i o n o f t r u t h , we u s u a l l y
i n t e r p r e t L i n hela-
-titiand Aybtemb c€ =
' $ 0 &ue
i n d'
.
I$
we can define,by
I n a system o f t h i s t y p e
everything
r e l a t i n g t o t h e symbols i n L i s determined. Thus, f o r e v e r y sentence o f L , @ i s t r u e o r f a l s e i n LJZ.
Systems r e p r e s e n t , i t may be s a i d , p o s s i b l e s t a t e s
o f t h e w o r l d . When we i n t e r p r e t L i n Ce, we determine c o m p l e t e l y which sentences a r e t r u e and which a r e f a l s e . When p r o b a b i l i t y statements apply, t h e p r e c i s e s t a t e o f t h e w o r l d which o b t a i n s i s n o t c o m p l e t e l y determined; t h e r e f o r e we must change t h e n o t i o n o f an i n t e r p r e t a t i o n o f t h e language L . Our i n t e r p r e t a t i o n s w i l l no l o n g e r be s i n g l e systems b u t c e r t a i n c l a s s e s o f systems w i t h some a d d i t i o n a l ties.
proper-
The C a l c u l u s o f P r o b a b i l i t y i s u s u a l l y a p p l i e d t o happenings o r o c c u r rences, which may be experiments, o b s e r v a t i o n s , o r n a t u r a l phenomena. i n t e r e s t s us i n a l l t h e s e cases i s t h e henu&
What
o r alLtcame o f t h e phenomenon.
The " t h e o r y " ( i n a wide sense o f t h e term) i s what determines which a r e t h e p o s s i b l e outcomes. I n some cases t h e r e i s o n l y one p o s s i b l e
outcome;
o t h e r s , many. I t i s i n t h e s e l a t t e r cases t h a t t h e C a l c u l u s o f
in
Probability
i s important. L e t us t a k e as o u r f i r s t example t h e choosing o f a sample S o f s i z e from a f i n i t e p o p u l a t i o n P . When we say 'S
m
has n r e d t h i n g s ' we mean t h a t
one o f t h e p r o p e r t i e s o f t h e outcome was t h a t t h e sample had n r e d objects. The same outcome has many d i f f e r e n t p r o p e r t i e s , which can be
described
in
many d i f f e r e n t wayswe can t h i n k o f a n i d e a l a p p r o x i m a t i o n - o f an outcome,nanel y a r e l a t i o n a l system t h a t r e p r e s e n t s a p o s s i b l e model o f t h e s i t u a t i o n in-
volved. I n t h e case we a r e l o o k i n g a t we can schematize t h e comes as systems
Rl,...,RH-l
a s =
,S>, where
possible
out-
P i s a f i x e d f i n i t e set,
a r e f i x e d subsets o f P t h a t r e p r e s e n t t h e p r o p e r t i e s w.e
are
i n t e r e s t e d i n ( f o r i n s t a n c e , ' r e d ' ) , and S i s any subset o f P o f m members ( t h e sample). For each subset S o f m members t h e r e i s a correspending system
as;
hence t h e s e t o f p o s s i b l e outcomes c o n s i s t s o f a l l models
o f t h e f o r m d e s c r i b e d above.
as
To speak about t h e s e outcomes we need a language L with n o n l o g i c a l s y m -
-
-
-
R1,...,Rn-l,S. We now use as i n t e r p r e t a t i o n s f o r L n o t systems, b u t classes, K o f r e l a t i o n a l systems a p p r o p i a t e f o r L . K, then, w i l l be t h e bols
s e t o f p o s s i b l e outcomes.
138
ROLAND0 6. CHUAQUI
t h e dimple
I n t h e case t h a t we s h a l l f i r s t c o n s i d e r , i . e . d-iXLLot(lheJ,
phobabLlLty
K t o be a s e t o f r e l a t i o n a l systems w i t h a common
we can t a k e
u n i v e r s e . These s i m p l e casesmay be c h a r a c t e r i z e d as t h o s e i n which t h e r e a r e no sequences o f outcomes. F o r i n s t a n c e , i n t h e example j u s t m e n t i o n e d , M i s t h e s e t o f systems
as. We
s h a l l l a t e r analyze cases i n which sequences o f
o u r p r o b a b i l i t y s t r u c t u r e s w i l l be more complicated.
outcomes appear; t h e n
However, i n o r d e r t o b u i l d t h e s e compound p h o b u b i l i t y n,iXuctwre~ we
shall
need t h e s i m p l e ones. P r o p e r t i e s o f outcomes a r e u s u a l l y c a l l e d ewe&.
These events
can
r e p r e s e n t e d by sentences. I f two sentences a r e l o g i c a l l y e q u i v a l e n t , r e p r e s e n t t h e same event.
be they
classes
Thus we may t a k e events as e q u i v a l e n c e
o f sentences determined by t h e r e l a t i o n o f l o g i c a l equivalence. Thus, i f we t a k e @ t o be t h e sentence ' t h e r e a r e n r e d t h i n g s i n S ' , t h e euent $J
o b t a i n s i s t h e c l a s s o f sentences l o g i c a l l y e q u i v a l e n t t o @ .
We
that shall
d e f i n e a p r o b a b i l i t y measure on t h e s e e q u i v a l e n c e c l a s s e s . However,
when
t h e r e i s no danger o f c o n f u s i o n we s h a l l speak o f t h e p r o b a b i l i t y o f Q meani n g t h e p r o b a b i l i t y o f t h e corresponding equivalence class. We know t h a t f o r r e l a t i o n a l system i s true i n
a.
( o r 6&e)
in M , if
For
a
and sentences @ we can d e f i n e @
K, we say t h a t Q ORhue M. i n K. We understand t h i s
simple p r o b a b i l i t y structures @
i s t r u e ( o r f a l s e ) i n e v e r y r e l a t i o n a l system i n
S i m i l a r l y , we s h a l l d e f i n e t h e phobabLlLty
06
@
p r o b a b i l i t y as s o r t of a measure o f t h e degree o f " p a r t i a l t r u t h " o f a sentence. Thus, i f @ i s t r u e i n
K,
i t s h o u l d be t h e case t h a t t h e p r o b a b i l i -
K, t h e p r o b a b i l i t y o f
ty o f @ i n M i s one; i f @ i s f a l s e i n
s h o u l d be z e r o ; and i n a l l o t h e r cases t h i s p r o b a b i l i t y s h o u l d
in
@
be
K
between
z e r o and one. T h i s degree o f " p a r t i a l t r u t h " i s vaguely p r o p o r t i o n a l t o t h e s i z e o f t h e s e t o f models o f set i s
K i t s e l f ; and i f
@
@ t h a t are i n
i s false i n
K. I f
@
i s t r u e i n M,
this
K, t h i s s e t i s empty. The n a t u r a l
t h i n g t o do, then, i s t o o b t a i n a p r o b a b i l i t y measure d e f i n e d on a c o l l e c t i o n o f subsets o f
K t h a t i n c l u d e s t h e s e t s o f models i n K o f sentences
of t h e a p p r o p i a t e language. Having done t h i s , we may d e f i n e t h e p r o b a b i l i t y of @ in
K as t h e measure o f t h e s e t o f models o f
o r eqLLiphobubXLty r e l a t i o n between events, i . e .
K.
@ that are i n
I n o r d e r t o d e f i n e t h i s measure I s h a l l i n t r o d u c e an ~ Q U &
likelihood
between e q u i v a l e n c e c l a s s e s
of l o g i c a l l y e q u i v a l e n t sentences. T h i s e q u i p r o b a b i l i t y r e l a t i o n i s d e r i v e d from a symmettry r e l a t i o n between s e t s o f models, i . e . t i v e l y , two s e t s o f models a r e nymmc?Ahd,
subsets o f
M. I n t u i -
i f t h e y a r e symmetrical w i t h r e -
s p e c t t o t h e laws o f t h e c o r r e s p e n d i n g phenomenon. F o r d e t e r n i n i n g
these
139
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
06 L ~ ~ a ~ d ~ o h m a t(ifoum n c t i o n s ) that atre "invahiant undw t h u e Lawb". Two subsets o f K w i l l be symmetrical i f one
symmetries, we consider the ghoup
can be transformed i n t o t h e o t h e r by one o f these functions. Then, two sentences ( o r , more accurately, two equivalence classes o f l o g i c a l l y equivalent sentences) models i n
9 and $ are e q u a l l y l i k e l y i n K , i f t h e corresponding sets o f M o f @ and are symmetrical.
The laws o f the phenomenon determine t h e simple p r o b a b i l i t y
structure
M ; M, i n i t s t u r n , gives t h e group o f transformations. L e t us s e e h o w t h i s group, c a l l i t G K , i s obtained i n our example.
GK i s a subgroup
group o f permutations o f t h e universe, P. For a permutation
6
of
the
t o be i n GK ascan
i t must s a t i s f y some a d d i t i o n a l requirements. Each r e l a t i o n a l system
K,
be decomposed i n two p a r t s . One t h a t i s the same f o r a l l s y s t e m s i n namely the constant p a r t part
= cP,S>.
cK
=
and another, the
6
Now, t h e f i r s t requirement on
variable
i s t h a t i t should n o t
M o u t s i d e o f K. That i s , i f we apply 6 t o the v a r i a b l e , transforming i t i n t o another system asl,v ( S l i s p a r t o f a system, 02 s, v then the s e t o f elements d ( x ) f o r x i n S, i . e . the image o f S by d ) , take a system i n
s h o u l d a l s o be t h e v a r i a b l e p a r t o f
%,V
,...,Rn-,
, S'>
a
system i n
IK(i.e.
should be i n K ) . I n the p a r t i c u l a r case o f theexample,
since the o n l y c o n d i t i o n on S i s i t s c a r d i n a l i t y , a l l permutations
of
P
s a t i s f y t h i s requirement. I n general, however, t h i s i s n o t so.
A second c o n d i t i o n imposed on t h e f u n c t i o n s
6
i n GK appears when
there
are d i s t i n g u i s h e d elements (denoted by i n d i v i d u a l constants i n the language) i n the systems i n K. I f we transform the s e t o f d i s t i n g u i s h e d elements
by
6,
To
t h e new elements should s a t i s f y t h e same sentences as the o l d ones.
s t a t e p r e c i s e l y t h i s requirement we need some t e c h n i c a l machinery t h a t w i l l be i n t r o d u c e d i n t h e n e x t section.The l a s t c o n d i t i o n imposed i s t h a t t h e image
by 6 o f a s e t o f models o f a sentence should be the s e t o f models of (poss i b l y another) sentence. Having t h i s group G K , we can d e f i n e the symmetry r e l a t i o n between subsets o f K. L e t A ,
B be subsets o f K ; we say t h a t
cal i f there i s a function tems
d
6
A and B are syhmetri-
i n GK such t h a t t h e v a r i a b l e p a r t s Sv o f sys-
i n B are e x a c t l y those obtained by t h e a c t i o n o f
p a r t s Gev
o f systems
6
on t h e v a r i a b l e
in A.
We are now l e f t w i t h the mathematical problem o f o b t a i n i n g a p r o b a b i l i ty measure i n v a r i a n t under t h i s equivalence r e l a t i o n . Namely, a measure that
assigns the same number t o symmetrical sets o f models. I s h a l l s e c t i o n f o u r , p o s s i b l e s o l u t i o n s t o t h i s problem,
discuss
i n particular,
in
some
140
ROLAND0 B. CHUAQUI
necessary and s u f f i ci en t conditions f o r the existence of such a measure. The simple probability structures ar e, according t o my views, the basic structures. All probability measures ar e derived from probabilities defined on thesesimple s t r u ct u r es . I shall not analyze a l l cases of derived probab i l i t y . However a very important case appears when there are sequences of outcomes. For t h i s s i t u at i o n I introduce the compound p m b a b i L L t y A&UCRwzen. They are b u i l t from the simple ones by adding an ordered system t h a t represents the ordering in the sequence of outcomes. A t each point in the order, the preceding outcomes determine a simple probability structure and the next outcome i s from t h i s s t r u ct u r e. The probability measure f o r compound structures i s computed from the measures on the simple ones as i s usually done with conditional probability. These compound structures will be discussed in some de ta il in section five. However the need f o r them may be seen from the following example: Suppose we have two urns, urn one and u r n two. Urn one has one b l a c k b a l l , and u r n two has two white b a l l s , ball one and ball two. Suppose, furthermore, t h a t we choose an u r n a t random, and then a ball from t h a t urn also a t random. I t i s cl ear t h at the event of choosing a white ball is equiprobable t o the event of choosing a black b al l . However,the f i r s t event can be divided into two events while the second cannot. I t i s easy t o see tha t the definition of equal likelihood I have given for simple structures implies the following: I f A i s equally l i k el y t o B and A can be decomposed intotw odisjoint events A, , A , , then 13 can also be decomposed into disjoint events E L , 8 , th a t are equa.ily likely respectively t o A, , A 2 . In the case we have j u s t discussed t h i s i s not true , unless we a r t i f i c i a l l y divide the event of choosing a black ball into two events. The explanation for t h i s phenomenon i s t h a t we ar e confronted with compound o u t comes. We f i r s t choose an urn and then, depending on t h i s choice, we choose a b a l l . The probability structures we have t o consider a re , thus, more complicated. They should be of the following type: Since there are only two succesive outcomes, the basic order of sequence of outcomes can be given by the numbers 0 and 1 in t h e i r natural order. We have two s e t s of relational systems Mo and M,. KO consists of models %=
< U , C > where U i s the s e t of urns and C contains the one chosen. Thus, K O consists of uI1 ,where urn one i s chosen, and UL,, where urn t w o i s chosen. M 1 contains systems of the form < A , B , W , D > , where A i s the s e t of balls in an urn, B the black b a l l s , W the white b a l l s , and D t h e chosen
141
SEMANTICAL DEFINITION OF PROBABILITY ball.
!K1 i s d i v i d e d i n two p a r t s ,
one system
UZ
B1 =
urn one,
contains just and lKml where A1 contains j u s t t h e black b a l l i n
lKul
contains two models Ce2 =
and Ce3 =
contains the white b a l l s i n u r n two, B 2 i s empty, W 2 = A 2 , b a l l one, and 0,
Ul
contains
contains b a l l two.
A compound outcome here i s a f u n c t i o n
6
whose domain i s I 0 , 1
1
and
such t h a t
6(0) E Mg
and
6(1)
E
Md(o)
*
To speak about these outcomes we need a language t h a t allows us t o t a l k about sequences o f elements o f t h e order. This i s n o t d i f f i c u l t
to
do as
w i l l be seen i n s e c t i o n f i v e . Once we have t h i s language we have ourevents. For instance, t h e event o f choosing a w h i t e b a l l has as models the two functions
6,g
such t h a t : d(0) = g ( 0 ) = U Z 2 ,
d(l)
= UZ2
and g ( 1 ) = (n3
.
I n order t o d e f i n e a p r o b a b i l i t y measure we proceed as f o r t h e p r o b a b i l i t y s t r u c t u r e s and d e f i n e i f f o r KO,
M m 1 , and lK
simple
independently.
a2 Then by usual c o n d i t i o n a l p r o b a b i l i t y procedures, we d e f i n e i t f o r the com-
pound events. For instance, l e t F be the s e t o f models c o n s i s t i n g o f the 1 function 6 qiven above. Then, as the p r o b a b i l i t y i n K O o f U 2 i s 7 and 1 the p r o b a b i l i t y o f d 2 i n !,M i s a l s o 7 , we have t h a t the p r o b a b i l i t y of F i s
Y1 - 1T =1 ~ .
2, SET-THEORETICALA N D
2
METALOGICAL P R E L I M I N A R I E S ,
We use throughout various f a m i l i a r s e t - t h e o r e t i c a l notions and symbols. U , n , - , 5 w i l l denote unian, iYLte)Lneot.ian, b e t - MevheLLcaQ. d i 6 6 e h e n c ~ and the incLuhion hel.aLLan. By w we denote t h e set 0 6 & vlatwlae numbem
(nonnegative i n t e g e r s ) , which a r e i d e n t i f i e d w i t h
the f i n i t e o r d i n a l s .
denotes the f i r s t uncountable o r d i n a l . Each o r d i n a l i s i d e n t i f i e d w i t h
w1
the
s e t o f a l l smaller numbers:
a=IB:B
a < f3 i s thus e q u i v a l e n t t o t h e c o n d i t i o n a € f3 and a , B are
142
ROLAND0 8. CHUAQUI
ordinals. I n p a r ti cu l ar , the number zero, 0 , i s the empty s e t 0, 1 = I 0 1 , 2 = {0,1 1 , etc... A' denotes the A& 06 & @nCtion6 on 6 ivcto A ( i . e . with domain 8 and range included in A ) . I f 6 i s a function, Uo6 denotes i t s domain, 6 - l i t s c o n w m e , U o 6 - l i t s hange, and f o r any x~ U o 6 , 6*x i s the h u g e 06 x undm 6 , and 6 I x i s 6 ha.thioted i n domain .to x The value
.
~ u n C t i o n 6 at t h e
element i of i t s domain i s denoted by d(i) or di. F E I A, we write in some cases F = < F i : i € I > and call i t a A@tem. The members of wA are w h a t are called nhimpte i n d i n i t e Aequencu w i t h are referred t o as 5i& t m n i n A . In case n E w , the members of A' vLite n - t m i e d nequencen again w a h t m in ~ A. If 6 E ' A , we write 6 = whose <do,. ..,6n-1> ; in particular < a , b> i s the two - termed sequence terms are a , b . The symbol S A denotes the A& 06 & d i n i t e hequencu w L t h t m i~ n A ( i . e . $A = u {'A: n E w 1 ) . An n - a h y h w o n R on A i s any subset of ' A . For n-ary relations R we write Rxo x n - ] instead of < x o,..., xn-l> € R. If R i s binary, aRb stands f o r < a , b> € R. A heRatioMae A Y A ~i s ~ a system a of the form d = < A , R , a > , where: A i s a nonempty s e t called the domain or u n i w m e of the system ul, R =
If
a
,...,
aI
K =
a.> j
.
L E I ~ K j, € J n K '
,
We now consider the concept of iAomohphiAm of two s i m i a r a J a n d & = <6,Si, b j > i Let oZ= < A , R i ,
systems. be t w o
similar relational systems. Then & and.% are isomorphic i f t h e r e i s
a
143
SEMANTICAL D E F I N I T I O N OF PROBABILITY
6
one-one f u n c t i o n
(i)
f o r every
A
from
i
I
E
onto B such t h a t :
x o ,..., x m , - , € A we have, RixO
and
,... , 6 ( x m
i f and only i f S i 6 ( x 0 )
(ii) f o r every j € J ,
6
Under these conditions
q)= h j .
,...,x
1
~ , - ~ h
h
i s c a l l e d an Domoxpkiom and & the homohpkic &I-
age 06 Ce undetl {, i n symbols ,$ = 6 * a . I t i s c l e a r t h a t i f d i s a system and 6 a one - one f u n c t i o n w i t h domain A , then there i s one and o n l y
one r e l a t i o n a l system
such t h a t
C$
8 = 6 *Ce. 6
systems w i t h a comnon universe A , and
A , then Bd denotes the b e 2 i n symbols: p16 = { d * Q
06
I f B i s a set o f
relational
i s a one-one f u n c t i o n w i t h domain
nyotems obfained by .tlze a o t i o n
06 6
in
B,
:Ce EB}.
We a l s o use some terminology from General Algebra.
F o r more
details
about t h i s and r e l a t i o n a l systems see Henkin, Monk, and Tarski 1971,Prelimi n a r i e s and Ch. 0. For any
or
a E w
a = w , an a - atly p c & t i d
o p w o n Q on a s e t A
i s a f u n c t i o n w i t h domain included i n “A and range i n c l u d e d i n
.
A If
002 = “A , then we say t h a t Q i s a t o t i d a - u h y opehation, o r s i m p l y an a - m y op&an, on A. A t o t a l 0 - a r y o p e r a t i o n on A represents a d i s t i n guished element o f A . A pahtiae @ e b m i s a system d o f the form CZ=
Q =
where A i s a nonempty s e t and e r a t i o n s on A.
i s t o t a l . An equivalence r e l a t i o n R on A i s a congmencc h M o n on
a
if
such t h a t Q. i s a - ary, aiRbi f o r every ’iE a , and J Qju and Q.b e x i s t , we have Q.a R Q.b, f o r every j € I . It i s well-known J J J t h a t i t i s p o s s i b l e t o obtain, f o r each congruence r e l a t i o n R , the y u o f i e n t a,b E “A
f o r every
(pw5.a.t)
Cel R
@ebm
.
Ile a l s o consider c e r t a i n kinds o f algebras and p a r t i a l B o o k z n @ebha
(BA) i s an algebra
B
=
the b i n a r y operations o f j o i n and meet, pleme*on,
A
-
i s the unary operation o f
corn-
and 0 and 1 are d i s t i n g u i s h e d elements o f B . B A ‘ s a r e c h a r -
a c t e r i z e d by w e l l
A { x : x a A}
algebras.
- known axiom systems.
t h e nupmnmum o r
I f A _c B , we designate by V I x : x E A } ,
A i f they e x i s t i n B . When we x € %, we w r i t e V I x , : i E w } , A { x i : i E d . I f a l l suprema and i n f i m a e x i s t f o r countable subsets A , we say t h a t B is coumkbty complete. The notions o f atom, ,id&, and countably c o m d e t e
have an i n f i n i t e sequence
in@num
of
144
ROLAND0 B. CHUAQUI
i d e a l i n a BA w i l l a l s o be needed. I f 1 i s an i d e a l i n a BA B , we w r i t e B / I f o r the q u o t i e n t algebra determined by I . For more d e t a i l s about BA's see S i k o r s k i 1969.
A p a r t i c u l a r k i n d o f BA i s a 6 i e L d 0 6 n u b n e t n 0 6 a n e t X, lF = - , 0 , X > , where F contains subsets o f the s e t X , and t h e op-
the empty net, n e A i s a f i e l d o f sets t h a t contains a l l countable unions and i n t e r s e c t i o n s . A ghoup 06 Rham6omcLtioy~d
e r a t i o n are union, i n t m e o t i o n , complement h M v e t o X, and X i t s e l f . A c o u w k b l y compLete 6ie.td 06
IF i s a subset
06
t h e ghoup 06 pehmLLtatiom ( o n e - one, o n t o ~ u n c t i o m ) o 6 elements o f F i n t o elements o f F.
06
X, t h a t transform
The s e t o f nonnegative r e a l numbers extended by adding
R . [ O , l] i s the closed i n t e r v a l between 0 and 1
. We
i s denoted by
use a d d i t i o n +,and
countable a d d i t i o n C among r e a l numbers.
A BA B i s isomorphic t o t h e c l o n e d - open n u 2 06 a t o t a U y d i s c o n n e c t e d compact t o p o l o g i c a l npace, t h e Stone npace 06 B; i n symbols S ( B ) . I f X i s a Stone space, C ( X ,z) denotes t h e n e t 06 covLti~uoud~unc-tiom wLthcompact nuppoht on X and v&es in R. Einary a d d i t i o n + o f f u n c t i o n s i s pointwise a d d i t i o n . Countable a d d i t i o n C i s the continuous l i m i t o f t h e f i n i t e sums, which d i f f e r s from the pointwise l i m i t on a s e t o f f i r s t category. A genehaeized cahdhul algebha (GCA) i s a p a r t i a l algebra with a p a r t i a l binary operation
+
a =< A ,
+ ,C >,
and a p a r t i a l countable operation C(see
Tarski 1949). A mdid algebha (CA) i s a GCA i n which t h e p a r t i a l operat i o n s are t o t a l . A p a r t i c u l a r k i n d o f GCA i s t h e d i s j u n c t i v e BA ed w i t h t h e BA B
.
E
=
and o n l y i f a V 0 = c and
+,
C>
a
~ =b 0,
"xi
CiE
= c
h associat-
a+ b = c
i s obtained as f o l l o w s :
and
if
: i E d = c and xi A x . = 0 f o r a l l i < j E w . J (See Tarski 1949, def. 15.14). By a 6 i n i t e L y a d d i t i v e ( p h o b a b L L t y ) meabuhe o r a BA IB we
if
only i f
v{x,
understand,
as usual, a f u n c t i o n p s a t i s f y i n g the f o l l o w i n g c o n d i t i o n s :
8
and DO p-1 5 l o , 1 ]
1.
DO
2.
i f a,b E B and a A b = 0 , then
3.
p ( 1 ) = 1.
p =
;
p(a
+ b)
= p(a)
+ p(b);
We say t h a t p i s a countable uddditive ( p m b a b U y ) meabue or,simply, a mecuue, if i n a d d i t i o n p s a t i s f i e s : 4.
I f x E "B
and
Ciao
xi e x i s t s , then
A measure 1.1 i s s t r i c t l y p o s i t i v e i f :
p(CiEwxi)
= ZiEw
p(xi)
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
5.
~ ( a =) 0 implies
a = 0,
f o r every
145
a 8 B.
Throughout t h i s paper we s h a l l consider two d i f f e r e n t f i r s t - order languages of s i m i l a r i t y type A = <<mi: i E I > , J > , a 6irZitatg language LA and . The language L* has a d e n u m e r a b l e s e t of an in,$ivtitahg language L* W1W
vuhiublw, and
1"
WlW
has a s e t of v a r i a b l e s of c a r d i n a l i t y
wl.
The con-
n.tatant6 a r e divided i n t o l o g i c a l
and nonlogical ones. Both languages have logical constants A , V , I , v , 3 and = , standing f o r (finite)conjuncLLon, dibjunOtiOn, negation, u n i v e u d and e x b t e n t i d quuntibication, and idena y . In a d d i t i o n the i n f i n i t a r y language LA has l o g i c a l constants A and WlW
v standing
f o r denumerable conjunctions and d i s j u n c t i o n s , r e s p e c t i v e l y . The nonlogical constants f o r both languages a r e the phedicatw ( o r heXation q t n b o b ) and the i n d i v i d u a l comRavLtn. W i t h every p r e d i c a t e a n a t u r a l number i s c o r r e l a t e d , which i s c a l l e d t h e hank of t h e symbol. For every i E 1 there i s a p r e d i c a t e of rank mi For each j E 3, an individual constant. The correspondence i s such t h a t d i f f e r e n t indices correspond t o d i f f e r e n t symbols. The i d e n t i t y symbol, though regarded a s a l o g i c a l c o n s t a n t , is i n cluded i n the set of binary p r e d i c a t e s ( p r e d i c a t e s of rank two). The v a r i ables and t h e individual constants a r e t h e i n d i v i d u a l h p b o b . The exphunion.4 of L* a r e f i n i t e sequences of symbols; t h e expressions f o r L;, a r e t r a n s f i n i t e sequences of symbols of length l e s s than wl.Among
.
expressions we d i s t i n g u i s h t h e domula. The s i m p l e s t , s o - c a l l e d atomic bomLLean, a r e obtained by combining M a r b i t r a r y individual symbols by means of a p r e d i c a t e of r a n k n ; compound formulas a r e b u i l t from simpler ones by means of s e n t e n t i a l connectives and y u a n t i @ x exphwniokih ( i . e . q u a n t i f i e r s followed by v a r i a b l e s such as y u o r 3 v ) . An occurrence of a v a r i a b l e i n a formula may be e i t h e r &tee o r bvund; a formula in which no v a r i a b l e occurs f r e e i s c a l l e d a nentence. The s e t of sentences of L* i s designated by n Sb , and t h e s e t of sentences of L* b j Solo . 01
We assume known T a r s k i ' s d e f i n i t i o n of n a t i s 6 a c A o n and .thLL.th (for s e e K e i s l e r 1971). When t h e r e i s nodanger L* s e e Tarski 1954, and f o r L* 01W
of confusion we s h a l l use the same metalogical symbols f o r both languages. An a n i g n m e n t i n o2 i s a function from the v a r i a b l e s i n t o A . For x an assignment i n a r e l a t i o n a l system CL of s i m i l a r i t y type b and $I a formula
of
LA o r LA
W1W
we.assume defined:
cx c
@
1x1
( i . e . x nuh>di,w @ i n
a);
146
ROLAND0 E.
CHUAQUI
and, also, i f @ i s a sentence we have:
le TWO
(oh L
L*
1W
)
have,
0 h e in
(i.e. @
a,$
r e l a t i o n a l systems
a).
o f s i m i l a r i t y type
etemev~tahiey
are
e q u b d e n t , ifthey s a t i s f y e x a c t l y the same sentences o f L* ( o r
); i.e.
W 1W
b= @
P
02:
if and o n l y i f f o r every sentence @ o f L'
i f and o n l y i f
OZ C @
&
!=
( o r Lilw)we
@.
For any class lK o f r e l a t i o n a l systems o f s i m i l a r i t y type common universe A , any formula @ of i n A , we define:
blodK,x($) =
{ a :a
L*
M, d
E
( o r L*
W1W
!=$
I f @ i s a sentence, we w r i t e ModK(@)f o r Mod
mod&
06
me .in
@ &that
M.
6 and w i t h ) , and any assignment x
[XI} . K, x
06
( @ ) ; ModK(@)i-4 t h e
I f C i s a s e t o f sentences and @ i s a sentence, then @ i~ a c o n h e -
quence 06 C i f
@ holds i s a l l systems i n which a l l sentences o f
and we w r i t e
C C $
and we w r i t e
C @
.
.
01
system o f deduction, and we w r i t e C t- $ i f $ h dehivabte @om C
Aheohem i f i t i s d e r i v a b l e from the empty s e t , and we w r i t e c o n s u l t K e i s l e r 1971). By the w e l l first
- order.
- known
. This
.@0a L*
I- $ ( f o r
W1W
Completeness Theorem o f
l o g i c we have f o r every C 5 S* and every @ 8 S,'
and o n l y i f C t- @
hold,
C
0 v a l i d i f i t i s a consequence o f t h e empty s e t , For both languages L* and' 1 we choose a standard @
i s n o t t r u e f o r L*
W1W
"weak" completeness i n t h e sense t h a t f o r every @ E Stlw,
C @
if
still
have
C
; however we
finitary
I=@ i f and
only
if I-@.
We c a l l two sentences eqLLivden.t i f I-
@
<-
$ (where, as usual,@-$
stands f o r ( $ A $) V ( l @ A 11)). I t i s w e l l - known t h a t t h i s i s an equivalence r e l a t i o n , and t h e equivalence classes form a B A , t h e s o - c a l l e d
Lindenbawn-Tmki d g e b m
06
nentencu. I n the i n f i n i t a r y case, t h i s BA i s
countably complete. We denote, f o r any sentence @, by @ / I class o f @
, and
Sh
/
I- and S'
W1W
/ t-
t h e equivalence
are t h e corresponding BA's.
By
weak Completeness Theorem these algebras are isomorphic t o a f i e l d o f o f models. We s h a l l d e f i n e p r o b a b i l i t y measures on So / I- and S*
/ I-
wlw
.
the sets
By the
isomorphism mentioned above, t h i s i s e q u i v a l e n t t o d e f i n i n g them on a f i e l d o f sets o f models as Carnap does i n Carnap and J e f f r e y s 1971.
147
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
For SA /I7.1): THEOREM 2 . 1 ditive.
we have the f o l l o w i n g theorem ( S c o t t and Krauss 1966,
.
Evwy ,$inLteby a d d i t i v e meMWte on Sb /
3, SIMPLEPROBABILITY
Lemma
countably ad-
I- i.b
STRUCTURES,
As explained i n s e c t i o n one, t o introduce p r o b a b i l i t y we have t o change I n s t e a d o f i n t e r p r e t i n g our languages
the concept o f an i n t e r p r e t a t i o n .
r e l a t i o n a l systems, we do i t i n more complicated s t r u c t u r e s .
in
I introduce,
now, the simple p r o b a b i l i t y s t r u c t u r e s :
A 6 h p t e pobabLLLty ~ t ~ ~ u c t w i sr ea s e t lK o f r e l a t i o n a l systems w i t h a common s i m i l a r i t y type, say 6 = < < m i : i E I > , J > , t h a t s a t i s f i e s the f o l lowing conditions:
.
(i) For each d,g€ lK, CR 1 J = $ I J (ii) L e t K be the l a r g e s t subset o f I U J
dl K
=$I
such t h a t
such t h a t f o r any
a,&€ K,
K ; then i f there i s a formula w i t h one f r e e v a r i a b l e
6
there i s a j E
J such t h a t b
$I
161 , then
i s the only element o f A w i t h C n l K I= @ = a
j * I t i s c l e a r t h a t c o n d i t i o n (i) i m p l i e s t h a t a l l systems i n
JK have the to
same universe and t h e same d i s t i n g u i s h e d elements. I t i s a l s o easy
see
t h a t there i s a l a r g e s t s e t K w i t h t h e p r o p e r t i e s r e q u i r e d f o r (ii). Cnl K w i l l be c a l l e d the common paht 06 lK; f o r f u r t h e r reference we s h a l l desig-
B E lK , By
=
06
-
dl ( I U J) K ;- a n d , f o r any {CRY : G! E B). Condition (ii) expresses the f a c t t h a t any de-
nate by CRY ( t h e vahiable paht
Cn)
f i n a b l e element i n the common p a r t o f lK i s designated by an i n d i v i d u a l constant. I t i s t o be remarked t h a t the only c o n d i t i o n e s s e n t i a l f o r the r e s t
our work i s t h a t a l l systems i n lK have a common universe.
The
other
of re-
quirements seem t o be n a t u r a l , b u t they are n o t indispensable. I n the f i r s t place, i t would n o t be n a t u r a l t o have as one o f our p o s s i b l e choices designation o f an element o f t h e universe, Hence, the i n d i v i d u a l
the
constants
should have an i n v a r i a n t i n t e r p r e t a t i o n i n a l l t h e systems. The naturalness o f t h e second c o n d i t i o n w i l l be c l e a r from the d e f i n i t i o n o f the transformations t h a t preserve the "laws o f t h e phenomenon". This ghoup
06 &mmtiom GK,
contains a l l permutatiors
6
group
o f the
of
common
148
ROLANDD B. CHUAQUI
universe A t h a t satisf.y simultaneously the f o l l o w i n g c o n d i t i o n s :
(1)
For any 1 7 7 ,= < A,Ri,
a ,> j
jsj
(a) < A , R i $ a j > i E I ,
< A , R ~ t ~ - ' ( a j ) >i (b)
(2)
< A , R i p d(aj)>LE 1,
J
5
j E j= < A , R i , b j > L E I , j € J
'
f o r some permutation g o f J
5
s * ( u Z y ) and d-'*(Ca,,)
E
My.
I f B = ModK($)
E
er, jeJ
If < A , R i , a j > i E I ,
b. = n f g(j)
(c)
L E I , j E J E IK we have:
then t h e r e are sentences $
,0
, then
6(bj)>iEI, j E J
, Cy
f o r some sentence $
=
By6
such t h a t
where
*
anti
1
,
Oy = Z$
and
C = ModK($)
P = ModK(8).
I t h i n k t h a t because o f requirements ( a ) , ( b ) above, c o n d i t i o n (ii) i n t h e d e f i n i t i o n o f a simple p r o b a b i l i t y s t r u c t u r e i s n a t u r a l . I t i s easy t o prove t h a t GK i s always a group o f transformations.
We need t o d e f i n e a r e l a t i o n o f symmehg between s u b s e t s o f M o f t h e form M o d K , x ( $ ) . We can do i t f o r any subsets B
G B
+
By6
,C
C
i f and only i f
C
i f and o n l y i f t h e r e i s an
Cy =
and $ / I -
@/I-
2
6
E GK
S*
/
L e t us now consider t h e f i e l d s o f subsets o f
form M o d K ( $ ) f o r $ i n S*
1W
or
family o f sets o f the form
x an assignment.
M
W1W
have
K, x
M'
W1W
on S*
WlW
/ I-
/I-
C.
1. We now
M:
ModK($) = I b d K ( $ ) .
M : M, Mwlw, M' , sets
Bl' ,
( $ ) f o r $ a formula i n L* o r L*
and of
the
have
the
Wl W
, and
a r e countably complete.
Suppose we have a measure p defined on p r o b a b i l i t y measure PK
B
l.lodK($).
, then
r e s p e c t i v e l y ; and
W1
and
-
W1W
as universes the f a m i l y o f
S'
Mod
I- ( o r S*
ModK($)
This i s w e l l - d e f i n e d as i f $ i s e q u i v a l e n t t o $
MAlw ; where M and M
such t h a t
a r e ~QUULQ fihiheeq in
i f and o n l y i f
$/I-
-6
;
Events a r e considered t o be elements o f d e f i n e when two cvents $ / I -
M by:
of
by:
M
Wl W
.
Then we
can define
a
SEMANTICAL DEFINITION OF PROBABILITY
PK(@/I- 1
u ( M o d K ( @ ))
=
149
*
As there i s no danger o f confusion we may w r i t e :
%$for
ii,
$/I-
$/I-
, and P,(@) f o r
PK(@/k).
S* /I- , then we o n l y need, because o f Theorem
I f we want a measure on
M. M'
2.1, a f i n i t e l y a d d i t i v e measure on I f we have a measure defined on
,
WIW
we can extend the
probability
measure t o a language t h a t has a new i n d i v i d u a l constant .ta f o r every e l e ment
o f the common universe A . We c a l l t h i s language
a
i t s s e t o f sentences
s6
W10
( T K ) . We extend PK
to
S:lW(TK)
L*
(TK)
W1W
/I-.
Let
and @ 8
be a l l t h e constants from T K t h a t appear i n n- 1 by new 4; l e t ii, be the formula obtained from $ r e p l a c i n g .ta ,...,-t o 'n-1 ~ l e t x be an assignment such t h a t variables v ~ , . . . , v ~ ;- and x ( v i ) = ai f o r a l l i < k. We define:
Sb ( T K ) and l e t t, ,...,fa 0
W1W
PK($
/I-
1
=
V ( M ~ ~ K , ~ ( 1J I* )
S i m i l a r considerations can be a p p l i e d t o
S*/I-
and
M'
.
I n order t o have an adequate p r o b a b i l i t y measure, i t must be
ci(
under
. That
If @ Thus,
u
invariant
is: ii,
, then
PK(@) = P K ( i i , ) .
a l s o has t o be i n v a r i a n t under +
, i . e . f o r B , C i n one o f our
f i e l d s o f sets, we must have: If
8
-
C,
then
p(B) = p(C).
Our problem, then, i s t o f i n d a measure p i n one o f t h e f i e l d s or
M'
W1W
t h a t has t h i s p r o p e r t y . I s h a l l use
M ,M'.MwlU,
IF= CF, U , n , - , 0 , lK>
for
any o f these f i e l d s .
I now reformulate these ideas i n order t o have a more perspicuous mathematical problem.We consider t h e f i e l d o f subsets o f E$,JFy=
-
given by,
Fy
=
{By: 8 E F}.
Fy i s obviously isomorphic t o IF. Thus i f we d e f i n e a measure ? on IF by,
we can o b t a i n p on
~ ( 3 =) !(By).
Fv
150
ROLAND0 6 . CHUAQUI
L e t G; =
{6*: 6
€ GK}
the isomorphic image o f formations o f
.
IF,,
The group G;
B
-GK*
, where A* by 6,
a,,
a p p l i e d on an element
determines an equivalence r e l a t i o n C
i f and o n l y i f
t i c e t h a t if 6 € G; Also we have f o r €3,
C €
F,
B-
C
Thus, what we need i s a measure if
-
8
-Gt
C, then
=
-Gt
of
M y gives
Fv by:
On
6
f o r some
d*(B) = C
, then 5
a,,
i s , then,a group o f t r a n s -
G;
6*((R,,).
(no-
€ G;
g* f o r some g € GK and j * ( B ) = B g ) .
i f and o n l y i f on
L(B) = v(C),
IF,,
such
B,,
-Q
.
C,,
that:
for all B ,
C €
F,,
.
i s what i s c a l l e d a measure on IF,, imatLiant undm t h e g m u p 06 ,ttam 60hmatiom G; (see Tarski 1949, p. 229). We s h a l l discuss i n the n e x t s e c -
p
t i o n necessary and s u f f i c i e n t c o n d i t i o n s f o r the existence o f such measures.
4, EXISTENCE OF
MEASURES,
Our task now i s t o f i n d a m e a s u r e on t h e f i e l d
U
of
IF,,
sets
, n , - , 0 , M y > i n v a r i a n t under the group o f transformations
Necessary and s u f f i c i e n t c o n d i t i o n s f o r t h e existence o f f i n i t e l y
measures i n v a r i a n t under a group o f f u n c t i o n s are given i n Tarski 1949. 231. Hence I s h a l l concentrate on (countably a d d i t i v e ) measures. To discuss t h i s k i n d o f measures i t i s convenient t o have complete f i e l d o f sets. I f our language i s
L*
w1
(i.e.
a
=
.
G; additive
p.
countably
i t contains countable
IF,, i s countably complete. But i f our l a n L* ( f i n i t a r y ) we work w i t h t h e s m a l l e s t countably complete f i e l d o f subsets o f M y generated by IF,, . conjunctions and d i s j u n c t i o n s )
guage i s
Thus, the mathematical problem, i n a l l i t s g e n e r a l i t y , i s t o measure p on a countably complete f i e l d o f sets
IF
=
-
obtain
a
,0 ,X>,in-
v a r i a n t under a group o f permutations G o f the u n i t s e t X ;namely p should satisfy:
(*)
if A, B €
F
and
A = 6*B
f o r some
6
€
G, then
p(A) = ~ ( 8 ) .
A measure p t h a t s a t i s f i e s ( * ) i s c a l l e d G - i n w a h i a n t . A l l r e s u l t s about these measures presented i n t h i s s e c t i o n are a l s o v a l i d f o r quasi -groups i n s t e a d o f groups (see Tarski 1949, p. 142). I n order t o solve t h i s problem o f existence o f measures, I s h a l l
prove
151
SEMANTJCAL DEFINITION OF PROBABILITY
B i s a BA and R an B t h a t i s R - invahiant,
some theorems t h a t apply t o a r b i t r a r y B A ' s . Thus, i f equivalence r e l a t i o n on B we need a measure
on
i.e., a, b E B
2
aRb
and
u(a) = u(b).
imply
To f i t our problem i n t h i s framework, we define the equivalence r e l a t i o n on
IF by: A
i f and o n l y i f t h e r e i s an
B
By ( * ) our measure 1-1 must be
6
8 G such t h a t 6*A = B .
%-invariant. I f
u
is
-G-invariant,
then i t a l s o has t o be i n v a r i a n t under the f o l l o w i n g equivalence r e l a t i o n : A
dG
i f and o n l y i f t h e r e are Y,Z E F'
B
6 = C.
L < U
A measure
u
Z.
L'
is
Yi +G Zi
and
>-invariant
such t h a t A = C i < o
i
f o r every
Yi,
< w.
i f and only i f i t i s
zG- i n v a r i a n t .
This equivalence r e l a t i o n cG has t h e f o l l o w i n g p r o p e r t i e s :
(i)
a cangkuence & W o n 06 IF
is
=G
i f A , 8 , C,
(a)
i f Y , Z E wF
(b)
all
(ii)
D
A t B
then
i
E j E
€
F with
ei;C
A
< F , t , Z>, namely,
n B=0=C n D
and A
+ D,
n
w i t h Yi
Y . = 9, = Zi
then
U,
=
j
Yi
ZG
n Z J. and Yi C i < U Zi .
i s ( ~ i n i t c L g ke&inLng, ) namely, i f A , Y o , Y,, A = Yo + Y, , then there are Z o , Z1 E F
LG
and
B = Zo
+
,
Z,
and Y,
Yo eG Z o ,
=G Z1
6C,
.
CG
B cGD, for
Zi
B E F, A cG B such t h a t
(For a discussion and p r o o f o f these p r o p e r t i e s see Tarski 1949, pp.
144,
145, 222). There are some elements o f t h a t an element elements o f
F, Y
A € F €
F t h a t must have measure zero.
L e t us say
i s G - n e g l i g i b l e i f t h e r e i s a sequence o f d i s j o i n t
wF, such t h a t A
zG Yi
f o r every
be i n v a r i a n t a l s o f o r the equivalence r e l a t i o n
i<
w. Now,
u
has t o
5G defined by:
sG B i f and o n l y i f t h e r e are G - n e g l i g i b l e e l e m e n t s o f F , C , D , C ' D' and elements A' , €3' E F such t h a t A ' C G 8 ' , A u C = A' U C ' , and B U D = B ' u D' . i s , again,a r e f i n i n g
A
congruence r e l a t i o n o f
t.
So, f i n a l l y , we get: 1-1 i s G - i n v a r i a n t i f and o n l y i f 11 i s
zG
sG- i n v a r i a n t .
As we r e q u i r e the measure p t o have p r o p e r t y ( 3 ) i .e.
u( X
) = 1
,
then
152
ROLAND0 B. CHUAQUI
f o r an i n v a r i a n t measure t o e x i s t , X there should n o t e x i s t d i s j o i n t s e t s
3
should n o t be G - n e g l i g i b l e , t h a t i s A , I: € F such t h a t X = A u B and
XzG A B ( i t i s easy t o see t h a t t h i s c o n d i t i o n i s e q u i v a l e n t f o r X t o t h e d e f i n i t i o n o f n e g l i g i b l e given p r e v i o u s l y ) .
The s e t o f n e g l i g i b l e e l e m e n t s I i s a countably complete i d e a l i n lF;thus,
IF / I
i s again a countably complete
BA.
From t h e r e l a t i o n
sG i t
i s pos-
s i b l e t o o b t a i n a corresponding equivalence r e l a t i o n on F / I defined by: A / I sG B / I
i f and o n l y i f
ASG
B.
This new r e l a t i o n sG i s now a r e f i n i n g , congruence r e l a t i o n on (lF/7), which has t h e a d d i t i o n a l p r o p e r t y (iii)i f and o n l y i f X i s n o t G - n e g l i g i b l e :
(iii)sG i s n-OLLctRy p o n X u e , i.e. i f a i s a nonzero element o f F / 7 , then there i s no sequence o f d i s j o i n t elements x € w ( F / I ) such t h a t a sG xi f o r every i < w . For a p r o o f o f these f a c t s see Chuaqui 1977. Thus, we are l e f t w i t h the general problem o f t h e existence o f ameasure
on BA B t h a t i s R
- invariant
under a s t r i c t l y p o s i t i v e , r e f i n i n g , congru
ence r e l a t i o n R on
i. In
t i o n , r e f i n i n g , and
s t r i c t l y p o s i t i v e f o r a r b i t r a r y equivalence
-
what f o l l o w s we use the terms congruence r e l a relations
R on BA's. We s h a l l f i r s t discuss t h i s problem when
B i s a finite
BA.
Although
the general s o l u t i o n given l a t e r includes t h i s case, f i n i t e BA's w i l l able us t o o b t a i n some examples o f measures and see t h e i r
en-
characteristics.
For the f i n i t e case, c o n d i t i o n s (i)and (iii)f o r the equivalence
relation
may be s i m p l i f i e d :
(i') R i s a congruence r e l a t i o n o f
B i s f i n i t e there always e x i s t s a s t r i c t l y p o s i t i v e measure i n -
v a r i a n t under a s t r i c t l y p o s i t i v e , r e f i n i n g , congruence r e l a t i o n on B.Moreover, the measure p might have the a d d i t i o n a l p r o p e r t y : if p(a) = ~ ( 6 ) then a R b .
Before p r o v i n g ' t h i s , we o b t a i n t h e f o l l o w i n g :
LEMMA 4.1. 16 B 0 a &&Lte BA and R n n-OLLctey p o n X u e , hedining conghuwce heeation on i , then: aRb impLieies &at t h a e ahe n < w and oequences oh atom6 06 B,
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
153
PROOF : Suppose a = C i
l a t i o n un
SUppUoC
6; nuppooc
B 0 a 6inL.t~8 A and R a hedining congauence hethat a . / R,
@&hen
. .. ,an-,
/R
uhe t h e
didduent
equivalence ctabnen 06 atomo that ahc not equal t o O l R , and mi t h e numbeh &mh i n a,/%!. Lot po,...,pn-l be any hequence 0 6 nonnegative he& numbehn nuch t h a R Xi nmi * 1 2 =~ 1 . Then t h e m 0 an R - inuahiant meabuhe
06
nuch t h a t (*)
43
p(ai)
=
i
doh
pi
< n
and o d y if not OR1. MoheoUeh, not OR1, t h e mcabwLc ndti46ying (*) 0 u L q u e .
PROOF:
Theorem 4.2 i s obtained immediately from Lemma 4.1 and properties
of B A ' s . THEOREM 4 . 3 . Suppone B 0 a 6inL.te BA and R a he6ining conghuence &e&-
.tion on k. Then a necennw~yand hu66ioien-t condition a b.thicteg poo&ue meabuhe on B huch t h a t (+)
ifi and on.k?g id
p(a) = y(b)
.LA that R be b.thi&y
604
t h e e xhte nc e
06
aRb
pohL~%ue.
Suppose R i s s t r i c t l y positive and l e t U ~ / R , . . . , U , - ~ / R be the different equivalence classes of atoms. Suppose, furthe r, tha t ai/Rcontains mI. atoms f o r every i < n. Define p ' recursively on the atoms of B by:
PROOF:
if
b E ao/R,
then
if
b E a /%, P +I Obtain p on the atoms by, u ( b ) = v ' ( b ) / (xi
p'(b)
then
=
1,
p ' ( b ) = Ei
p'(ai).
- u'(ai)).
Then extend p t o B by Theorem 4.2. I t i s n o t d i f f i c u l t t o prove, using Lemma 4.1 and simple properties of natural numbers, t h a t y has the desired
154
ROLAND0 B. CHUAQUI
properties. The converse i m p l i c a t i o n i s obvious.
I n a sense, t h e measure p d e f i n e d above i s q u i t e u n n a t u r a l as a probab i l i t y measure. I f a and b a r e atoms t h a t a r e n o t e q u i v a l e n t , t h e n
p(a)
p ( 6 ) and we may choose p ( a ) > p ( b ) o r p ( 6 ) > p ( a ) .
may be v e r y d i f f e r e n t f r o m
To a v o i d t h i s problem we need t o impose a s t r o n g e r r e q u i r e m e n t t h a n i n v a r i ance on t h e measure p :
We say t h a t a measure p i s n f i o n g l y R-inuatLiant i f f o r any
a, 6 E B
we have, c € 8 such t h a t aRc and c
p ( a ) 5 p ( b ) if and o n l y if t h e r e i s a
5 6.
Here, t h e second 5 i s t h e usual p a r t i a l o r d e r i n g i n B .Me w r i t e ' a 5 b ' f o r 'there i s a
c
e 8 such t h a t aRc and c 5 b ' . We have t h e f o l l o w i n g :
R
THEOREM 4 . 4 . Let B be a BA and R a 4.thiotey 13un&ue, 4 e 6 i n i n g , c o n g u e n c e helation on 6 . Then t h e m 0 a n f i o n g L y R - invahiant memute on B id and ovtey is 6 0 4 m y atom4 a , 6 E B, we have aRb. hloheoue4, id
t k i b kkmt condition 0 6LLe6.ieeed t h e 4 f i o n g L y R - inuahiant meanuhe
0
unique.
PROOF:
By Lemma 4.1 i t i s c l e a r t h a t t h e measure d e f i n e d i n t h e p r o o f
of
4.2 i s s t r o n g l y i n v a r i a n t i f a l l atoms a r e e q u i v a l e n t ( i t i s enough,in t h i s case, t o d e f i n e
p(a) = 1 / m
, where tn
i s t h e number o f atoms,for any atom
a )* Suppose, now, t h a t p i s s t r o n g l y R - i n v a r i a n t , and with
p ( a ) -< ~ ( b ) Then. . aKc 5 b
a,b
are
atoms
f o r some c . B u t c has t o be an atom and,
so, n e c e s s i t y f o l l o w s . I t i s easy t o see t h a t t h i s measure i s R
- invariant
even
if
not
all
atoms a r e e q u i v a l e n t , though i n t h i s case i t i s n e i t h e r s t r o n g l y R - i n v a r i a n t n o r s a t i s f i e s (+) o f 4.3.
Notice, also, t h a t the condition
l e n c e f o r a l l atoms c o u l d be r e p l a c e d by
'2
of
equiva-
i s a simple ordering',
i.e.
' ~ 5 o ~r b6 5 R a f o r a l l a , b E B ' . L e t us now c o n s i d e r t h e g e n e r a l case. IF i s a c o u n t a b l y complete of subsets o f a s e t X , G
a group o f t r a n s f o r m a t i o n s g f
IF, and I
field
a subset
F . ble want a measure p t h a t i s G - i n v a r i a n t and t h a t vanishes e x a c t l y . ble c a l l measures t h a t s a t i s f y t h i s l a s t r e q u i r e m e n t I - p o n h i u e ( i . e . p i s I - p o s i t i v e i f , p ( A ) = 0 i f and o n l y i f A E I ) . I n
of
on t h e elements o f I
o r d e r t o f o r m u l a t e necessary and s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f such measures we s h a l l use t h e requirements o b t a i n e d i n K e l l e y 1959,for t h e
155
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
existence o f a s t r i c t l y p o s i t i v e measure on a Boolean Algebra. We need some d e f i n i t i o n s , which apply t o any countably complete
x
Let
=
<xi
:
i < n> be a f i n i t e sequence i n
where tn i s t h e l a r g e s t i n t e g e r k 5 n
io < i, < a6
... < ik-l<
n. Then, i f A
such t h a t
c_
B:
BA
8 . We d e f i n e
... A xi
xi
i(x) =tn / n # 0 for
0 k- 1 B we d e f i n e the i n t m e d o n numbetl
A, i n symbols i ( A ) by, i(A) = in6{i(a)
We say t h a t
B has the K&ey
:
a E nA
doh home n <
phopehty i f B - { O }
W}
.
i s a countable union o f
sets w i t h p o s i t i v e i n t e r s e c t i o n number. B i s weakey countably dint&blltive i f f o r every double sequence x € W x W 8such t h a t xi .+ 5 x.. f o r every rJ 1 .cj i , j < w , we have
V{ A {x..
: j E LJ We have the f o l l o w i n g :
W}
: i € a} =
:i A I V I x i,$I(i)
8
W}
: $I E Wu}.
let IF be a countably complete 6ield 06 nubneh5 06 a s e t F, and 1 a hubhet 06 F . Then t h e doeloluing conditionh me necunnatry and j o i n f l y nu6~icientdoh t h e exhtence 06 a countably a d d i t i v e , G - inv&nt, and 1 - p O b i t i V e tneanuhe on IF:
THEOREM 4 . 5 . X , G a ghoup
06
R h a ~ n , 3 o m ~ o n06 b
(i) 1 0 a countably complete phopeh id& i n IF; (ii) dee G - n e g f i g i b l e h e t n belong t o 7; (iii) i6 A € 1 and A +G 8, t h e n B E I; (iv) lF / 'I has t h e K&ey phopehty and .& weakey countably din-thibu
-
five. A complete p r o o f o f t h i s theorem i s t o o long t o be included h e r e
w i l l appear elsewhere (Chuaqui 1977).
and
I s h a l l g i v e o n l y a b r i e f sketch.The
necessity o f t h e c o n d i t i o n s i s c l e a r , given K e l l e y ' s necessary
conditions
f o r the existence o f a s t r i c t l y p o s i t i v e measure on a BA (see K e l l e y
1959,
and Horn and Tarski 1948). I n order t o c o n s t r u c t the measure 11 w i t h t h e r e q u i r e d p r o p e r t i e s proceed as f o l l o w s : We s t a r t w i t h the d i s j u n c t i v e BA B the r e f i n i n g congruence r e l a t i o n R on
, where B = IF/ 1 6 defined by:
aRc i f and o n l y i f t h e r e are A , C 8 F and A c
G
C;
we o b t a i n a new GCA
a =B / R
.6
i s a GCA.Using
such t h a t a = A / l ,
(see Tarski 1949, Th. 6.10).
we
c = C / I, We close
156
ROLAND0 B. CHUAQUI
to
d ( T a r s k i 1949, Ch. 7 ) . Using t h e p r o p e r t i e s o f B assumed by t h e theod = < A , +, C > :
rem, i t i s p o s s i b l e t o p r o v e t h e f o l l o w i n g f o r (1)
i f a , b € A, t h e n
(2)
i f a € A,
then
a A b exists, a 5 m h, where
h = (X/I)/R
i s finite.
, By a theorem o f F i l l m o r e 1965, i t i s p o s s i b l e t o p r o v e t h a t d i s i s o morphic t o a subalgebra o f < C ( S ( I D ) , R ) , + , C > where D i s t h e BA o f i d e m m u l t i p l e elements o f
d . From c o n d i t i o n (iv) on B , we deduce t h a t
ID
a l s o has t h e p r o p e r t i e s i n c l u d e d i n t h a t c o n d i t i o n . Hence, by K e l q e y ' s 1959 measure on
ID. We t r a n s f e r t h e mea-
s u r e t o t h e o p e n - c l o s e d s e t s o f t h e Stone space
S(ID), e x t e n d t h e measure
there i s a
strictly
positive
t o t h e c o u n t a b l y complete f i e l d o f s e t s generated by them, and
define
the
11 on t h e measurable f u n c t i o n s . II i s d e f i n e d on a l l o f C(S(D),Z). T r a n s f e r back, now, II t h r o u g h t h e isomorphism t o 6i and then, u s i n g 16.11 o f T a r s k i 1949, t o IF. T h i s measure on IF has t h e d e s i r e d p r o p e r t i e s . integral
The n o t i o n o f a s t r i c t l y p o s i t i v e measure s h o u l d be m o d i f i e d when want t h e measure G - i n v a r i a n t . A measure t h a t i s G - i n v a r i a n t must on a l l G - n e g l i g i b l e s e t s . B y a r e s u l t o f B r a d f o r d 1971, p. 194, we
we
vanish can
see
t h a t most group o f t r a n s f o r m a t i o n s o f i n t e r e s t produce non - empty G-negli
-
g i b l e s e t s . T h i s i s t h e c a s e , f o r i n s t a n c e , when f o r e v e r y n E w t h e r e a r e d i s Yo,... such t h a t X = CiCnYi and Yi z' Y . f o r e v e r y i , j < n. ,Yn-l G j To t a k e i n t o account t h i s f a c t we i n t r o d u c e t h e f o l l o w i n g n o t i o n :
joint
We say t h a t a measure
A €
F, Let
p(A) = 0
u
on
IF
positive i f for
i s G- strictly
any
implies t h a t A i s G - negligible.
NG be t h e s e t o f G - n e g l i g i b l e elements. I n Chuaqui 1977,
proved t h a t NG i s a c o u n t a b l y complete i d e a l i n
IF.
it i s
Thus, f r o m 4.5.we
get:
THEOREM 4.6. Le* IF be a c o u n t a b l e coi~iple*e d i e l d 06 oubnetn od a b e i X, and G a ghoup 0 6 Ltanhdornunationh 06 IF Then t h e d o U o w h g conditionh
.
and j o i n t l y nuddicient 604 t h e exbintence 06 a countably addiLive, G - &vatLiant, and G - 6.thicLLy pob&iUe rneame on IF: (i) X f NG (ii) IF / NG h a .the K&ey phopehty and b weahey c o u n t a b l y d i n M b u Live. I t s h o u l d be p o i n t e d o u t t h a t (i) i s n o t enough f o r t h e e x i s t e n c e o f a G-in-
ant
MtCedAdhY
v a r i a n t measure. F o r a couterexample see Chuaqui 1973. S i m i l a r l y as f o r f i n i t e B A ' s , t h e measures o b t a i n e d by
these
theorems
a r e n o t , i n g e n e r a l , unique. We need t h e c o n d i t i o n o f s t r o n g i n v a r i a n c e . measure
p on
IF
i s strongly G
- invariant
i f f o r every
A
,
B
€
F
A
with
157
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
A,B
1 NG , we have: u(A) ~ p ( 8 i) f and only i f there i s a C 8 F such t h a t A gGCcB.
The f o l l o w i n g theorem
can be obtained from 2.10 o f Chuaqui 1969:
THEOREM 4 . 7 . Lct B be a countably complcte dield 06 nubseZ5 0 6 a n c t X, and G a ghoup 06 .tham~omnatiom06 B . Then, doh t h e e x h t e n c e o d a counta b 4 a d d i t i v e , n.thongly G - i n v a h i a n t , and G - n&k,t4?y p o n U v e meawe p on F , t h e 6oUowing c o n d i t i o m me n e c e s o m y and nu6@,&2nt: (i) X B NG (ii) doh any
, A, B E
F,
$hue 0 a C E
F,
nuch t h a t
2
A
B S G C ~ A . Mokeoveh, undeh t h e s e conditioMd t h e
4.7.
oh
C 5B
meauhe p 0 unique. Condition ( i i ) i n
can be replaced by:
n~ztih~ies t h e countable chain c o n d i t i o n ( i . e . e v a y n e t 0 6 dinjoint elements 0 6 F 1 NG 0 at mast c o u n t a b l e ) , and 608 any a E F / NG , a # 0 , t h u e 0 a sequence 06 d i n j o i n t e l e m e n t s x 8 wF 1 NG and a 6equence y E wF 1 NG buch t h a t X / NG = Ci < wxi and xi sG yi 5 a 6ok e v m y i < w .
(ii F'/) NG
There a r e several o t h e r a l t e r n a t i v e s t o (ii'), t h a t are
obtained, j u s t
as i s (ii'), from p r o p e r t i e s of simple c a r d i n a l algebras (see Tarski p.
1949,
1 1 7 - 120 and Chuaqui 1 9 7 + ) . The main problem t h a t needs f u r t h e r study i s t h e r e l a t i o n
l o g i c a l theory and t h e measure. I b e l i e v e t h a t the methods
between
the
developed
in
S c o t t and Krauss 1966, might be useful i n t h i s connection, because once
we
have a measure p on
F we can p u t some o f our simple p r o b a b i l i t y s t r u c t u r e s
i n the framework o f t h e i r p r o b a b i l i t y systems w i t h s t r i c t i d e n t i t y . Unfortunately, t h e c o n d i t i o n on t h e i r p r o b a b i l i t y systems t h a t requires the measure on the BA t o be s t r i c t l y p o s i t i v e , excludes some o f the mostnat u r a l simple p r o b a b i l i t y s t r u c t u r e s from t h e i r framework. For instance, l e t
us take the simple p r o b a b i l i t y s t r u c t u r e K formed by r e l a t i o n a l systems le = < C , + , a , U > a E C,where i C , + > i s a c i r c l e i n t h e planeandOchooses 0 i s t h e constant p a r t o f M . G K contains one element o f C.
ROLAND0 B. CHUAQUI
158
PK ( 3 v
a(,)) = 1
-
and nupFEC( w ) PK(Vue
O(ta))= 0,
C ( w ) i s t h e s e t o f a l l f i n i t e subscts o f
where
s e n t i n g 0, and ta
5, COMPOUND
C
, 3 i s the symbol repre-
the i n d i v i d u a l constant t h a t denotes
a
.
PROBAB I LITY STRUCTURES.
I n t h i s s e c t i o n I s h a l l f o r m a l i z e the s t r u c t u r e s t h a t appear when t h e r e are sequences o f outcomes, as i n the u r n example o f s e c t i o n one. The o f succesion
o f outcomes plays a fundamental r o l e i n these cases.
order In
urn model, there were o n l y two succesive choices, so t h a t the b a s i c was the system < {O
, 11, 5
P
=
IT
:
>. I n the general case, we may have any a r b i t r a r y
. For t h i s system, l e t z = i n : n < t3 t E T I , where n < t means n 5 t and n # t. The p o s s i b l e ‘IT= < T , 5 >
p a r t i a l l y ordered system and
the
order
succesive outcomes are determined by a f u n c t i o n
M w i t h t h e f o l l o w i n g prop-
erties :
Do K = { b : i s a function, Do6 E P, and f o r each R: e Dad, d ( 4 E “ 5 13) (ii) For each E Do M , M i s a simple p r o b a b i l i t y s t r u c t u r e
(i)
6
n = <<m4: i
s i m i l a r i t y type
e J, z
ements < a4 : j j
(1
nI d g (iv) I f j 8 J h n J (iii) I f i 8 I
9
If
6
1
.
€ 1 z
d
,J
d
>
, and
distinguished e l -
for
6,
g E Do
M, then
mk =
for
6,
g E Do
M, then
a4 = u? J
with
m? j
. .
Do M , M6 i s the s e t o f p o s s i b l e outcomes provided t h a t the
E
IK d i2
should be i n
6
SUC-
6
E DoK , t h e s e t o f p o s s i b l e cutcomes determined by the preced-
Lesive choices given by
a r e r e a l i z e d . The choice a t t o f an
i n g choices. Requirements (iii) and (iv) are c o n d i t i o n s o f coherence:
the
same index should represent the same symbol, and thus the corresponding r e should
l a t i o n should have the same a r i t y ; a p a r t i c u l a r i n d i v i d u a l constant always represent t h e same i n d i v i d u a l .
A compound p r o b a b i l i t y s t r u c t u r e i s , then, a p a i r <‘IT, M >
with the
p r o p e r t i e s s p e c i f i e d above. The s e t o f p o s s i b l e (compound) outcomes set
H,,.,
,
K>
‘
=
IF : f
Intuitively,
i s the
defined by: E
U
QoM-l
and f o r each t € T, F ( t ) E KFl
.
U represents the order o f succesive outcomes ( i n general,
159
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
i t w i l l be time) and each f u n c t i o n
H,
F €
,a
T, K >
sequence o f
such t h a t each outcome depends on a l l preceding ones. For each F E H
,F ( t )
i s a r e l a t i o n a l system i n K F J z
outcomes,
. The
uni-
F ( t ) i s denoted 1 F ( t ) ( . The universe o f F i n symbols 1 F ( , i s the union o f I F(t)l f o r t E T . The universe o f < 71, K>, i n symbols IIKl,
verse o f
%
i s the union o f
IF 1
F E PI
for
.
The languages we have t o consider f o r these compound p r o b a b i l i t y s t r u c tures must be more complicated than f i r s t - o r d e r languages. We must be able to talk
about f i n i t e sequences o f outcomes. In order t o do t h i s ,
f i n i t e sequences o f elements o f T
. Thus,
our language has f o u r
we
need
types
of
VL4.VLl={uo,v1, . . . I contains the v a r i a b l e s used t o r e f e r t o o b j e c t s i n I K ;1 V12 = {to,tl, ...I
vatLiabLes given by t h e denumerable sets VLl,VL,,V13,and
r e f e r t o e l e m e n t s o f T ; VL3 = {no,fi,,,...l r e f e r t o n a t u r a l
VL,, = C x o , x ,
%)
of
,... 1
numbers ;
and
r e f e r t o f i n i t e sequences o f elements o f T (i.e.elements
. The L o g i d
corn&&
, V,
Y ) the t e r beand T , which stand f o r
i n c l u d e (besides 1 , A
=
nary r e l a t i o n s S and P I which stand f o r a d d i t i o n and m u l t i p l i c a t i o n tween n a t u r a l numbers; the i n d i v i d u a l constants
a
L , which stands f o r 5 between elements o f T ; and the t e r n a r y r e l a t i o n V a l , such t h a t V a l ( x , n ,t) i s i n the numbers 0 and 1 ; the b i n a r y r e l a t i o n
t e r p r e t e d as: t h e n a t u r a l number n i s i n t h e domain o f t h e sequence x and x evaluated a t n i s t. The 6 h d k t L t g .type f o r a language c o r r e s p o n d i n g t o
i E I > , J>, where I = U {J : 6 € D O M I . 6 For each i E I , we have an
6 = <<mi:
J
=
U
{ I d : d E UoK}
mi
+ 1 - ary
, mi
= m$
predicate
< for
T ,K>
is
i E I and
6
Pi ,and f o r each j EJ,
an i n d i v i d u a l constant c . . The s e t o f i n d i u i d d con,5mh i s denoted CN, J Ul, U CN i s denoted IS1 , and VL3 U IU,'T) i s denoted IS3. The n o t i o n o f dtomic
born&
i s defined by:
(a)
I f a , B € IS1 u VL2 U I S 3 U V 1 4
(b)
I f t E V L , and ao,.
. . ,a mi-l
, then a
E IS1
= 8 i s an atomic formula.
then P . t a o ...am,-l i s an I
atomic formula. (c) (d)
,p E
If n , m
IS3
, then Smp
and Pmp are atomic formulas.
I f x € V L 4 , n € I S 3 , and .tEVL2, then
Val(xnt) i s an atomic
formula. (e)
If R1,
R 2 E VIZ,then L i l t 2 i s an atomic formula.
160
ROLAND0 B. CHUAQUI
For the connectives we define, as usual, -I@ , (@A $), and (4 v $)
as
formulas i f @ and J, are formulas. For s i m p l i c i t y we consider o n l y f i n i t a r y languages. We a l l o w q u a ~ 6 i c ~ over a n any type o f v a r i a b l e w i t h t h e f o l lowing r e s t r i c t i o n : Vu@ i s a formula i f and o n l y i f @ o n l y
For u E V l p and I$ a formula,
contains i n d i v i d u a l symbols i n IS1 and a t most one v a r i a b l e i n VL, o n l y appearing f r e e i n @
. For
the o t h e r types o f v a r i a b l e s t h e r e are no r e s t r i c -
tions. We d e f i n e n a d h d a o t i o n
,
F € H,T,K> Vll
u
u V13 u V14
Vl,
t E Vl,,g(f)
6orniueabin m e m b m 06 M,T,K, . For each F i s a f u n c t i o n g w i t h domain if such t h a t : i f v E VLl g ( v ) E IFI;
06
an a n n i g n i i l e n t Liz
I
if n € V13,
E T;
g(n) €
and i f
W ;
, g(x)
x E V14
EYT
.
I f g i s an assignment i n F and a a v a r i a b l e , g ( y ) stands f o r t h e assignment t h a t coincides w i t h g i n every v a r i a b l e except, p o s s i b l y , i n a where
.
i t assigns T I f a i s an i n d i v i d u a l constant o r a v a r i a b l e , we have: a F = g ( a ) , if a i s a v a r i a b l e ; aF = a . , i f a i s c.; aF = 0 , i f a i s 9 4 J J 9 U; aF = I , i f a i s i 9 I f t e T, and Pi. i s a p r e d i c a t e , we d e f i n e :
.
PI(')
= Ri,
P!()'
=
I
where
Ri i s t h e i& r e l a t i o n o f F ( t ) , i f i
E IF,T ;
0 , otherwise.
Now we can define by r e c u r s i o n g n a d h & h @ i n F, for any formula @, assignment g , and I$ [g] F E H,T,K> ; i n symbols t h i s w i l l be F
F t=
F @[g] i f and o n l y i f a = B g
1) I f @ i s
a = 6 , then
2)
If @ i s
Piaho...an7i-1
3)
I f $ i s Smnp ( o r Pmnp), then F != @
F
, then F I=@
22
5)
nF = pgb). 4 I f @ i s V d ( x n t ) , then F C @ [g] i f F F main o f x and x evaluated a t nF i s 9 9 4 I f @ i s Ltlt2,then F !=@;9] i f and
6)
If @ i s l$
4)
,
then F I=
*
[g] i f and o n l y if
i f and o n l y i f m
( o r mF
4
.
F 9
and o n l y i f nF i s i n F 9
+
F F n g=pg the
t
9'
F
F
o n l y i f tl 5 t2
4
@ rq1 i f and o n l y i f n o t
9
F I=
. $b].
do-
161
SEMANTICAL DEFII.IITlON OF PROBABILITY
If @ i s ($ A
7)
F b e Cgl
e),
F b @ r g ] i f and o n l y i f F!=$b]
then
*
F b @ [g]
I f @ i s ($ V O ) , then
8)
F
0
Cgl
*
, then
If @ i s Ww$J with
9.1)
There i s a v a r i a b l e t € V t 2 f r e e i n $ ; i n t h i s F b@ [g] i f and o n l y i f f o r every T € IF(g(.t))l
w 8 Vtl
Otherwise we define,
IF1
T €
F C @ [g]
with
and o n l y i f f o r every
t h e r e are two cases: case
we
, F I=$ Lg(,")]
i f and o n l y i f
a € Vt2 (Vt3 or T
E T (U o r
For an assignment g and a formula @ f o r every
or
for
have,
.
every
, FC$J[g(:)].
If @ i s W a$
10)
F b $ b ]
i f and o n l y i f
9)
9.2)
and
F € M,T,K>
, F !=@ [g] .
I n order t o d e f i n e
P
Vt,),
then F!=@[g]
if
3) F b $[g(:)].
,<1T , M >
I=@ [g]
i f and o n l y i f
the p r o b a b i l i t y o f a sentence @ ( o r ,
more, accurately, o f the equivalence c l a s s o f sentences l o g i c a l l y equivalent to
@) i n
of
K>
< H , K > , we have t o d e f i n e a p r o b a b i l i t y measure on t h e subsets o f the form: IAad
IF:
H
F !=@I.
and
With t h i s measure, we d e f i n e P
lH< T , K > *
For each
6
€
DoM,K
6
i s a simple p r o b a b i l i t y s t r u c t u r e and
subsets then,
if
possible, we can proceed as i n s e c t i o n t h r e e t o g e t a measure on an approp r i a t e f i e l d o f i t s subsets. This f i e l d o f subsets o f M i s obtained a s f o l -
6
lows: We take a l l formulas t h a t have j u s t v a r i a b l e s i n VL1, one v a r i a b l e a i n V t 2 , and symbols i n the s i m i l a r i t y type o f
IK
6'
O m i t the v a r i a b l e a a n d
we get a sentence @ o f a f i r s t - o r d e r language. Then t h e f i e l d o f o f the form ModK ( @ ) f o r t h i s type o f sentence i s t h e one needed.
6
subsets L e t US
c a l l t h i s f i e l d B16 and t h e corresponding measure !.I 6' I do n o t have a general s o l u t i o n f o r passing from these measures !.I
to
a measure on the subsets o f ,& ,,
in
>,,
.
It i s easy t o do i t , however,
6
two important cases:
CASE I. The order type o f finite.
r
i s f i n i t e o r w , and f o r each
6
€
Do M,K6 i s
162
ROLAND0 B. CHUAQUI
Consider
B the f a m i l y o f subsets o f M,T,K>
o f t h e form A=IF:F E
H
we F I Poj = j} f o r some j E Do IK. The f i e l d o f subsets o f M
be extended t o t h i s f i e l d . We d e f i n e
u ( A ) as the product o f t h e
numbers
u j l (Ij(t)l) ~ for
a l l R: E P a j . Proceeding as i n t h i s case, we can a p r o b a b i l i t y measure f o r the u r n model discussed i n s e c t i o n one. CASE 11.
6,
of
g E Pa
B
is arbitrary,
K, M6
=
IKg
P ,T times, i . e .
.
but simply ordered,
L e t us c a l l
for
every
IK6, P. Then H,T,K, i s t h e d i r e c t power
= TP = I F : DaF = T
P.l,T,K>
and
define
s i r e d measure i s the product measure o f t h e
u
6
and D 0 F - l c P}.
The de-
' s . By the usual considera-
t i o n s i n P r o b a b i l i t y Theory, we can o b t a i n B e r n o u i l l i ' s law
of
l a r g e nun-
bers.
6,
EXAMPLES A N D
APPLICATIONS,
We s h a l l f i r s t study a few p r o b a b i l i t y models and
a n a l y z e l a t e r how
they may be applied. If2 have already seen the model f o r choosing
a sample
from a f i n i t e population. For f u r t h e r reference, I s h a l l c a l l t h i s example, Example 1. EXAMPLE 2.
Distribution o f
b a l l s i n t o n c e l l s . There are several
h
ways
o f s e t t i n g up t h e simple p r o b a b i l i t y s t r u c t u r e s depending on the s t a t i s t i c s t o be obtained. Which o f these a p p l i e s i s determined by the evidence. 2a) Maxwell-Boltzmann s t a t i s t i c s . I n t h i s case each p a r t i t i o n ( i d e n t i f i a b l e ) b a l l s i n t o the
/r
of
the
c e l l s i s e q u a l l y l i k e l y . The s e t lK can be
considered as c o n s i s t i n g o f a l l s t r u c t u r e s Cno = < A,Po,. contains a l l p a i r s o f numbers < i , m > w i t h
i<
h
.. ,Pn-l,
U > where A
and m < n ( i . e . A = ) L X n ,
i f we adopt t h e usual s e t - t h e o r e t i c a l conventions)'; Pm c o n s i s t s o f a l l parts, w i t h m as second coordinate, and 0 i s any subset o f A t h a t i s a f u n c t i o n w i t h domain t h e s e t o f a l l numbers l e s s than h . Each elenents o f
...
A, < i , m >
represents t h e f a c t t h a t t h e b a l l
i is in
c e l l m. Po, Pn-l are t h e n c e l l s and 0 i s the p a r t i c u l a r p a r t i t i o n chosen. I t has t o be a f u n c t i o n , because each b a l l can be i n o n l y one c e l l . The group GK c o n s i s t s o f a l l permutations f u n c t i o n f o r every f u n c t i o n 0 w i t h domain n o t c o n t a i n a l l permutations o f
A.
4 .
6
o f A such t h a t
Thus, i n t h i s case
6*0 GK
is a does
2b) B'dse - E i n s t e i n s t a t i s t i c s . I n t h i s case t h e b a l l s are n o t i d e n t i f i a b l e
163
SEMANTICAL D E F I N I T I O N OF PROBABILITY
The systems
a0 i n
K , f o r t h i s s t a t i s t i c s , a r e o f the same s i m i l a r i t y type
as before, i. e . d o = < A,Po,.
.. Plz- 1 , 0 > . The
A, such t h a t i f < i , m > € 0 ,
i s a l s o the same, b u t 0 i s a subset o f every < j , m > f o r some
j < i i s also i n 0 (i.e.
for
.. Pn-1'
common p a r t < A , P o , . f o r each m <
then
m1 = i
lz,O-l*{
i < a).
GK i s n o t the group o f a l l permutations o f A , b u t contains o n l y
those
t h a t preserve t h i s p r o p e r t y o f 0. 2c) Fermi
- Dirac
s t a t i s t i c s . In t h i s case, p a r t i t i o n s may have a t
most
one b a l l i n each c e l l . The systems here are s i m i l a r t o those above, w i t h t h e c o n d i t i o n t h a t 0 should be a one-one f u n c t i o n .
EXAMPLE 3.
We s h a l l analyze now a more complicated example. We
shall
not
be able t o g i v e a d e t a i l e d a n a l y s i s because o f t h e complicated physics i n volved, b u t , I hope, the discussion w i l l be s u f f i c i e n t
for
understanding
how t o proceed. Suppose we have a c i r c u l a r r o u l e t t e w i t h i n f i n i t e l y many p o i n t s . For s i m p l i c i t y , the r o u l e t t e s t a r t s from a f i x e d p o s i t i o n
and
a
v a r i a b l e f o r c e w i t h constant d i r e c t i o n i s applied. Each outcome r e s u l t s from the a p p l i c a t i o n o f a p a r t i c u l a r f o r c e . The systems i n lK may be taken t o be o f t h e form
Cn r = < C U F , C , F ,
tc, tF,6
, a , I >a € c
where C i s the s e t o f p o i n t s i n the c i r c l e , F resents t r a n s l a t i o n s i n the c i r c l e ,
tF
,
i s the s e t o f forces,
a d d i t i o n o f forces,
6
tc rep-
i s t h e con-
tinuous f u n c t i o n t h a t associates each i n i t i a l f o r c e w i t h a f i n a l
position,
and I i s t h e s e t c o n t a i n i n g the i n i t i a l f o r c e ( I contains one elements o f
F 1. In t h i s example, i f g E GK and the d i s t i n g u i s h e d elements a r e t r a n s formed by
CJ
to
g ( a ) ( a € C), then these new elements should
s a t i s f y the
GK s a t i s f i e s c o n d i t i o n s
l ( a ) and l ( b )
same sentences as the o l d ones j i . e .
i n s e c t i o n 3). These transformations i n GK are isometries
o f the
Their e f f e c t on t h e semigroup o f forces i s more complicated. the functions g i n GK
6(x)
do n o t d i s t i n g u i s h elements x , y
= 6(y). Thus, g t r e a t s
a l i k e . As
6
i s continuous,g
of
circle.
In p a r t i c u l a r , F such t h a t
F as a c i r c l e by d e a l i n g w i t h these two p o i n t s i s a l s o an isometry o f t h i s new
"circle".
Hence, i f one takes two i n t e r v a l s A , B o f t h e same length o f t h i s " c i r c l e " o f forces, the class o f s t r u c t u r e s t h a t has 1 i n A i s e q u a l l y l i k e l y t o the class t h a t has I i n
B. Which f i n a l p o s i t i o n s a r e equally l i k e l y depends
164
ROLAND0 9. CHUAQUI
on t h e f u n c t i o n
6.
I f t h e r o u l e t t e i s balanced, we should have an
sends equal i n t e r v a l s o f forces i n t o equal i n t e r v a l s o f p o i n t s ,
d
and
that then,
equal i n t e r v a l s i n t h e c i r c l e o f p o i n t s are equiprobable. I f t h i s i s n o t the case, we might have p o i n t - i n t e r v a l s o f d i f f e r e n t l e n g t h e q u a l l y l i k e l y . When we add a v a r i a b l e i n i t i a l p o s i t i o n o r
a variable
direction
of
force, t h e a n a l y s i s i s s i m i l a r , though more complicated.
A s i m i l a r method can be a p p l i e d t o the case o f t h e throw o f a c o i n o r a How-
die, because t h e outcome depends on the f i n a l p o s i t i o n o f t h e o b j e c t .
ever, the s i t u a t i o n i n t h i s case i s even more complicated so we w i l l n o t a t tempt a d e s c r i p t i o n . I t i s u s e f u l t o n o t i c e , t h a t i n t h e case o f a t r u e c o i n o r d i e , t h e models can be s i m p l i f i e d thus:
c A , I > where A i s t h e
set
of
faces and 1 contains the one t h a t obtains. We pass now t o examples o f compound s t r u c t u r e s . We have
a l r e a d y dis-
cussed an instance o f th,e u r n model. I s h a l l c a l l i t Example 4. A l l o t h e r cases o f u r n models can be represented by s i m i l a r p r o b a b i l i t y s t r u c t u r e s . For these examples, we d e f i n e a p r o b a b i l i t y measure by Case I o f s e c t i o n 5.
EXAMPLE 5.
An example o f much i n t e r e s t i s when we have the same experiment
repeated several times. To be s p e c i f i c , l e t us take the t o s s i n g o f coin
M
times. We take
Cel = < A , C1>
I
a
true
KO w i l l be t h e s e t o f j u s t two models where A = { h , t}, C1 = {lz} and C2 = It 1
= < n , 5 >.
and U l 2 = < A , C >;
5 E Ua lK, M = K O . 5 To d e f i n e a p r o b a b i l i t y measure f o r t h i s example, we use the method d i s -
( h = heads, .t = t a i l s ) . For every cussed f o r Case I 1 i n s e c t i o n 5.
I s h a l l now discuss b r i e f l y how these p r o b a b i l i t y models can beapplied. My views here are very much i n f l u e n c e d by Lucas' views(see Lucas 1970)
es-
p e c i a l l y i n Chapter 5. When applying P r o b a b i l i t y 'co p a r t i c u l a r cases,
we
have t o d i s t i n g u i s h two f a c t o r s : t h e framework and the evidence. I n my way o f s e t t i n g up things t h e framework i s given by the p a r t i c u l a r
probability
s t r u c t u r e s chosen, and t h e evidence i s what leads us t o these s t r u c t u r e s . We s h a l l see f i r s t the case analyzed by J e f f r e y s 1961, p. 15.,discussed a l s o by Lucas 1970, p. 50.: "Suppose that
I know
that S m i t h i s a n Englishman, b u t otherwise
n o t h i n g p a r t i c u l a r a b o u t him. He i s v e r y l i k e l y , on t h a t e v i d e n c e , have a b l u e r i g h t eye. But s u p p o s e t h a t I am i n f o r m e d t h a t h i s
know to left
eye i s brown - t h e p r o b a b i l i t y i s changed c o m p l e t e l y " .
I agree w i t h Lucas t h a t
t h i s i s n o t a p r o b a b i l i t y o f asingular
165
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
p r o p o s i t i o n whose subjects i s Smith. Instead, i t i s t h e p r o b a b i l i t y of t a k i n g a sample o f one element o f a population. In t h e f i r s t case, t h e s e t lKE o f possible outcomes c o n s i s t s o f s t r u c t u r e s <E,B,S > where E i s t h e
set o f
Englishmen, 6 The b l u e - e y e d Englishmen,and S a s e t o f one element. I n the
ME
second case, i . e . when we know t h a t Smith has a brown l e f t eye, t h e s e t o f possible outcomes c o n s i s t s o f s t r u c t u r e s
€3h
i s the class
o f Englishmen w i t h a brown l e f t eye. L e t I$ be t h e sentence " t h e r e i s in S
and x i s i n
a b l u e eye i s
x
an
E " . I n the f i r s t case t h e p r o b a b i l i t y o f Smith having
PK (I$), w h i l e i n t h e second E
it i s
PK
probabi l i t i e s a r e q u i t e d i f f e r e n t .
E
(I$),a n d
t h e s e two
The case o f the throw o f a c o i n can be analyzed as f o l l o w s . From p r e v i -
ous experiences, physical laws, e t c . ... we suppose a t f i r s t t h a t t h e c o i n used i s a t r u e c o i n w i t h equal p r o b a b i l i t i e s , as explained i n E x a m p l e 3. Further evidence might l e a d us t o the conclusion t h a t t h i s i s n o t the r i g h t model and we change i t . Among the important items o f f u r t h e r
evidence i s
t h a t provided by s u c c e s s i v e t r i a l s a t throwing the coin. Here,using compound p r o b a b i l i t y models, as those i n Example 5,
we may use B e r n o u l l i ' s
theorem
I f a sequence
obtains
and estimate t h e P r o b a b i l i t y o f a c e r t a i n sequence.
t h a t would be very improbable according t o our i n i t i a l assumptions,we r e j e c t t h i s i n i t i a l model and look f o r another. My procedure terms,
w i t h t h a t appearing i n Lucas 1970 Chapter V
agrees,
i n general
which t h e reader may
consult f o r f u r t h e r d e t a i l s . The o n l y s t a t i s t i c a l methods t h a t I have n o t y e t thought are some types o f Bayesian s t a t i s t i c s .
Prima
facie,
out c a r e f u l l y
they seem n o t
t o be
j u s t i f i e d , because they i n v o l v e p r o b a b i l i t i e s o f p r o b a b i l i t y hypotheses t o gether w i t h p r o b a b i l i t i e s o f events and t h i s , i n my system, would i n v o l v e a mixture o f o b j e c t - language and metalanguage.However there might be methods o f dealing w i t h t h i s mixture. A p o s s i b i l i t y i s given by procedures
similar
t o those appearing i n S c o t t and Krauss 1966, s e c t i o n 6. I b e l i e v e t h a t the methods I have presented i n t h i s
adequate d e f i n i t i o n o f a p r o b a b i l i t y measure t a k i n g i n t o
paper provide account t h e
an main
c h a r a c t e r i s t i c s o f p r o b a b i l i t y statements.
REFERENCES, Bradford, 1971,
R.
C a h d i n d a d d i t i o n and t h e axiom 111 - 196.
0 6 c h o i c e , Ann. Math. Logic,
vol.3,
166
ROLAND0 6 . CHUAQUI
Carnap, R. 1950, Carnap, 1971,
Logical foundations of probability, Chicago U n i v e r s i t y Press.
R. and J e f f r e y s , R. Studies in inductive logic I , u n i v e r s i t y o f C a l i f o r n i a
Press,
Berkeley and Los Angelas. Chuaqui, R. 1965, A definition of probability based on equal likelihood,Ph. D . D i s s e r t a t i o n , U. o f C a l i f o r n i a , Berkeley. 1969,
Cahdind & e b m
and meautu invatLiant undm eqLLivdenceheWonb, 142, 61 - 79.
Trans. Amer. Math. Soc.,Vol. 1973,
The exintence ad an &uatLiant meaute and pahadoxicd decompab~ovls, Notices Amer. Math. Soc.,Vol. 20, A - 636, Abstract 73T-8313.
1975,
A m o d e l - theohe,ticd de&hit-ion a d ptobabLkLtq, Contributed papers,
5th I n t e r n a c i o n a l Congress o f Logic, Science
Methodology and Philosophy
of
London, Ontario, Canada, V I 7-8.
06 R / L a n b d v ~ m a t i v n b ,
1977,
Measwren invahiant undm a ghoup Pac. J. o f Math.
197t,
Simple catrdind d g e b m and ththein. appUcc&Lvnb t o inuatLiant b u t e n , To appear.
To appear
in
mea-
F i l l m o r e , P. A. 1965,
The dOnenbion t h e o t y
06
c
a
n catldind d g e b n a , Trans.Amer.
Math.
S O C . , V O ~ . 117, 2 1 - 3 6 . Henkin, 1971,
L., D. Monk and A. Tarski Cylindric algebras, Studies i n Logic, North - H o l l a n d Amsterdam.
Pub.
CO.
Horn, A. and A . Tarski 1948,
!.leasutu i n Boolean d g e b m , Trans.Amer.Math.
SOC., Vol. 64,467-497.
J e f f r e y s , H. 1961, Theory of Probability, 3rd. ed. ,Oxford U n i v e r s i t y Press, Oxford. K e i s l e r , H. J. 1971,
Model theory for infinitary logic, Studies i n Holland Publishing Co. Amsterdam.
Kelley, J. L. 1959, Meawreo on Baadean d g e b m
Pac. J. o f Math.,Vo
.
Logic., North-
9, 1165-1178.
SEMANTICAL OEFlNlTlON OF PROBABILITY
167
Lucas, J. R. 1970, The concept of probability, Oxford U n i v e r s i t y Press. Oxford. Scott, 0. and P. Krauss 1966, A b b i g n i n g pho b a b i d i t i e n t o dogicae b o t m u e a b ,
inductive logic, 3. H i n t i k k a and P. Suppes editors., Logic.,North Holland Pub. Co. Amsterdam.
-
S i k o r s k i , R. 1969,
Boolean algebras , 3rd. ed. Springer
- Verlag,
Aspects Studies
of in
-
B e r l i n Heidelberg.
Tarski, A. 1935,
D e h W a h h h e i t n b e g h i b b i n d e n 6ohmadibiehten Spmchcn, Studia Philos. (Warsaw).Vol. 1, 261 - 405 (English t r a n s l . i n Logic,Semantics and Metamathematics, Oxford U. Press, 152-278).
1949,
Cardinal Algebras , Oxford U n i v e r s i t y Press, New York.
1954,
ConttLibwtionb t o t h c t h e m y VOI.
06
mod&
I,II, Indagationes Mathematicae,
16, 572-588.
l n s t i t u t o de Matemitica Universidad C a t o l i c a de C h i l e Santiago, C h i l e .
and Departamento de M a t e m i t i c a
Universidade Estadual de Campinas Campinas. Sao Paulo, B r a z i l .
Non-Classical Logics, Model Theory and Computabi l i t:!, A.I. Arruda, N.C.A. da Costa and R. Chuaqui (eds.) 8 North-Holland Publishing Company, 1977
THE EQUIVALENCE OF SOME
AXIOMS
OF
STRONGIWIFIITY
b y L U l Z PAUL0
DE ALCNJTARA
I , I NTRODUCTI O N , As pointed o u t by Mostowski 1967 t h e e x i s t e n t i a l assumptions
known as
'axioms o f i n f i n i t y ' can be j u s t i f i e d by two general p r i n c i p l e s : (I)
P r i n c i p l e o f t r a n s i t i o n from p o t e n t i a l t o a c t u a l i n f i n i t y .
(11)
P r i n c i p l e o f existence o f s i n g u l a r sets.
An e a r l y a p p l i c a t i o n o f t h e f i r s t p r i n c i p l e i s Dedekind's argument
for
the existence o f i n f i n i t e sets. Also, t h e f i r s t p r i n c i p l e allows us t o formulate the axiom o f inaccess i b l e numbers and L g v y ' s r e f l e c t i o n schema f o r Zermelo
- Fraenkel
s e t theo-
ry. The second p r i n c i p l e allows us t o formulate s t i l l stronger axioms. L e t us suppose t h a t i n c o n s t r u c t i n g sets by means o f t h e usual set-theo r e t i c operations we o b t a i n o n l y sets w i t h a p r o p e r t y P. I f good reasons why a l l s e t s should have the p r o p e r t y P we
t h e r e a r e no
are f r e e t o a d d t o
the axioms a statement i m p l y i n g the existence o f sets w i t h o u t the
property
P.
For example we apply t h i s second p r i n c i p l e i n t h e f o r m u l a t i o n
of
the
axiom s t a t i n g the existence o f medsurable c a r d i n a l s . I n t h i s paper we study some statements r e l a t e d w i t h the p r i n c i p l e o f e x istence o f s i n g u l a r sets, which are e q u i v a l e n t t o r e f l e c t i o n p r i n c i p l e s . We work i n t h e syStem BG o f Bernays t i o n and choice. 169
-
Gddel w i t h t h e axioms o f founda-
170
L U l Z PAUL0 DE ALCANTARA
We employ the terminology and t h e n o t a t i o n o f Drake 1974.
I n t h i s s e c t i o n we given an account o f some r e s u l t s a n n o u n c e d
i n de
Alcantara 1974.
A. L6vy 1960 proposes an axiom schema (M) t o be added t o t h e Z e r m e l o
-
Fraenkel axioms which p o s t u l a t e s t h e existence o f a t l e a s t one i n a c c e s s i b l e number i n the range o f every normal f u n c t i o n d e f i n e d id.f.a.0.)
THEOREM 1 .
(M)
h equivalent t o .the 6oUow&g Achema :
Euehy nvtonal dunction d.6.a.o. wkich atre inaccennible.
THEOREM 2. (M")
a l l ordinals
(Lgvy)
(MI)
(M)
for
and proves t h e
has cYrb&Wy
h g e @xed pointn
(Lgvy)
h e q u v d e n t t o t h e conjunction 06: Evetly notonal d u n d o n d.6.a.o. h a at t e a t one heguRatl nwnbm i n .i& hange.
@ Thehe I) exht
a t r b M y
h g e inaccennible n w n b m .
LGvy's p r o o f o f theorem 2 i n v o l v e s a n o t i o n o f inaccessible number whose equivalence w i t h the usual one seems demonstrable o n l y on the assumption o f the axiom o f choice (see Shepherdson 1952). Assuming the axiom o f choice i t i s p o s s i b l e t o prove t h a t (M)is equiva-
.
l e n t t o (M")
We need a lemma:
LEMMA 1 .
(M") hpk%A: (3) Evehy namal d u c t i o n d.6.a.a.
has a2 l e a t one hegULah
6ixed
point. PROOF:
6,
5
Let
6
be a normal f u n c t i o n d.f.a.0.;
let
6'
be t h e d e r i v a t i v e o f
the normal f u n c t i o n which counts t h e f i x e d p o i n t s o f 6 . Since 6 i s d.f.a.0. then 6 ' i s a l s o d.f.a.0.. By (M") t h e r e e x i s t s
i.e.,
such
that
6'(5)
= f3
i s regular.
a
171
AXIOMS O F STRONG I N F I N I T Y
THEOREM 3 .
(M)
.iA
eqLLiude& -to (MI').
PROOF: Obviously (M) i m p l i e s (M"). I n order t o prove t h e converse we g i v e some p r e l i m i n a r y d e f i n i t i o n s : L e t 5 be a c a r d i n a l , and -t5
t (0) = x 5
a sequence defined by
tp.
t ( n + 1 ) = 2
5
We p u t
L e t g be a f u n c t i o n defined on the c l a s s
g(5) =
of
c a r d i n a l numbers by
*
5
Finally l e t
O n + O n b e a f u n c t i o n defined by
6:
for all
5
.
We have g ( 5 ) >
6(0) = g(0)
6(5
+
1) = g(d(5))
d ( X ) = sup
P
d(@),
I t i s t r i v i a l t o see t h a t
f o r X a l i m i t ordinal.
6 i s normal.
Now, we can prove (MI. Let h be a normal f u n c t i o n . Then hod
i s normal and by (M") there ex-
i s t s a r e g u l a r B such t h a t
h(b(B)) = B But,
B = h ( 6 ( B ) l L 6(8) 2 B And we have
d(B)
=
B
.
F i n a l l y , we have t o prove t h a t B i s inaccessible. I n fact, i f
172
L U l Z PAUL0 DE ALCANTARA
5 ( 5 < 6 ) such t h a t
then there exists
d(5) > p
.
Thus,
v
< 9(P) < g(d(5)) =
=
Then, t h e r e e x i s t s M ( n E w )
uwt6(E)(n).
nE
such t h a t
~5 t
6(5 1(')
*
And we have, 2' < ;d(c)(n)
=
t d b ( E ; ) +( n1 ) < g ( d ( 5 ) ) = d ( 6 -+ 1 ) 5 d ( 6 ) = B
.
The p r o o f i s now complete. L e t us c o n s i d e r t h e f o l l o w i n g schemata:
(L)
h h
evehy incheaning dunction
6 d.6.a.u. t h e m eXin-tn an i n a c -
censibte numbeh a nuch that
(WB) (6 < a + d(6) < a ) .
(L') F a t euehy incheaning d u n d o n 6 d . 6.a.o. and euehy ohdind -thehe exh& a n inaccennibLe numbeh a , a > y nuch that
(WB) (B < a
+
d(B)
-)
d.6.a.u.
thehe
e x h a a heg&WL
d(B) < a ) .
Arguments which a r e v e r y s i m i l a r t o t h o s e o f theorem 3 can be u s e d o r d e r t o p r o v e t h e e q u i v a l e n c e between (L) and (L"). Assuming t h e axiom o f c h o i c e we p r o v e t h e
THEOREM 4 . h eyLLivdent .to each one ud (L) and
(M)
(L').
PROOF: We show f i r s t t h a t (L) and (L') a r e e q u i v a l e n t . L e t 6 be an i n c r e a s i n g f u n c t i o n . We d e f i n e a f u n c t i o n
dy(B) = d ( Y
6,
+
,
< a).
(L") Foh euehy incneaning & ~ n d o n numbeh a nuch .that
(WB) (B < a
y
B).
i s i n c r e a s i n g and then by (L)
(W6)
( B < a + d ( B ) 5 d ( +~ 0 ) = d y ( B ) < a ) .
Y 'Y
+
A1 so,
6 5 d ( +~ B ) <
8 .
We prove now t h e e q u i v a l e n c e between (M) and (L).
Y
by
in
173
AXIOMS OF STRONG INFINITY Let
6*
tion
6
be an i n c r e a s i n g f u n c t i o n . We can associate t o defined as f o l l o w s :
d*(O)
= d(0);
d*(A)
= sup
B<
6(f3),
6
a normal func-
f o r h a l i m i t ordinal;
5 such t h a t 3r1(5 = ~ ( T - I ) )and 5 > d * ( p ) .
6*(p t 1) = the l e a s t
And we have,
5 6 ( 9 ) 5 d*(B
6*(B)
t 1) f o r a l l
f3
.
By ( ~ 4 ) there e x i s t s an i n a c c e s s i b l e a such t h a t
Let
. Then,
< a
5 6*(B + 1) < 6*(a)
d(B)
6
Conversely l e t
= a
6*(a)
= a
.
.
be a normal f u n c t i o n . By (L) there e x i s t s an inacces-
s i b l e a such t h a t
(B < a -td(B) < a ) .
(W B ) Therefore,
a and
5 d(a)
=
u d(S) 5 a
s
d(a) = c1
3, SOME NEW
A X I O M S OF STRONG I N F I N I T Y ,
I n t h i s s e c t i o n we i n t r o d u c e some new statements e q u i v a l e n t t o 04).
DEFINITION, LeX 6 be a
b d
v d u d @ m X o n dedined on t h e
&abb
06
&
o&k&.
6
0 V - n a m d id and onty id
(i) ~a
I
B(B < a + d ( 0 )
(ii) 6 ( A ) =
U 6(5),
B
B(a));
ifi h 0 a L i m i t o h d i n d .
I n analogy w i t h (L) we consider t h e f o l l o w i n g statement:
(LV) Fat evmy V - n o m d &w.5on bee a buch 2 h . t (WB) (B < a
-S
d. 6. a. 0. fithetre exA.th an i m c c ~ d-i
6(B) € Va)*
174
LUlZ PAULD DE ALCANTARA
.
(LV) 0 eqcLi~dentto (L)
THEOREM 5. PROOF:
Let
6
d,(B)
be a V
- normal where
= p(d(B)),
6,
f u n c t i o n . We d e f i n e
p(x) i s t h e rank o f x
: On + O n
by
.
i s increasing, s i n c e if B < a , d(B) E 6 ( a ) andp(d(B)) < p ( 6 ( a ) ) .
6,
By (L) there e x i s t s an i n a c c e s s i b l e a such t h a t ( W B) ( B < a Then, i f
6,(B)
+
< a).
B < a , p(d(B)) < a
6
Conversely, l e t
: On+On
6
We can associate t o
and
.
d ( 0 ) E Va
be an i n c r e a s i n g f u n c t i o n .
a normal f u n c t i o n
6*
defined as i n
the
second
p a r t o f theorem 4.
By (Ly)t h e r e e x i s t s an i n a c c e s s i b l e a such t h a t ( W B ) (B < a and we have
B
+
+
fi*(B)
1< a
E
and
Val : 6*(B + 1) E Va
.
Therefore,
6 ( 8 ) E Va
d(B) < a
and
.
I n order t o g i v e a new f o r m u l a t i o n o f (L") we c h a r a c t e r i z e t h e standard complete models o f BG w i t h o u t t h e power s e t axiom. This w i l l be done using i n Kruse 1965 and Shepherdson 1952.
some o f t h e ideas developed
DEFINITION, LeX y 6e.a cahdivlaR nwnbm. V(0
I Y )
=
0
;
V(a + 1, y ) =
? V(a ,y ) ,
whehe
Y TY(x)= { t i Z c x A c m d ( x ) < y } ;
175
AXIOMS OF STRONG INFINITY
I f x has power h e r e d i t a r i l y l e s s than y , c a h d ( T C ( x ) ) 5 y
.
Thus
and the c l a s s o f a l l sets whose power i s h e r e d i t a r i l y l e s s cam! ( p ( x ) ) 5 y than y i s a s e t denoted by H(y). The f o l l o w i n g lemma can be e a s i l y proved.
By t r a n s f i n i t e i n d u c t i o n on a , we show
16 y 0 an i n ~ i n i t eheguhm catrdinat! .then H ( y ) c V
LEMMA 4.
PROOF:
If
x
c
x
such t h a t
V and d ( x ) < y . As Y c V . Thus x E V Y ' Y'
.
y i s regular
16 y .i~ an .in&inLte h e g W c~V~divrae, then H(y) = V(y ,y).
COROLLARY.
Foh y a hegueah c a n d i d numbU~we have
LEMMA 5.
(i)
x € H(y), then y' < y
there e x i s t s
Y
( v x ) ( x E U + U x E U)
(ii) ( v x ) ( x c U A ( 3 g € U ) ( c a h d ( x ) 5 c a h d ( g ) ) + x E U)
id
u
=
v
u
oh.
a d ovZeg
= if(y).
For y a r e g u l a r c a r d i n a l number i t i s p r o v e d i n
md(H(y)) =
4
1 Zp.
B o f f a 1970 that
P
From the lemmas 3 , 4
and 5 we deduce
0 a bfandahd cornplkte n i o d d 06 BG* i d and o d q id THEOREM 6 . Al = ?U U = V o h U = V(y ,y) whem y 0 a hegueah cahdivlae g h & a than x o
.
I n analogy w i t h (L") we formulate t h e statements
(L;)
Foh evmy
cmdid a
V - nohnid d u n d o n
, b U C h .that
(W R) ( R < a
+
d(R)
, d. d.a.o., &thehe e h f . 4
2,
a
E
Val.
and
a htgdah
176
L U l Z DAULC DE ALCANTARA
THEOREM 7 .
(dv) 0
eqLLiudeizt
3 :)
(
fo
The proof follows e a s i l y from theorem 6. THEOREM 8. PROOF:
(gv)
6
Let
:
on
( $ :)
implien +
be a V - normal function.
V
By (&;) t h e r e e x i s t s a standard complete model u of BG* s u c h t h a t ( W D ) (4 E
+
d(B)
u).
B u t then, t h e r e e x i s t s a r e g u l a r y > X o and
u = V(y ,y),
i.e.
for
B < Y
4 As
€ V(y ,y) +
V(y .y) c Vy
6(0)
we have
E V(Y
,y).
i ( 0 ) E Vy
We can e a s i l y see t h a t ( L ~ implies )
. ( g v )because
a ,
H(a) = Vu
T h u s \ve have t h e following diagram :
L
c
> L'
(
> L"
f o r an i n a c c e s s i b l e
AXIOMS O F S T R O N G I N F I N I T Y
177
REFERENCES, Boffa, M. 1970,
de pLLinsance .indEtieune cmdinul inhini donnT, B u l l e t i n de l a S o c i g t e Mathgmatique d e
Sun L’enhenible den en5embLen h&Edit&aiien.t UM
B e l g i q u e , 22, 3 8 9 - 392. de A l c a n t a r a , L. P. 1974,
On kiew
ctxioiii
06
oche~iic&t
nthomj L n 6 i ~ L i t yin axiotiiatic
nef
theaay
( a b s t r a c t ) , The J o u r n a l o f Symbolic L o g i c , 39, 410. Drake, F. 1974,
S e t Theory,
North
- Holland,
Amsterdam.
Kruse, A . H. 1965,
G?utliendiccl: i i i i i v w e s cciid .the sups Compositio Mathematicae, 17, 96 - 101.
COlllJJ~e*~ riiodeb
06
Shephadoun,
Lgvy, A. 1960,
Axiatii
nclwnic~tct 0 5 n t r i u i i g i i i i i i i t y
iii
uxioittcLtic net theutry,
Pacific
J o u r n a l o f Mathematics, 10, 2 2 3 - 238. Mostowski, A. 1967,
Recerd t r e ~ d ! Ain ne,t theatry, i n Problems i n t h e Philosophy of
Mathematics, N o r t h
- Holland,
Amsterdam.
Shepherdson, J . 1952,
l n n m I?lad& 225
- 237.
doh Se.t
Tlieotry, The J o u r n a l o f S y m b o l i c
Logic,
Departamento de Matematica Universidade Estadual de Campinas Campinas, Sao P a u l o , Brazil.
14,
Non-Classical Logics, Model Theory and Computability. Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
A.I.
PRESENT PROBLEMS ABOUT INTERVALS IN RELATION-THEORY AND LOGIC, b y ROLANU F R A Y S S E .
0 1NTRODUCTI ON I
Everybody knows
AND SUMMARY ;
t h e two
t o t a l ordering r e l a t i o n .
d e f i n i t i o n s o f an
Firstly, the abhotde
i n t e r v a l i n a chain,
intehvd
c l o s e d under i n t e r m e d i a r i t y : i f x and g > x belong every z such t h a t
to
, which
the
or
i s a Set
interval,
then
x < z < y belongs t o i t . Secondly t h e h e h t i v e i n t e h v d :
g i v e n two elements a and b > a , whose s e t Ia,bl relative interval i s the set o f t i o n o f the r e l a t i v e interval o f
bound,
i s called the
the
x ' s such t h a t a < x < b . Analogous d e f i n i x ' s such t h a t x < a, o r such t h a t
the
More g e n e r a l l y , g i v e n an a r b i t r a r y subset F o f t h e base E o f
x > a. ordering
F w i l l be a subset o f E - F, D such t h a t , i f t belongs t o F, t h e n a l l x ' s o f U a r e
~ ~ d a t i wintehwae e w d h bound
maximal by i n c l u s i o n among s e t s
v a l i s o b v i o u s l y c l o s e d by i n t e r m e d i a r i t y , But t h i s e q u i v a l e n c e
t h e n i s an a b s o l u t e
between d e f i n i t i o n s w i l l
not subsist i n
interval
.
generaliza-
tions t o arbitrary relations. I n t h e y e a r s 1950, t h e a u t h o r proposed t h e (see, f o r i n s t a n c e , F r a f s s e 1973, p. 107). tions
A and
following
generalizations
Recall that, given
B o f common a r i t y n , a b i j e c t i o n
6
two
base I A l , t o a subset G o f I B I , i s c a l l e d a t o c d hoamohpkism o f A 8, when
6
i s an isomorphism o f t h e r e s t r i c t i o n A / F o n t o B/G. F o r
l o c a l isomorphism, i t i s s u f f i c i e n t , and o b v i o u s l y necessary, s t r i c t e d t o every s e t o f < n
elements o f i t s domain F , be a 179
rela-
F of
f r o m a subset
being
that local
the
towards
6
a re-
isomor-
180
ROLAND
phism. R e c a l l t h a t
6
FRA'I~SSE
i s c a l l e d a L o c d aLLtomo/Lpkinm o f
6
A, when
lo-
is a
c a l isomorphism o f A towards A. These n o t i o n s , as t h e f o l l o w i n g n o t i o n s o f i n t e r v a l s , e x t e n s i b l e t o any muLt.iheL&on,
which i s a f i n i t e
are
sequence
immediately of
relations
t h a t a r e c a l l e d i t s c o m p o n e ~ ,w i t h a common base; t h e n t h e a h i t y
of
the
m u l t i r e l a t i o n w i l l be t h e maximun o f t h e a r i t i e s o f i t s components.
Gener-
a l l y we w i l l c a l l them r e l a t i o n s , by a b b r e v i a t i o n ,
compo-
except when new
n e n t s a r e e x p l i c i t l y added. Given a r e l a t i o n A o f base E = I A l , a subset D o f E i s c a l l e d an A-ubboLLLte i n t e n u d , o r s i m p l y an A - i n t e r v a l , when any l o c a l automorphism o f t h e r e s t r i c t i o n A/D, extended by t h e i d e n t i t y on E - U , g i v e s a l o c a l automorphism o f A: we s h a l l say s i m p l y t h a t i t i s e x t e m i b e e by t h e i d e n t i t y on E - 0. Note t h a t i t i s s u f f i c i e n t t o c o n s i d e r l o c a l automorphism o f A/U on domains o f p 5 n-J elements, and t o e x t e n d them by i d e n t i t y on n - p e l e ments o f E - D .
If A
I f A i s a chain, we r e f i n d t h e usual i n t e r v a l .
is
a
U i s an A - i n t e r v a l i f f , f o r any element t o f E - D , e i t h e r a l l elements o f D a r e < t , o r a l l a r e >.t, o r a l l p a r t i a l o r d e r i n g r e l a t i o n of base E , t h e n
a r e incomparable t o 2 . Given a r e l a t i o n A o f base
E, t a k e a subset F o f E. Then a s u b s e t D o f
E - F i s c a l l e d an ( A , F ) - i n t e , t ~ ~ L , o r an A - h e h L i w e intwvul bound, when
with
F as
a
D i s a maximal s e t , by i n c l u s i o n , among s e t s 0' such t h a t any
l o c a l automorphism o f A/D',
extended by i d e n t i t y on
F, g i v e s a
local
au-
D' have, f o r any e l ement t o f F, a l l t h e i r elements < t o r a l l >t;and by t a k i n g f o r D a maxi m a l D' , we g e t a s e t c l o s e d f o r i n t e r m e d i a r i t y , and so a u s u a l interval. Any a b s o l u t e i n t e r v a l D i s o b v i o u l y a r e l a t i v e i n t e r v a l , by t a k i n g t h e tomorphism of A. I f A i s a chai'n, t h e p o s s i b l e s e t s
bound E - D .
However t h e converse i s f a l s e , a l r e a d y f o r
a
partial
i n g r e l a t i o n A. F o r i n s t a n c e , g i v e n u a n d b > a (modu1o.A) t h e s e t X'S
such t h a t a < x < b i s a r e l a t i v e i n t e r v a l P , w i t h
Now suppose t h e r e e x i s t x,y i n incomparable t o y and
{u,b}
as
D, and t o u t o f D, w i t h u < x <
6. Then t h e l o c a l automorphism f r o m
{ X I t o {yl% i s e x t e n s i b l e by t h e i d e n t i t y o n {a,b}, on Ctl.
a ,t
the
orderof
all
.
bound and
t
singleton
but notbythe identity
D u r i n g t w e n t y y e a r s , o u r knowledge about i n t e r v a l s ,
extervals
a r e t h e i r complementary s e t s ) , and r e l a t e d t o p i c s , was p r a t i c a l l y
( which reduced
t o t h e s t a t e m e n t o f t h e e x e r c i s e 6, page 107, i n F r a f s s g 1973. Some r e c e n t researches o f F o l d s s , G i l l a m , and Pouzet, s e r i o u s l y i n c r e a s e d t h i s
knowl-
edge. Moreover t h e author? r e c e n t l y became conscious o f s e v e r a l l o g i c a l p r o -
181
PRESENT PROBLEMS ABOUT INTERVALS
blems involving i n t e r v a l theory. For i n s t a n c e , l o g i c a l problems about i n t e r v a l l a r y extensions o f a r e l a t i o n , which a r e natural g e n e r a l i z a t i o n s of the c l a s s i c a l extension of t h e r a t i o n a l ordering by t h e r e a l ordering relat i o n . In connection t o t h a t , the notion of a compact s e t of u l t r a f i l t e r s , and of a compact r e l a t i o n ( s e e 5.1 bellow), leads t o l o g i c a l problems. Fiand nally we s h a l l r e c a l l Ehrenfeucht’s r e s u l t s about t h e comparison, sum, product theory among o r d i n a l s , r e p o r t e d i n Feferman 1957, where the notion of i n t e r v a l i s deeply connected with l o g i c ; andwe s h a l l propose several possible g e n e r a l i z a t i o n s of t h e s e r e s u l t s t o comparison, sun, product theory among ordering r e l a t i o n , o r among a r b i t r a r y r e l a t i o n s .
1, ELEMENTARY PROPERTIES
OF ABSOLUTE INTERVALS
AND WTERVALS ,
I t i s obvious t h a t t h e empty s e t , t h e t o t a l base, and t h e singleton of any element, a r e i n t e r v a l s . There e x i s t r e l a t i o n s in which these a r e t h e only ones; f o r i n s t a n c e t h e c o n s e c u t i v i t y C on natural i n t e g e r s : C ( x , q ) = + when y = x + I , and C ( x , g ) = - in o t h e r c a s e s . Indeed i f a s e t P of inx € D, t e g e r s has a t l e a s t two elements, and i s not t h e whole base, take x’ 6 P and 6 E - 0 (where E i s t h e b a s e ) , b u t g = x + I o r 4 = x - 1;then t h e transformation of x i n t o x ’ i s a l o c a l automorphism of C , unextens i b l e by t h e i d e n t i t y on y . A mlLetimBuLivn i s a f i n i t e sequence of r e l a t i o n s on the same base, c a l l e d t h e campvnnnt r e l a t i o n s . The notions of local isomorphism, automorphism, i n t e r v a l a r e immediately extended t o m u l t i r e l a t i o n s . A unahy multir e l a t i o n i s a m u l t i r e l a t i o n whose composant r e l a t i o n s a r e unary ones ( t a k i n g value + o r - f o r each element i n t h e base). 16 A b a unahy m u L t h e h t i a n , t h e n any nubnet ad t h e b a e b a n
A-in-
RULVCd.
1.1 Recall t h a t a r e l a t i o n B i s 6 h e e - i n t a n p h e t a b l e i n a r e l a t i o n A on the same base, when any l o c a l automorphism of A i s an automorphism of 8. I t i s equivalent t o say t h a t t h e r e exists a f r e e l o g i c a l formula, e . g . , a formula without q u a n t i f i e r , with a p r e d i c a t e s u b s t i t u a b l e by A , and a number n of f r e e v a r i a b l e s equal t o t h e a r i t y of B , taking the value B(xI. ..., xn) when t h e p r e d i c a t e i s A and x I , xn a r e elements of t h e common base.
...,
16 A and E ahe each 6hee-intehphetable i n t h e a t h e k , t h e n t h e A - i n t e h v&
and t h e B-LntehvaCn m e t h e name.
182
ROLAND
FRA'ISSE
Consequently,i6 thehe e d . 1 2 a unahy mLLetitleeation B nuch &a2 A and B
ahe muahdXy 6hee-intapheXab&, t h e n a&? 4ubneA 0 6 -the b a e I A l ahe Aintaw&. Example: concatenate a unary r e l a t i o n Uand t h e b i n a r y r e l a t i o n o f equivalence w i t h two classes, defined by t h e values + and - taken by U. The converse i s n o t t r u e : take t h e chain on two elements; see a l s o
bellow
1.9
.
Given a r e l a t i o n A, consider as e q u i v a l e n t any two elements u,v o f
IAl
when t h e t r a n s p o s i t i o n (u,w) extended by t h e i d e n t i t y on a l l o t h e r elements, i s an automorphism f o r A. Thehe dined doh A, id6 t h e m
ahe 6 i n i t d y many e q u i w d e n c e . c l a n e 4 no de-
.ih a unahj
m L L e t i t l d d o n i n which A
dhee-ideh-
i.6
p e t a b L e . Indeed f o r a unary m u l t i r e l a t i o n U, l o c a l automorphisms a r e
ex-
a c t l y those b i j e c t i o n s which transform any element i n t o an o t h e r g i v i n g t h e t h e same value t o each component o f U ; each l o c a l automorphism domain i s generated by some t r a n p o s i t i o n s between e l e m e n t s value
t o U, i n t h e sense t h a t i t i s obtained
by
of
finite
g i v i n g Same
composition
of
these
t r a n s p o s i t i o n s and then by
r e s t r i c t i o n t o i t s domain. Coming back t o A,
t h e same argument proves t h a t
A i s f r e e - i n t e r p r e t a b l e by t h e unary
r e l a t i o n whose components are defined from
multiof el-
each equivalence c l a s s
ements o f I A l . Now note t h a t , i f a r e l a t i o n 6 i s
f r e e - i n t e r p r e t a b l e i n A, i t i s pos-
s i b l e e i t h e r t h e r e a r e more A - i n t e r v a l s than more B - i n t e r v a l s than
B-intervals, o r
+
and once the value
equivalence r e l a t i o n . I f A(x) = A ( x ' ) = an A - i n t e r v a l ,
+
- , and
and A(y) =
are
intervals
A - i n t e r v a l s , o r t h a t none o f both sets o f
i s included i n t h e other. F i r s t example: A i s a unary l e a s t t w i c e t h e value
there
r e l a t i o n taking
at
B i s t h e corresponding
- , the
p a i r Ix,y)
is
as a l l t h e subsets o f t h e base, and i s n o t a B - i n t e r v a l , t h e
transformation o f
x i n t o y being an automorphism o f B, unextensible by t h e
i d e n t i t y on { x ' } . Second example: A i s t h e chain o f n a t u r a l i n t e g e r s , and B i s t h e unary r e l a t i o n t a k i n g always t h e value
+.
= (N,O)
T h i r d example: A
where N i s t h e chain o f n a t u r a l integers, and 0 i s t h e u n a r y
relation,
s i n g l e t o n o f zero; B i s t h e equivalence r e l a t i o n w i t h t w o c l a s s e s , s i n g l e t o n o f zero and t h e s e t o f p o s i t i v e i n t e g e r s . Then
B is
the
free-inter-
p r e t a b l e i n 0,then i n A. But t h e s e t C0,l) i s an A - i n t e r v a l w i t h o u t
being
a 5 - i n t e r v a l , and the s e t {1,3} i s a B - i n t e r v a l w i t h o u t being an A-interval.
1.2. Given a hek,t.iOn
A, any i n t e M c o t . i o n
06
A-intehvd
0
an A-in-
tthvd.
0 Consider a s e t o f A - i n t e r v a l s D ,i
and t h e i r i n t e r s e c t i o n U. L e t
6
be a
183
PRESENT PROBLEMS ABOUT INTERVALS
local automorphism of the r e s t r i c t i o n A / U . In order 6 be e x t e n s i b l e by t h e i d e n t i t y on I A l - U , i t i s s u f f i c i e n t t h a t 6 be e x t e n s i b l e by t h e i d e n t i t y on any f i n i t e subset of IAl - U . For such a f i n i t e s e t H , we can consider an a r b i t r a y sequence of the elements u I , ..., u h of H , and extend 6 by the i d e n t i t y on {ul}, then on {ul,u2}, and so on. For each element ui of H , t h e r e e x i s t s a U i such t h a t ui 6 IAl - U i , so t h a t the addition of the i d e n t i t y on Cuil transforms t h e local automorphism of A i n t o another local automorphism of A; note t h a t i t i s i r r e l e v a n t t h a t previous u j ( j < i) be elements of Ui o r elements of IAl -Ui. 0 1.3. Giwen a be.t 0 5
A-intehuakLn,
@5i2hing
bq i n c t w i v n , then
.theit union 0 an A - i r i t e h v d .
be these i n t e r v a l s , and U t h e i r union. Let 6 be a local autoUi. morphism of the r e s t r i c t i o n A / U , with a domain and a codomain which can be supposed f i n i t e . Then t h e r e e x i s t s a t l e a s t a Ui including t h e domain and the codomain of 6. Consequently, 6 i s e x t e n s i b l e by t h e i d e n t i t y on I A I - U i , thus by t h e i d e n t i t y on IAl - U . 0 0 Let
06
1.4. L e i n be t h e &.ty a mLLetiheLcction); L e t U , V
A ( t h e maximun a h i t q 06 cumpone&
be Auv A - i M R e t w a A ; huppobe t h a t
v h A / [ LI u V ) v 6 c a t d i n d _< IP - 1, admix2 an h e ~ z 7 ~ L c L L o oah A / ( U n V ) . Then t h e union
LLvn
homohphic
the
U U V
any
hetntivn
id A .LA he~thic-
among
0 an A-intehwd.In t h e
case o f a chain A , o r more g e n e r a l l y of a r e f l e x i v e binary r e l a t i o n we refind t h a t t h e u n i o n of two i n t e r v a l s w i t h a common element i s an i n t e r v a l . 0 Let 6 be a local automorphism of A , with i t s domain and codomain i n cluded i n U U V . I t i s s u f f i c i e n t t o suppose these domains a r e each of c a r d i n a l i t y m 2 n - 1 , and t o prove t h a t 6 i s e x t e n s i b l e by t h e i d e n t i t y on n-m a r b i t r a r y elements o u t of U U V . !.et us note ul, u the elements o f the domain of 6 , and wl, ... , wm t h e i r transformed elements by 6 . By the hypothesis, t h e r e e x i s t i u l , . .,wm belonging t o t h e i n t e r s e c t i o n U n V , transformation of ui i n t o igi ( i = 1, ..., m ) and the transformation of u. i n t o w . being local autonorphisms of A . I t i s s u f f i c i e n t t o prove t h a t t h e f i r s t transformation i s e x t e n s i b l e by t h e i d e n t i t y on n - m arbitrary elements out of U U V . The analogous proof works f o r t h e second transformation. We may always suppose t h a t t h e r e i s a pzrn with ul, u belonging t o P U , a n d U ~ + ~ , . . . , U ~belonging t o V and o u t of U . The transformation of u. i n t o tu.L (i = 1, . . . , p ) i s a l o c a l automorphism of A , with domain and codomain included i n U. T h u s i t i s e x t e n s i b l e by t h e i d e n t i t y on u . ( j = p + l , J m ) , and moreover by the i d e n t i t y on n - m a r b i t r a r y elements out of
...,
.
...,
...,
184
ROLAND FRA'I'SSE
U u U . On a n o t h e r s i d e , t h e t r a n s f o r m a t i o n o f iui ,...,wp, upl ,...,um into t h e wi (i = I , ...,m) i s a l o c a l automorphism o f A , s i n c e i t i s composed o f two l o c a l automorphisms, as i t i s seen by g o i n g t h r o u g h t h e u4 (i= 1,. . .,m). I t s domain and codomain a r e i n c l u d e d i n U ; t h u s i t i s e x t e n s i b l e i d e n t i t y on n - m a r b i t r a r y elements o u t o f U U U . F i n a l l y t i o n o f ui i n t o 1.5.
L U ~
(i = 1. ...,m) i s so e x t e n s i b l e . 0
Given a r e l a t i o n A , l e t us c a l l an A - e x t e t l v a l
the the
by
the
transformacomplementary
s e t o f an A - i n t e r v a l . A AubheX 0
6
17.5
t h e bane I A l 0 an A - e x t e h v a e
i66,
doh any 4acalautomoh-
A I U , a n d any nubneXG od P*= I A l - P eLtheh 6 0 n o t ex.tev&ble by t h e i d e d t y on G , oh ,{ .LA e x t e n h i b e e b y any local aLLtaniohpkism 0 6 AID* pkinm
06
with domain G . 0 Suppose
D i s an e x t e r v a l , t h e n D*=
l o c a l automorphism of tomorphism
6
of
A/D
A/D*
IAl
w i t h domain G
-D
i s an i n t e r v a l . L e t y
. Now
i s e x t e n s i b l e by t h e i d e n t i t y
6 U l G i s a l o c a l automorphism o f A. The s e t
union
be
a
suppose t h a t t h e l o c a l au-
l G on G , P* b e i n g an
so
the
interval,
g i s e x t e n s i b l e by t h e i d e n t i t y I F , on t h e codomain F' o f 6. Thus t h e uni o n 6 U g i s a l o c a l automorphism, as o b t a i n e d by c o m p o s i t i o n o f 6 u l G and IF'
u
9 * Suppose
D
i s n o t an e x t e r v a l , t h e n U*
i s n o t an i n t e r v a l . Thus
there
e x i s t s a l o c a l automorphism g o f AID* and a subset F o f D , such t h a t I F U g i s n o t a l o c a l automorphism. Then t h e i d e n t i t y I F on F i s o b v i o u s l y extenb l e by t h e i d e n t i t y on G , w i t h o u t b e i n g e x t e n s i b l e by 9; a g a i n s t o u r c o n d i tion.O 1.6. Given a 4eLatian A and ZM' V din joint A - i n t e h v a & F and G , and a LocaL automatlpkinm 6 03 t h e t l u t h i c t i o n A / F and anothetl g o6 A / G , t h e n t h e u n i o n 6 U g 0 a Locd automotlphi~m06 A . I t i s s u f f i c i e n t t o e x t e n d d by t h e i d e n t i t y on t h e domain o f g, and g by t h e i d e n t i t y on t h e domain o f 6 , and t o compose. L e t A be a r e l a t i o n , E i t s base, n i t s a r i t y . L e t us share E i n t o d i s j o i n t A -interval
Pi
Di.
To each
Ui, l e t us a s s o c i a t e a f i n i t e subset di o f Pi, such t h a t each r e s t r i c t i o n o f AIDi
t h a t we c a l l t h e hcphe~cntanto f
w i t h c a r d i n a l i t y < n - l a d m i t s an i s o m o r p h i c r e s t r i c t i o n o f A / d i .
L e t us c a l l
i n t e h v a f l a h y image, o r more s i m p l y image o f A, t h e r e s t r i c t i o n o f A t o union o f t h e representants
the
di.
Thehe exin& o n l y one a e l a t i o n A
06
g i v e n bane Ephahed i n t o dinj o i n t A-
PRESENT PROBLEMS ABOUT INTERVALS
inte4v& Ui, 06 g i w n 4enttLicLioni, AIDi and A/U d i lohehe tach di h a heptiencnaknt o h Di.
loah
185
a given intc4wdLahy.image conditions;
0 Suppose t h e r e e x i s t two r e l a t i o n s A and A ' s a t i s f y i n g o u r
E, with cardinality z n ,
i t i s s u f f i c i e n t t o prove t h a t , f o r any subset F o f
we have
A/F = A ' / F . T h i s i s obvious i f F i s i n c l u d e d i n aDi
i o n Ud.i
Suppose t h e c o n t r a r y : each i n t e r s e c t i o n
o r i n t h e un-
Fi = F n D . i s o f
n a l i t y 5 n - 1 . By h y p o t h e s i s , t h e r e e x i s t s a n isomorphism o f A/Fi
. These
s t r i c t i o n o f A/di
main i s i n c l u d e d i n a Di
onto a re-
isomorphisms have d i s j o i n t domains s i n c e each do; same remark f o r codomains; by
, and
statement, t h e i r u n i o n i s a l o c a l automorphism o f A b e i n g i d e n t i c a l t o A ' / udi,
A/Udi
cardi-
t h i s automorphism g i v e s
1.7. GLJCVI a kicl'rttiuii A , a hubnot U
05
the
preceding
also o f
A'.
So
A/F = A'/F. 0
i h bane I A I and an A-intehwal U,
thc iti,ttehbect,iuvi D n U i~ UYI ( A / D ) - i n t e h w d . Note t h a t , g i v e n an ( A D ) - i n t e r v a l V , t h e r e does n o t n e c e s s a r i l y e x i s t an A - i n t e r v a l
U such t h a t U = 0
n
U. F o r i n s t a n c e , t a k e f o r A t h e
o r d e r i n g r e l a t i o n , Boolean l a t t i c e r e p r e s e n t e d by
a
usual
partial
3-dimensional
cube whose minimun u and maximun w a r e two o p p o s i t e v e r t i c e s ; t a k e f o r D t h e s e t o f t h e t h r e e v e r t i c e s x , y, z i m m e d i a t e l y p r e c e d i n g w , and t a k e V = Ix , q l. Then an A - i n t e r v a l i n c l u d i n g V has n e c e s s a r i l y t h e element immediately ant e r i o r t o x and z ( a n d contradiction.
incomparable t o y ) , and t h e n n e c e s s a r i l y z i t s e l f :
me exactey t h e A - i n Given A and an A - i n X e h w d D , t h e (AID1-iMRehv& te4v& i n c e u d t d i n U , OK tquiwalentey t h e inteaecLitioni, i u i t h D 0 6 aU Ainte4vat~. Le* E be t h e he..t
03
and N t h e w u a l chain on E . d h e e - i ~ t e 4 p h e t a b l ei n N and i 6 any poi& 0 6 i n t e g e u h an A-ivdehwal, .then a l l pehmcLtatioMn 0 6 E me aLLtoma41.8.
nntwlae
16 a 4ela.tion A on t h e name bane €
pldrnb
604
ivU%gC.hb,
;A
A. ( T h i s statement and t h e f o l l o w i n g a r e communicated
by
PI.
Pouzet. ) 0 L e t u < v < w be t h r e e i n t e g e r s . As A i s f r e e - i n t e r p r e t a b l e i n N , transformation
the
o f u i n t o w i s a l o c a l automorphism o f A; as t h e p a i r Iu,lol
i s an A - i n t e r v a l ,
t h i s l o c a l automorphism i s e x t e n s i b l e by t h e i d e n t i t y on
v. Always by f r e e - i n t e r p r e t a b i l i t y o f A i n N
,
any
transformation w h i c h
saves t h e o r d e r o f i n t e g e r s i s a l o c a l automorphism o f A :
by
composition,
t h e t r a n s p o s i t i o n between any two i n t e g e r s i s a l o c a l automorphism o f A. I t f o l l o w s t h a t any p e r m u t a t i o n o f E i s an automorphism o f A. 0
186
ROLAND FRAI'SSE
06 i h e bane I A l 0 a n then: ( 1 ) the4e 0 a p a h t i f i o n 0 6 -the bane. into d i n i t e l y many c l a n n u , t h e thannpon.&ion belween any .iWu element4 0 6 a hUme clanb, e.xtended by t h e i d e n t i t y , giving a n aLLtamo4pkinm 504 A; (2) -the4e a u n a y mu&L4&27%on i n wkich A 0 & ~ c e - i n t e t p ~ ~ e t a b l e ; m 0 4 2 0 U e h i6 .thehe 0 at moni one c l a n i n conclubion ( 1 1 , iukich aeducu t o a n i n g l e t o n , then thene 0 a u n a y mu&Lteeation B nuch .that A and B ahe rnutuaLLy ~4ee-intchphetable. 1.9.Given a 4eLation A, i6 any p a i 4 06 dement4
A-intckud,
0 Consider as e q u i v a l e n t any two elements
u,v
o f t h e base I A l when t h e
t r a n s p o s i t i o n ( u , v ) i s a l o c a l automorphism f o r A; s i n c e t h e p a i r lu,v}
an A - i n t e r v a l ,
t h i s t r a n s p o s i t i o n extended by t h e i d e n t i t y
on
other
is ele-
ments, g i v e s an autonorphism f o r A. I n o r d e r t o g e t o u r c o n c l u s i o n ( 1 ) suppose t h e r e a r e i n f i n i t e l y many c l a s s e s i n o u r e q u i v a l e n c e . Using
t h e axiom
o f c h o i c e , t a k e a denumerable sequence o f elements ai (i n a t u r a l
integer),
n u t u a l l y not equivalent. Call N the
i5
w-chain o f a i l s , w i t h N ( a . , a . ) = L
J
+
when
L e t n b2 t h e a r i t y o f A; c o n s i d e r as e q u i v a l e n t any two s e t s w i t h t h e same c a r d i n a l p 5 n , when t h e t r a n s f o r m a t i o n o f t h e one i n t o t h e o t h e r , j .
which p r e s e r v e s t h e o r d e r i n g N , i s a l o c a l a u t o m o r p h i s n o f
A.
Using
Ramsey's theorem, t a k e a denumerable s e t U o f elements ai, t h a t we r e - n u m e r a t e , so t h a t any l o c a l automorphism o f N/U, on < n elements, q u e n t l y any l o c a l automorphism o f
and
N / E , i s a l o c a l automorphism
i n o t h e r words A/U i s f r e e i n t e r p r e t a b l e i n
N / U . By
1.7
any
of
conseA/U :
pair o f el-
ements o f U i s an ( A / U ) - i n t e r v a l . By 1.8 a l l t r a n s p o s i t i o n s i n U a r e
auto-
morphisms f o r A/U, and t h e n f o r A s i n c e any p a i r o f elements i s an A - i n t e r Val: c o n t r a d i c t i o n , proving o u r conclusion (1)
.
We know f r o m 1.1 and o u r c o n c l u s i o n ( l ) , t h a t A i s
free-interpretable
i n t h e unary m u l t i r e l a t i o n R whose components a r e d e f i n e d each by an e q u i v alence c l a s s o f elements o f A
, it
I A l . To p r o v e t h a t 8 i s f r e e - i n t e r p r e t a b l e
i s s u f f i c i e n t t o t a k e any two elements u,v
in
i n d i f f e r e n t classes,and
prove t h a t t h e t r a n s f o r m a t i o n o f u i n t o v i s n o t a l o c a l
automorphism
for
A. Suppose t h e c o n t r a r y f o r u and v , and suppose t h e r e e x i s t s u' f u a n d e q u i v a l e n t t o u. The t r a n s f o r m a t i o n o f u i n t o v b e i n g e x t e n s i b l e ,
since
Iu,v) i s an A - i n t e r v a l , t h e t r a n s p o s i t i o n ( u ' , ~ )i s a l o c a l automorphism f o r A: so u,u' and v a r e e q u i v a l e n t : c o n t r a d i c t i o n . 3
2, THE
FINITE-VAL,
A
BOOLEAN NOTION,
Consider a r e l a t i o n A and a subset
D
o f i t s base I A l . F o r each p o s i t i v e
187
PRESENT PROBLEMS ABOUT INTERVALS
i n t e g e r p , consider a s e q u i v a l e n t two p-uples ul, ...,u and u l , . . . , w P P' where t h e u ' s and w ' s a r e elements of D , when t h e transformation o f u i i n t o w . ( i = 1 , . . . , p ) i s a l o c a l automorphism of A , e x t e n s i b l e by t h e i d e n t i t y on I A l - 0. Then we c a l l D an A - d i n i t e - w d , when t h e r e a r e f i n i t e l y many c l a s s e s i n t h i s equivalence. I f n i s the a r i t y of A , i t i s s u f f i c i e n t t o l i m i t the length of sequences t o p 5 n - 1 and t h e i d e n t i t i e s t o n - p elements. Foh any h e e a t i o n , avy
nite-vat below).
[the
union 0 6 dinjo.int i P l t Q h w C d A i 6 a 6id i s j o i n t n e s s w i l l be cancelled a f t e r 2.3
diiUt4
condition o f
cl Two p-uples a r e e q u i v a l e n t i f f t h e i r f i r s t terms a r e both
i n a same i n t e r v a l , t h e i r second terms both in a same i n t e r v a l , and so on, and obviously i f t h e transformation of t h e one i n t o t h e o t h e r i s a l o c a l automorphism. This gives a f i n i t e number of equivalence c l a s s e s . 0 For a chain, t h e f i n i t e - v a l s a r e e x a c t l y a l l f i n i t e unions of i n t e r vals. However t h i s i s n o t t h e general case. For instance,consider the cyc l i c ordering, obtained by s t a r t i n g from a chain A , and giving t o any 3uple x , l j , z t h e value + i f f x 5 y 5 z o r y 5 z 5 x o r z 5 x 5 y (modulo A ) . Take f o r i n s t a n c e A = usual chain of natural i n t e g e r s . Then t h e only intervals o f the c y c l i c ordering a r e t h e empty s e t , t h e base and s i n g l e t o n s . Indeed f o r any o t h e r set D , t h e t r a n s p o s i t i o n between two d i s t i n c t elements x , y of D , i s a l o c a l automorphism, and i s not e x t e n s i b l e by t h e i d e n t i t y on any element out of D. However f o r any i n t e g e r u , t h e s e t of i n t e g e r s 2 u i s a f i n i t e - V a l , a s i t i s seen by considering a s equivalent any two p-uples of such i n t e g e r s , when t h e transformation of t h e f i r s t p-uple i n t o the second i s a b i j e c t i o n saving the ordering.
As a consequence of t h e previous statement, any d i n i t e oubnet bane 0 a ~ i n i t e - u d . .LA
2.1. The Comp&meF&Lhy 4e.t u diniRe-~aL.
06
06
the
afly d i n i t e - u d (.in he6ehQMCe t o the. band
0 Let E be the base and D a f i n i t e - V a l ; l e t n be t h e a r i t y of t h e r e l a t i o n . For each p o s i t i v e i n t e g e r p 2 n - 1, consider any two p-uples i n 0 as equivalent when the transformation of t h e one i n t o t h e o t h e r i s a local automorphism, e x t e n s i b l e by the i d e n t i t y on € - 0 : by hypothesis, t h e r e a r e f i -
n i t e l y many c l a s s e s of t h i s equivalence. Now f o r each p-uple ( p 5 n - 1 ) i n D, l e t us take a unique representant belonging t o the same c l a s s , and c a l l H the f i n i t e subset o f 17, union of a l l these r e p r e s e n t a n t s . For each
posi-
188
ROLAND FRA'I'SSE
t i v e i n t e g e r 4, l e t us c o n s i d e r as e q u i v a l e n t any two q-uples i n E - V , when t h e t r a n s f o r m a t i o n o f t h e one i n t o t h e o t h e r i s a l o c a l automorphism e x t e n s i b l e by t h e i d e n t i t y on H : as H i s f i n i t e , t h e r e i s o n l y
a
f i n i t e number
o f equivalence classes. I t i s now s u f f i c i e n t t o p r o v e t h a t , i f g i s such a l o c a l
automorphism
i n E - U , e x t e n s i b l e by t h e i d e n t i t y on H , t h e n g i s e x t e n s i b l e by t h e i d e n t i t y on
U. L e t F be any subset o f U , o f c a r d i n a i i t y 5
n-1: i t i s sufficient
t o prove t h a t g i s e x t e n s i b l e by t h e i d e n t i t y on F, s i n c e i n t h i s
case,
extended by t h e i d e n t i t y on 0 , and t h e n r e s t r i c t e d t o any subset w i t h dinality
5
g
car-
n, w i l l g i v e a l o c a l automorphism. By h y p o t h e s i s , t h e r e e x i s t s a
subset o f F ' o f H and a l o c a l automorphism
6
w i t h domain F and codomain
F',
e x t e n s i b l e by t h e i d e n t i t y on E - V , t h u s e x t e n s i b l e by t h e i d e n t i t y I G on G.
IG and g U I F ' ( i d e n t i t y on F') a r e l o c a l automorphisms, so i s 6 U g 6 i s e x t e n s i b l e by t h e i d e n t i t y on E - V , t h u s b y t h e i d e n t i t y I G , on G', t h e u n i o n 6 - l u I G t , i s a l o c a l automorphism, and by c o m p o s i t i o n T F u g i s a l o c a l automorphism.0 Consequently, t h e compLementahy n e t 0 6 any Q X t e n u b n e t 0 6 t h e bnne 0 a ~ i n i t e - v d ;any & i n i t e i n t e u e c t i o n a6 e x t e h v u h 0 a d i n i t e - v d . As
6
U
b y composition. As
2.2.
16 a h e h t i o n B 0 64ee-ivLtehptleXabLc i n A, t h e n any A - 6 i n i t e - v d
0 a B-&Lte-vd.
Indeed, i f two f i n i t e sequences a r e t r a n s f o r m e d one i n t o
t h e o t h e r by a l o c a l automorphism o f A, t h e n t h i s t r a n s f o r m a t i o n i s a l o c a l automorphism o f B . So i f t h e y a r e e q u i v a l e n t f o r an A - f i n i t e - V a l equivalence subsists f o r V
U ,t h i s
and €3.
Consequently i f t h e r e e x i s t s a u n a r y m u l t i r e l a t i o n i n which A i s f r e e i n t e r p r e t a b l e , t h e n a l l subsets o f t h e base a r e A - f i n i t e - v a l s ;
f o r t h e con-
verse statement, see 2.5 below. Given a heLation B and a 8 - 6 i n i t e - u d V , t h e m e x h h a h e L a t i o n A i n w h i c h B 0 6kee-intehphetabLe,
V b e i n g an A - i n t e 4 v d .
0 L e t n be t h e a r i t y o f 8. Consider t h e e q u i v a l e n c e among t h e n-uples i n V , p u t t i n g two n-uples i n t h e same c l a s s when t h e t r a n s f o r m a t i o n o f one i n t o on t h e o t h e r i s a l o c a l automorphism o f B , e x t e n s i b l e by t h e i d e n t i t y I B I V . F o r each c l a s s U, t a k e t h e n-ary r e l a t i o n w i t h base IBI h a v i n g t h e v a l u e + f o r n-uples o f U and t h e v a l u e o t h e r w i s e . T h e n t h e f i n i t e sequences o f t h e s e r e l a t i o n s and o f a l l t h e components o f B , c o n s t i t u t e an na r y m u l t i r e l a t i o n A i n which 8 i s o b v i o u s l y f r e e - i n t e r p r e t a b l e . Moreover, i f two n-uples i n V a r e t r a n s f o r m e d by a l o c a l automorphism o f A, t h e n t h e y belong t o t h e same c l a s s p r e c e d e n t l y d e f i n e d f r o m B and V , then t h i s l o c a l
-
-
PRESENT PROBLEMS ABOUT I N T E R V A L S
189
automorphism i s e x t e n s i b l e by t h e i d e n t i t y on IBI - U , giving another automorphism of A .O Consequently, giwen a rnLLetihelation B,a n u 6 5 ~ tU 06 t h e b a e .i~ a B-din i t e - v d i d 6 thehe e d h a rnlLetihelation A i n wkich 8 & bhee- i n t e h p h e t able, U being an A - i n i e h w d (M. Pouzet, 1975, not published). 16
06
Q i~ a t j i n L t e - v d , t h e n Q incheaned
O h
dimininhed by a dinLte nub-
t o suppress, take i t s s i n g l e t o n r e l a t i o n (unary r e l a t i o n taking the value + f o r t h e considered element) and add i t t o t h e m u l t i r e l a t i o n A i n t h e previous s t a t e ment.
set
t h e bane, ,LA a 6 i n i t e - w d . For each element t o add o r
2 . 3 . The union and t h e i n t e m e d o n
0 6 auo
(jinite-vah 0 a dinite-wd
( M . Pouzet, 1976, not published). O I t i s s u f f i c i e n t , i n view of 2.1, t o prove i t f o r t h e union.Supposethe contrary; l e t ti and V be two f i n i t e - v a l s such t h a t t h e union U U V i s not a f i n i t e - V a l . Let p be t h e l e a s t p o s i t i v e i n t e g e r f o r which t h e r e e x i s t s an this i n f i n i t e sequence of p-uples i n U u V , mutually not equivalent , i n sense t h a t the transformation of any one p-uple i n t o another, extended by the i d e n t i t y out of U u V , i s n o t a local automorphism. For each p-uple, l e t us c a l l u the terms belonging t o U and not t o V , c a l l w the terms b e longing t o V and n o t t o U, and w the terms belonging t o U n V . We can a l ways suppose t h a t , f o r a given index h 5 p, the term of index h i s always a term U , o r always V , o r always w ; so t h a t , simplifying the presentation without l o s s of g e n e r a l i t y , we can suppose p = 3 with one term u , one w.and one w i n each 3-uple. For any i n t e g e r i, l e t us c a l l uh. , w .h, w .h these t h r e e terms. (Since U i s a f i n i t e - V a l , each sequence u . w . i s i n U, these seh 4 quences belong t o a f i n i t e number of c l a s s e s , f o r the equivalence d e f i n e d by local automorphisms e x t e n s i b l e by t h e i d e n t i t y out of U . T h u s we can suppose t h a t a l l sequences ui V J ~ belong t o t h e same c l a s s : consequently f o r any i, t h e transformation of ui wi mi i n t o uo wi w o is a l o c a l automorphism. On another s i d e , s i n c e V i s a f i n i t e - V a l , t h i s s e t V diminished of the f i n i t e s e t of terms w,, i s a finite-Val ( s e e 2 . 2 ) . So the terms w belonging t o this f i n i t e - V a l , and t h e terms uo and w o being out of i t , we can suppose t h a t , f o r any i n t e g e r s i, j , the transformation o f u o wi luO i n t o u w . w o i s a local automorphism. T h u s , f o r any i n t e g e r s i, j , we have loO J cal automorphisms from ui wi wi t o u o w . w o , then u o w . w o , then t o J u . w . w . : contradiction.0 J
J
J
190
ROLAND FRA'I'SSE
E b e t h e n e t 06 n a t u h d i n t e g e m , N t h e w u d c h a i n on E. 16 N and b any n e t 06 i n t e g e m b a n A - & L n i t e - w d , t h e n d L pemvnLLtdo~n0 6 E ahe acLtomohpkinmn doh A ( t h i s s t a t e m e n t and t h e 2.4. LeZ
A 0 6hee-intehphetabLe i n
f o l l o w i n g a r e due t o M. Pouzet).
o
Suppose t h e r e e x i s t s a p e r m u t a t i o n o f E which changes A; t h e n i f n
t h e a r i t y o f A, t h e r e e x i s t s a p e r m u t a t i o n on a t most 2n
integers,
changes A, and so a t r a n s p o s i t i o n betweentwo i n t e g e r s , which chqnges A t h e f r e e - i n t e r p r e t a b i l i t y o f A i n N , t h e r e e x i s t s an i n t e g e r
u
is
which
. By
such t h a t ,
f o r any i n t e g e r x . t h e t r a n s p o s i t i o n ( x , x + u ) changes A. F o r t h e same
rea-
son, t h e t r a n s f o r m a t i o n o f any x i n t o x t 2u i s a l o c a l automorphism o f A n o t e x t e n s i b l e by t h e i d e n t i t y on t h e s i n g l e t o n { x + d i d e n t i t y on i n t e g e r s < x o r
, but
e x t e n s i b l e by
> x + 2 u . Consequently, t h e s e t o f
all
the
multi-
p l e s o f 2u i s n o t a f i n i t e - V a l . 0
2 . 5 . Giwen a h e ~ a t i a nA, id any bubneX
06 ,the
b a e I A l b an A - d i n i t e -
w d , t f w M Rhe concLwion (1)06 1.8 0 w a e i d ; no b y 1 .l,thehe 0 a unmy m d L i h&7fiVn
i n A 0 ~ h e e - i n R e h p h e ~ a b e e . N o tteh a t (2)
o f 1.8 i s n o t
necessarily
v a l i d : example, a b i n a r y e q u i v a l e n c e r e l a t i o n w i t h f i n i t e l y many c l a s s e s .
0 Consider as e q u i v a l e n t any two elements
U,W
o f t h e base I A l when
the
, by t h e i d e n t i t y on a l l o t h e r elements o f I A l , t r a n s p o s i t i o n ( u , ~ ) extended
i s an automorphism f o r A . Suppose t h e r e a r e i n f i n i t e l y many
equivalence
classes. Using t h e axiom o f choice, t a k e a denumerable subset
Eo o f t h e Eo., extended
base, such t h a t t h e t r a n s p o s i t i o n between any two elements o f by t h e identity;ch'anges
A.Moreover,
t a k e an w-chain No
and by Ramsey's theorem, suppose t h a t t h e r e s t r i c t i o n
on t h e b a s e Eo, A/EO i s f r e e - i n t e r -
p r e t a b l e i n No. L e t n be t h e a r i t y o f A; f o r each p 5 n , c o n s i d e r a s e q u i v a l e n t any two subsets o f E~ w i t h c a r d i n a l p, when t h e unique
l o c a l automor-
phism o f No which t r a n s f o r m s t h e one i n t o t h e o t h e r , b e i n g a l o c a l automorphism o f A, i s s t i l l e x t e n s i b l e by t h e i d e n t i t y o u t o f Eo. Since Eo i s a a f i n i t e - v a l ,there a r e o n l y f i n i t e l y many e q u i v a l e n c e c1asses:using Ramsey's theorem, we g e t a denumerable subset E~ o f E~ i n which a l l s u b s e t s a r e e q u i v a l e n t , and t h i s f o r each p s i v e s e t s Eo 1 El 2 E2 2 I
5
.. . 2 En.
n . I t e r a t i n g n t i m e s , we g e t t h e
L e t u,w be two d i s t i n c t elements o f E n . By o u r h y p o t h e s i s , t h e
succestrans-
p o s i t i o n ( u , w ) changes A; s o ' t h e r e e x i s t elements x ~ , . . . , x ~i n- t~h e base IAI, such t h a t (u,w), extended by t h e i d e n t i t y on a l l x ' s , changes t h e r e s t r i c t i o n A/{u,w,xl, n ) such t h a t E h -
. . . , x ~ - ~ )There . e x i s t s a t l e a s t one i n t e g e r h (15 h
- Eh
5
does n o t have any x among i t s elements.By t h e p r e -
191
PRESENT PROBLEMS ABOUT INTERVALS
c e d i n g statement 2.4,
t h e t r a n s p o s i t i o n ( u , v ) extended by i d e n t i t y ,
automorphism f o r Ah= A/E,
-
Eh U ( I A l
Nh = NO/Eh on I A l
- Eh
is
but not f o r the r e s t r i c t i o n o f A t o t h e
an
union
Eh -l). However i n t h i s r e s t r i c t i o n , any l o c a l automorphism
of
i s a l o c a l automorphism o f Ah and i s e x t e n s i b l e by t h e i d e n t i t y -1
.
Consequently, g i v e n any t h r e e elements u < w < v
(modulo Nh)
i n E l l , t h e t r a n s f o r m a t i o n o f u i n t o v i s a l o c a l automorphism o f A,extensib y t h e i d e n t i t y on a l l elements o f ECL which a r e < u o r > v ,
ble
t h e i d e n t i t y on I A I
-
Eh-l;
and
by
b u t n o t e x t e n s i b l e by t h e i d e n t i t y on IAI-€h-l
i n c r e a s e d by t h e element w . So t h a t t h e s e t o f elements h a v i n g an even rank i n E h , f o r i n s t a n c e , i s n o t an A - f i n i t e - V a l :
c o n t r a d i c t i o n p r o v i n g o u r con-
c l u s i o n ( 1 ) . The c o n c l u s i o n ( 2 ) f o l l o w s f r o m 1.1. 0
5 , THE
SUBVAL,
AN INTERMEDIARY
BETWEEN INTERVAL
NOTION
AND F I N I T E - V A L , Given a r e l a t i o n A, a subset 0 o f t h e base I A l w i l l be c a l l e d a n
d
subvnL when, f o r any l o c a l automorphism
IAl
- D , either Q
i s n o t e x t e n s i b l e by t h e i d e n t i t y on G, o r
by t h e i d e n t i t y on any subset o f D*, phism o f 3.1.
6
A-
D*
o f A/D, and any subset G o f
=
i s extensible
o b t a i n e d f r o m G by any l o c a l
automor-
A/D*.
Any i n t e h v d a h e x t e h v d -in a n u b v d .
D be an e x t e r v a l and E t h e base, Q be a l o c a l automorphism o f AID, w i t h do-
D T h i s i s obvious f o r an i n t e r v a l . L e t so t h a t E - 0 i s an i n t e r v a l . L e t main F and codomain
F ' , and l e t g be a l o c a l automorphism o f A/(€-D),
domain G and codomain G ' . Suppose t h a t on G: we have t o p r o v e t h a t The u n i o n g U IF'
6
6
6
i s a g a i n e x t e n s i b l e b y t h e i d e n t i t y l G , on
U IG b e i n g a l o c a l automorphism, and
i s a l o c a l automorphism and so
u
6
U g, we g e t
lG G I .
E - D an i n t e r v a l ,the u n i o n
g by composition ( s e e a l s o 1.5).
Noreover g U I F and t h u s g - l U I F a r e l o c a l automorphisms: with
with
i s e x t e n s i b l e by t h e i d e n t i t y
6 u lG,. 0
by
composition
T h e r e e x i s t s a subval which i s n e i t h e r a n i n t e r v a l , n o r a n e x t e r v a l . T a k e t h e c y c l i c o r d e r i n g r e l a t i o n a l r e d y d e f i n e d i n 2,from t h e t o t a l o r d e r i n g o f n a t u r a l i n t e g e r s : f o r any i n t e g e r u, t h e s e t o f i n t e g e r s ? u i s a subval. cause i f u = 0, t h i s i s t h e whole base; i f u ? 1, among i n t e g e r s
any
local
Be-
automorphism
u , i s e x t e n s i b l e by t h e i d e n t i t y on t h e s i n g l e t o n
{Ol, i f f
i t preserves t h e usual t o t a l o r d e r i n g between i n t e g e r s ; and i n such a case, i t i s e x t e n s i b l e by t h e i d e n t i t y on a l l t h e i n t e g e r s < u.
Now
it i s
suffi-
192
ROLAND FRA'I'SSE
2
c i e n t t o take u
2
2, i n o r d e r t h a t t h e s e t o f i n t e g e r s
u be n e i t h e r
an
i n t e r v a l n o r an e x t e r v a l : we a l r e a d y n o t i c e d i n 2 t h a t t h e o n l y i n t e r v a l s o f t h e c y c l i c o r d e r i n g a r e t h e empty s e t , t h e base and t h e s i n g l e t o n s . Anq n u b v d b
3.2.
R
binite-ud.
0 L e t n be t h e a r i t y o f o u r r e l a t i o n A; we know t h a t , i n o r d e r f o r D t o i t i s s u f f i c i e n t t h a t sequences o f l e n g t h p 5 n f a l l i n t o
be a f i n i t e - V a l ,
a f i n i t e number o f e q u i v a l e n c e classes,each of t h e s e c l a s s e s b e i n g sodefined, t h a t e q u i v a l e n t sequences g i v e a l o c a l automorphism e x t e n s i b l e by on any s e t o f M - p elements o u t o f Q; and a l l t h a t f o r e a c h p pose
2
identity
M. Now sup-
D be a subval. F o r each q 5 n, c o n s i d e r as e q u i v a l e n t any two subsets (where E i s t h e base) w i t h c a r d i n a l q , when t h e r e s t r i c -
G and G' o f E - D
a r e i s o m o r p h i c . Take a r e p r e s e n t a n t i n e a c h
o f these
e q u i v a l e n t classes, and l e t H be t h e f i n i t e s e t , u n i o n o f t h e s e
represent-
t i o n s A/G and A/G'
ants. Now c o n s i d e r any two sequences u,u i n D, w i t h a same len,gth 5 M , as e q u i v a l e n t when t h e t r a n s f o r m a t i o n o f u i n t o u i s a l o c a l automorphism o f A , e x t e n s i b l e by t h e i d e n t i t y on H . As H i s a f i n i t e s e t , t h e r e a r e f i n i t e l y
D
many c l a s s e s o f e q u i v a l e n t sequences. I t i s s u f f i c i e n t now t o n o t e t h a t ,
b e i n g a s u b v a l , t h e t r a n s f o r m a t i o n o f u i n t o an e q u i v a l e n t s e q u e n c e u o f length
p 5 M,
if' i t i s
e x t e n s i b l e by t h e i d e n t i t y on H , i s a l s o e x t e n s i b l e
by t h e i d e n t i t y on any subset o f E - Q
with cardinal
a subset G t h e r e e x i s t a subset G ' o f H w i t h A/G'
n - p , since f o r i s o m o r p h i c t o A/G
There e x i s t s a f i n i t e - V a l which i s n e i t h e r a subval
nor
the
such
.0
comple-
mentary s e t o f a subval. To see i t , n o t e t h a t , f o r a t o t a l o r d e r i n g r e l a t i o n A
, the
A-subvals a r e e x a c t l y t h e A - i n t e r v a l s and t h e A - e x t e r v a l s .
Indeed
f o r any s e t D which i s n e i t h e r an i n t e r v a l n o r an e x t c r v a l , t h e r e e x i s t s x ,
q, i n D and z , R o u t o f D w i t h x < z < q < t o r R < x < z < q. I n any case t h e t r a n s f o r m a t i o n o f x i n t o q i s e x t e n s i b l e by t h e i d e n t i t y o n t h e s i n g l e t o n {t} b u t n o t b y t h e i d e n t i t y on { z } .
Now t h e u n i o n o f t w o
intervals
a d m i t t i n g a n o t h e r i n t e r v a l between them and a n o t h e r a f t e r , f o r a t o t a l d e r i n g , i s a f i n i t e - V a l which i s n e i t h e r a subval n o r t h e complementary
orof
a subval. There e x i s t s a subval whose complementary s e t i s n o t a s u b v a l . Indeed, t a k e t h e p a r t i a l o r d e r i n g r e l a t i o n o b t a i n e d f r o m two t o t a l o r d e r i n g s A and
8, w i t h any element o f I A l uncomparable t o any element o f 181. Take a median i n t e r v a l 8' o f 8, and c a l l D t h e u n i o n o f I A l and 18'1. Now Q i s a subVal, s i n c e any l o c a l automorphism i n D, e i t h e r t r a n s f o r m s an e l e m e n t o f I A l i n t o an element o f I B ' I , o r conversely, and t h e n i s n o t e x t e n s i b l e t o
193
PRESENT PROBLEMS ABOUT INTERVALS
any i d e n t i t y out of D; o r i t i s t h e union of a local automorphism of A and of one of B ' , and then i s e x t e n s i b l e t o the i d e n t i t y on E - D (where E i s the base). However E - D i s n o t a subva1,sincethe transformation of an element of I B I , a n t e r i o r t o B ' , i n t o an element p o s t e r i o r t o R ' , i s e x t e n s i b l e by t h e i d e n t i t y on t h e s i n g l e t o n of any element of I A l , b u t not on t h e singleton of any element of 18'1. F i n a l l y , note t h a t the notion of subval i s not preservedby f r e e i n t e r p r e t a b i l i t y , as i s t h e notion of f i n i t e - v a l . Let C be a chain, o r t o t a l ordering, A a unary r e l a t i o n on the base I C I , w i t h value + f o r elements of a C - i n t e r v a l , and B another s i m i l a r unary r e l a t i o n , corresponding t o a C-interval d i s j o i n t from the former, and even l e t t i n g a t l e a s t one element between them, and a t l e a s t one element a f t e r them. Mow t h e union of these Ci n t e r v a l s A and R i s n e i t h e r an i n t e r v a l nor an exterval f o r C , thus i t i s not a C-subval. However i t i s an ( A €3 C)-subval, where A 8 C i s t h e multirel a t i o n of components A , 8 , C. Indeed i f 6 i s a local automorphism of A €3 C i n s i d e t h e union of C-intervals A and 8 , then 6 cannot transform an e l e ment of t h e C-interval A i n t o an element of t h e C-interval 8 , b e c a u s e A takes value + in t h e former a n d - i n t h e l a t t e r . So 6 i s the union o f a local automorphism of C i n s i d e A and a l o c a l automorphism i n s i d e B , and then 6 i s e x t e n s i b l e by any i d e n t i t y o u t of t h e union of i n t e r v a l s A and B
.
of an
PROBLEM: I f A i s f r e e - i n t e r p r e t a b l e i n a chain, the complementary set A-subval i s i t an A-subval?
4 , THE
RELATIVE
INTERVAL,
Note t h a t t h e empty s e t i s a r e l a t i v e i n t e r v a l , with t h e whole base as i t s bound; t h e whole base i s a r e l a t i v e i n t e r v a l , the unique one w i t h t h e empty s e t a s i t s bound; t h e s i n g l e t o n of any element u , with t h e base minus u as a bound ( o t h e r bounds can e x i s t ) . Note t h a t f o r a given bound F, t h e condition about s e t s D ' in our introduction: "any local automorphism of A/D' i s e x t e n s i b l e by the i d e n t i t y o n F 'I, i s an inductive condition. P r e c i s e l y , i f f o r a same F, we have an ascending chain of s e t s D' ( f o r i n c l u s i o n ) , t h e i r union i s s t i l l a 0 ' . Indeed i f n i s t h e a r i t y of A , f o r being a l o c a l automorphism of A / D ' and being e x t e n s i b l e by t h e i d e n t i t y on F, i t i s s u f f i c i e n t t h a t a b i j e c t i o n 6 h a v e a l l i t s r e s t r i c t e d b i j e c t i o n s t o any s e t with cardinal p 5 M , s a t i s f y i n g the same property, reduced i t s e l f t o s u b s e t s of F with cardinal n - p , and
194
ROLAND
FRAYSSE
...,
that f o r p = 1,2, n. As an immediate Consequence, given a subset F o f the base, a n y s i n g l e ton o u t o f F being obviously a s e t 0’ w i t h t h e above mentionned p r o p e r t y , there e x i s t s an (A,F)-interval
which includes t h i s s i n g l e t o n . I n o t h e r words,
$he union 06 a&? (A,FJ-intenw&
0 IAl
- F.
Other consequence: id G_c F, any ( A , F ) - i n t e h w d 0 i n c l u d e d
i n an
(A,G) -int&t~d.
Foh a b i n m j h e l u t i o n A, and a divLite nubnet F
4.1.
06
L2.b bane, Mehe
arre divLiteRy many (A,F)-intehv&. 0 Consider as e q u i v a l e n t any two elements u,v o f I A l
-
F,when t h e t r a n s -
formation o f u i n t o w i s a l o c a l automorphism o f A, e x t e n s i b l e by t h e ident i t y on F. Since F i s f i n i t e , t h e r e a r e f i n i t e l y many classes i n t h i s equiv-
alence. Then any i n t e r v a l w i t h
F as a bound , as soon as i t contains an e l -
ement u , has t o c o n t a i n a l l the c l a s s o f u; as i t i s seen
6
bijection
because
a
i s a l o c a l automorphism o f A i f f i t s r e s t r i c t e d b i j e c t i o n s
to
any p a i r o f elements o f t h e domain, a r e themselves l o c a l automorphisms. 0 Note t h a t the statement does n o t extend t o t e r n a r y r e l a t i o n s . For i n stance, take the c y c l i c o r d e r i n g defined i n 2, from
the usual
chain
n a t u r a l i n t e g e r s , and take f o r F the s i n g l e t o n o f 0. Each s i n g l e t o n integer
uf- 0
s a t i s f i e s the condition o f e x t e n s i b i l i t y t o
o f
of an
F o f any lo-
cal automorphism ( n e c e s s a r i l y reduced t o t h e i d e n t i t y on u ) ; and t h e addit i o n o f another i n t e g e r
w
# Oand # u i s impossible,the
transposition
(u,w) being a l o c a l automorphism o f the c y c l i c ordering, i n e x t e n s i b l e
the i d e n t i t y on 0. So any s i n g l e t o n o f an i n t e g e r bound
+0
i s an
by
interval with
F.
4.2. Foh a binahy h e L d o n A, and a bu64et F 04 .LO b a e , i6 U, V ahe each a n (A, F) - i n t e t t v d , and id doh any dement x 0 6 $he union U U V , ,thehe e d t a g 06 $he i n t e M e c t i o n U il V , t h e t t a m ~ o h m a t . i o n06 x i n t o g being a Local automohpkinm 06 A, t h e n U = V.
o According t o the maximality o f r e l a t i v e i n t e r v a l s , f o r i n c l u s i o n , i t i s s u f f i c i e n t t o prove t h a t the union U u V i s an (A,F)-interval, o r simp l y s a t i s f i e s t h e e x t e n s i b i l i t y c o n d i t i o n . I t i s even s u f f i c i e n t t o p r o v e i t from a l o c a l automorphism o f A/(U U V ) , w i t h a domain o f o n l y one e l e -
ment
. Let
x and z be two elements o f the union U U V , t h e
transformation
o f x i n t o z being a l o c a l automorphism. By hypothesis t h e r e e x i s t s an e l ement g o f the i n t e r s e c t i o n , the transformation
o f x i n t o y being
a
lo-
PRESENT PROBLEMS ABOUT INTERVALS
195
cal automorphism, and so the transformation of y i n t o z. S i n c e x and y belong both t o U , o r both t o V , t h e f i r s t l o c a l automorphism i s e x t e n s i b l e by the i d e n t i t y on F; same r e s u l t f o r the second, and f i n a l l y f o r t h e transformation of x i n t o z. o PROBLEM: For a t e r n a r y r e l a t i o n , e x i s t e n c e of two d i s t i n c t i n t e r v a l s , with a same bound, each r e s t r i c t i o n in the union, with cardinal 2, h a v i n g an isomorphic r e s t r i c t i o n in t h e i n t e r s e c t i o n . More s t r o n g l y , each f i n i t e r e s t r i c t i o n i n the union, having an isomorphic r e s t r i c t i o n i n t h e i n t e r section.
Note t h a t 1 . 7 does not s u b s i s t f o r r e l a t i v e i n t e r v a l s . For i n s t a n c e , take a base E of four elements u , x , y, z; t h e s e t F = {u}; t h e unary r e l a tion A with value + f o r x, z, u and - f o r y; t h e binary symmetrical r e l a t i o n R with value + f o r ( u , x ) , ( u , z ) and t h e i r converses, value - i n o t h e r cases. Now the p a i r {x,yl. i s an (AB,F)-interval: t h e element z c a n n o t be added s i n c e the transformation of x i n t o z i s a local automorphism inextens i b l e t o the element U . Take U = {y,z,u} and consider the r e s t r i c t i o n t o 27 of A 8 , the bound F being unchanged. Now t h e i n t e r s e c t i o n of U and the i n t e r v a l { x , y l reduces t o t h e s i n g l e t o n { q l : i t i s n o t an (AB/U,F)-interVal, being not maximal f o r i n c l u s i o n , s i n c e iii t h e p a i r { y , z l , t h e t r a n s formation of y i n t o z i s not a l o c a l automorphism.
5, CLOSURES O F
A RELATION; COMPACT R E L A T I O N ,
These c l o s u r e s , already defined i n F r a f s s e 1974, p . 135, extend t o rel a t i o n s t h e c l o s u r e procedure whereby t h e chain of real numbers i s derived from the chain of r a t i o n a l s . Given a r e l a t i o n A of base E , we d e f i n e an A-@3e4 as a s e t 3 of nonempty A-intervals ( a b s o l u t e i n t e r v a l s ) s a t i s f y i n g the following c o n d i t i o n s : ( 1 ) any A-interval including an element of 3 i s a l s o an element o f 3 ; ( 2 ) t h e i n t e r s e c t i o n of two elements of 3 i s an e l c n c n t of 3 ( i t i s an A-interval , by 1 . 2 ) . A maximal A - f i l t e r w i l l be c a l l e d an A-uLtm6i-eteh. Any A - f i l t e r m a y be extended t o an A - u l t r a f i l t e r . An A - u l t r a f i l t e r i s s a i d t o be L k i u i d i f i t c o n s i s t s of a l l A-intervals including a s i n g l e t o n . I f t h e A - u l t r a f i l t e r i s not t r i v i a l , t h e i n t e r s e c t i o n of a l l i t s elements i s empty, and each e l ement i s i n f i n i t e . For a given A-interval 27 and A - u l t r a f i l t e r 3 , e i t h e r U i n t e r s e c t s each element of 3 , and i s t h e r e f o r e an element o f 3 , o r t h e r e
196
ROLAND
FRA'I'SSE
e x i s t s an element of 5 d i s j o i n t from D. Consequently, i f 3 and 3' are distinct A-ultrafilters,thereexist elementsD of 3 a n d D ' of 3' with D and 17' disjoint. Let us complete the base € by embedding.itin the s e t E* of a l l A - u l t r a f i l t e r s : by identifying each t r i v i a l A - u l t r a f i l t e r with the element of € generating i t . With each non-trivial A - u l t r a f i l t e r 3 l e t us associate a relation A ( 8 ) with base € ( a ) and the same a r i t y as A . The bases E ( 3) are assumed t o be d i s j o i n t from € and from one another. The relations A ( S) are subjected t o the following condition, which may always be s a t i s f i e d : f o r any f i n i t e subset F of € ( b ) and any element 0 of 3, there e x i s t s a t l e a s t one isomorphism of the r e s t r i c t i o n A ( 3 ) / F o n t o a r e s t r i c t i o n of A / D . Once the relations A ( 8 ) have been chosen (some of them may have empty bases), the c h u t e At of A i s unavbiguously defined as follows on the union €+ of E and the s e t s E ( 3) f o r a l l 3. Let It be the a r i t y of A a n d xl, ..., x , elements of €+. I f some x . belongs t o E , we replace i t by X I 4. = x4. and say t h a t i t i s fixed. Now g r o u p a l l xi lying in a same s e t € ( 3 ) t o gether, and consider t h e i r images under a local isomorphism of A (3) towards A/D, where 0 i s an element of 3. Denote these images by xIi; we s t i p u l a t e t h a t d i f f e r e n t members D of d i f f e r e n t u l t r a f i l t e r s 5 be pairwise d i s j o i n t and contain no fixed elements xi. We then s e t A+(xl, x,) = A ( x ' ~ , x ' ) ; t h i s value i s independent of the s p e c i f i c intervals D and isomorphisms chosen (by 1 . 6 ) . Let us recall two problems of the above reference:
...,
...,
PROBLEM 1 . Given a relation A and a nontrivial A - u l t r a f i l t e r 3 , assoc i a t e an empty relation with every other nontrivial A - u l t r a f i l t e r . Does there e x i s t a nonempty relation A ( 3 ) giving as closure a logical (element a r y ) extension of A ? PROBLEM 2. Let 5 and be two nontrivial A - u l t r a f i l t e r s , A ( 3 ) and A ( 6 ) relations each of which yields a logical extension of A via closure (the relations associated with a l l other A - u l t r a f i l t e r s being empty).Is the extension obtained by considering A ( 3 ) and A ( q ) a logical extension of A ?
5.1. ement of pact i f , each 3,
Starting from a s e t €, consider a s e t of f i l t e r s 2 on E , each e l such a f i l t e r 3 being a subset of €. Call t h i s s e t of f i l t e r s camf o r any choice function 6 such t h a t 6 ( 2 ) i s an element of 3 f o r there e x i s t f i n i t e l y many 3 ' s such t h a t t h e u n i o n of t h e s e t s
PRESENT PROBLEMS ABOUT I N T E R V A L S
d ( 3) i s
197
E . Fa4 any n e t E , t h e b e t oh aPl LLeRhndilReh) o n E i n a c o m p a c t
6 e R . 1 n d e e d s u p p o s e t h e c o n t r a r y : t h e r e e x i s t s a c h o i c e f u n c t i o n d such t h a t , f o r
a n y f i n i t e s e t U o f u l t r a f i l t e r s , E u = E minus t h e union of 6 ( 3) f o r a l l of U, i s nonempty. So t h e s u p e r s e t s of the € [ , I S c o n s t i t u t e a f i l t e r on E ; take a f i n e r u l t r a f i l t e r : i t would be an u l t r a f i l t e r on E , d i s t i n c t f r o m a1 1 ul t r a f i l t e r s (communi cated by FI. J e a n ) . Coming back t o r e l a t i o n s , we say t h a t a r e l a t i o n A i s c o m p a c t when the s e t of a l l A - u l t r a f i l t e r s i s compact. By t h e preceding r c m a r k , any unary r e l a t i o n o r m u l t i r e l a t i o n i s compact. By t h e same a r g u n e n t , i t i s seen t h a t any chain i s a compact r e l a t i o n . A hellLtion A L8 compact i56, camplemeiztahlj h e t vah.
604
any h i n i f e u n i o n 0 6 A - i n t e h u a h , t h e 06 A - i n t e h -
(hedehently t o t h e babel .in a & X t e uiuon
Let us prove f i r s t l y t h e following lemma. CnCe E die b a e
0 5 A;
nuppo4e t h a t , do4 any A-intehual 0, t h e
cample-
dinitc union o6 A-intehua&. T h e n a n y Ua,tmd.iet e a a n E, once heduced t o A-in-tehu&, g i v e n an A-uetha6iLteh. nieil*cutj 4 e L
E-
D
i 4
a
Note t h a t , f o r t h e c o n s e c u t i v i t y C on natural i n t e g e r s , t h e only C-int e r v a l s being t h e empty s e t , t h e whole base and s i n g l e t o n s , an:' non t r i v i a l u l t r a f i l t e r , once reduced t o C - i n t e r v a l s , gives only t h e base: t h i s i s not a C-ul t r a f i l t e r . 0 Let 3 be an u l t r a f i l t e r on E , and A t h e s e t of a l l A - i n t e r v a l s b e longing t o 3. Suppose A i s not an A - u l t r a f i l t e r : t h e r e e x i s t s ar A-interVal D with D B A and D i n t e r s e c t s any element of A. The c o m p l e m e n t a r y s e t E - U E 3. By hypothesis E - D i s a f i n i t e union of A-intervals: so one of them U* E 3 and consequently U* 6 A. F i n a l l y D and D* a r e d i s j o i n t s e t s ; on another s i d e U i n t e r s e c t s any element of A, and so D i n t e r s e c t s D*: c o n t r a d i c t i o n . o
Proof of the proposition :
0 Suppose t h a t , f o r any f i n i t e union of A - i n t e r v a l s , t h e c o m p l e m e n t a r y s e t i s s t i l l a f i n i t e union of A-intervals. I f A i s not compact, t h e r e i s a choice function 6 such t h a t , f o r any f i n i t e s e t U of A - u l t r a f i l t e r s , EU = E minus union of 6 ( 3 ) ' s f o r a l l 3 o f LI, i s nonempty. Then the s u p e r s e t s o f EUis c o n s t i t u t e a f i l t e r on E . Take a f i n e r u l t r a f i l t e r , and reduce i t t o A-intervals: t h i s gives an A - u l t r a f i l t e r , by t h e above lemma. Call i t V
ROLAND FRA'I'SSE
198 and n o t e t h a t
6(v) 6 v.
F o r each f i n i t e s e t U o f A - u l t r a f i l t e r s ,
i s by
EU
h y p o t h e s i s a f i n i t e u n i o n o f A - i n t e r v a l s : one o f them belongs t o V. T a k i n g U = {Vl, we g e t
EU = E -
€ - 6(V)
cluded i n
6 (v); t h e r e
v which i s 6(v)-: c o n t r a d i c t i o n ; so A i s
e x i s t s an element o f
, thus d i s j o i n t from
incom-
pact. o f A-
Conversely suppose A compact; suFpose t h e r e e x i s t s a f i n i t e s e t intervals
D, with
€ minus u n i o n o f D ' S b e i n g n o t a f i n i t e u n i o n o f A - i n t e r -
v a l s . To each A - u l t r a f i l t e r 5 , a s s o c i a t e
6( 3) b e i n g
a 0 belonging
to
5;
o r 6(5) element o f ? a n d d i s j o i n t f r o m each D. W i t h t h i s c h o i c e f u n c t i o n , no f i n i t e u n i o n o f 6(3) can g i v e €: c o n t r a d i c t i o n . 0
6
I
PROBLEMS ABOUT I N T E R V A L S t SUMS t PRODUCTS 8 I N C O N N E C T I O N
WITH
EHRENFEUCHT'SLOGICAL
STUDY O F ORDINALS
I
I n F e f e m a n 1357, s e v e r a l i m p o r t a n t r e s u l t s due t o E h r e n f e u c h t a r e r e lated. <
F o r i n s t a n c e , t h e c l a s s o f a l l o r d i n a l s , w i t h t h e usual
, admits
comparison
as a l o g i c a l ( e l e m e n t a r y ) r e s t r i c t i o n , t h e r e l a t i o n < o n
n a l s l e s s t h a n w".
Analogous r e s u l t w i t h
t h e comparison
<
ordi-
andusurn,
s m a l l e s t l o g i c a l r e s t r i c t i o n b e i n g (<,+) on o r d i n a l s l e s s t h a n a("
gous r e s u l t w i t h t h e comparison, sum and p r o d u c t , t h e s m a l l e s t l r g i c a l s t r i c t i o n being
the
Analore-
(< , +, ,) on o r d i n a l s l e s s t h a n w power ( w ( ~ ' ) ) , o r w su-
perpower 4 . Moreover E h r e n f e u c h t proved t h e d e c i d a b i l i t y o f t h e o r i e s
in
<
and i n ( < , +); t h e t h e o r y i n (< , +, , ) b e i n g o b v i o u s l y u n d e c i d a b l e s i n c e i t g i v e s e a s i l y a r e p r e s e n t a t i o n o f t h e a r i t h m e t i c w i t h + and on n a t u r a l I
integers. I n a general manner, g i v e n a c l a s s E and a ' r e l a t i o n ' A d e f i n e d on E t h e r e a d j u s t e d Lbwenheim-Skolem theorem always g i v e s a denumerable
,
logical
r e s t r i c t i o n . B u t i t i s n o t a t a l l obvious t o see what r e s t r i c t i o n i s conven i e n t , and t h e elements o f t h e base o f t h e denumerable l o g i c a l
restriction
a r e n o t themselves always denumerable s t r u c t u r e s . F o r i n s t a n c e , l e t us s t a r t f r o m t h e c l a s s o f a l l chains, w i t h t h e comparison 5
, defining
A 5 B when t h e r e e x i s t s a r e s t r i c t i o n o f B which i s i s o m o r p h i c a l t o A. T h i s comparison i s r e f l e x i v e , t r a n s i t i v e , b u t n o t a n t i -
symmetric, even i f we i d e n t i f y i s o m o r p h i c c h a i n s : i f w- i s t h e chain o f the o r d i n a l w, then a c h a i n i s o m o r p h i c t o w-,
is
symmetric
@(om),where each element o f w i s r e p l a c e d by
5
and
t o 1+ w(w-),
o f a f i r s t element. We c a l l equimohphic such c h a i n s
obtained by
addition
5 and 2 each t o
the
199
PRESENT PROBLEMS ABOUT INTERVALS
other. Also the chain
2
+ o.
o f r a t i o n a l s i s equimorphic t o Q + 1 o r t o Q
Now l e t us c o n s i d e r t h e t h e o r y o f comparison between chains, analogous t o E h r e n f e u c h t ’ s t h e o r y among o r d i n a l s . There a r e s e v e r a l t h e o r i e s , t h a t a r e p r e c i s e d i f we i n t e r p r e t t h e i d e n t i c a l symbol = as t h e i d e n t i t y
between
chains, o r as isomorphy, o r as equimorphy between c h a i n s . A f i r s t p r o b l e m i s t o know i f these t h e o r i e s a r e d e c i d a b l e . A second problem i s t o f i n d d e numerable l o g i c a l r e s t r i c t i o n s . It i s e x c l u d e d t h a t t h e c l a s s o f a l l d e n u merable chains g i v e s a l o g i c a l r e s t r i c t i o n : indeed t h e c h a i n O_ o f r a t i o n a l s is
each denumerable chain; so t h e model o f a l l denumerable chains
sat-
i s f i e s t h e f o r m u l a 3 if g 5 x , o b v i o u s l y unacceptable f o r t h e c l a s s o f a l l X L I chains. I t i s a l s o excluded t o t a k e o n l y f i n i t e c h a i n s , because we w o u l d have t h e f o r m u l a s a y i n g t h a t any c h a i n admits an i m m e d i a t e l y
inferior
chain. Furthermore i t i s e x c l u d e d t o t a k e o n l y f i n i t e and some
denumerable
chains. Indeed by a known theorem due t o Dushnik and M i l l e r 1940,
f o r any
c h a i n X s t r i c t l y s u p e r i o r t o O_ ( r a t i o n a l s ) and i n f e r i o r t o R ( c h a i n o f r e a l
. This
numbers), t h e r e e x i s t s a n o t h e r c h a i n < X and s t i l l >
irnpliesan i n -
f i n i t e s t r i c t l y decreasing sequence, t h a t i s f a l s e f o r denumerable c h a i n s
,
by Laver 1971. An analogous problem i s asked by c o n s i d e r i n g t h e c l a s s o f a l l
rela-
t i o n s o f a g i v e n a r i t y , w i t h t h e same comparison: e x i s t e n c e o f a r e s t r i c t i o n o f B i s o m o r p h i c a l t o A . Even t h e problems about a d d i t i o n a n d m u l t i p l i c a t i o n , which a r e o b v i o u s l y d e f i n e d among c h a i n s , can be asked f o r t h e c l a s s o f a l l r e l a t i o n s . Indeed we can say t h a t a r e l a t i o n C i s a
SUR
o f A and B
if
the
base C i s t h e u n i o n o f A and 8 , and i f any u n i o n o f a l o c a l automorphism o f A and a l o c a l automorphism o f B , i s a l o c a l automorphism o f C (see
Frafssc
1973, p. 108).
REFERENCES, Dushnik, B . and E. W . M i l l e r 1940, Concatlning
4 M a h i t c j & ~ 7 ~ n ~ u h m a t i 06 o ~ nU
Amer. Math. SOC., v o l . 46
n e d y ohdehed
4eh,
Bull.
, 322-326.
Feferman, S. 1957, Some hecent wo&
06
Ehhen6eucht and
b o l i c L o g i c , mimeographied, 201-209.
F&4Z,
Summer I n s t i t u t e o f Sym-
200
ROLAND FRA'I'SSE
Foldgs, I . 1973, Retktiom demen e l dinpeh;l&~; ex.temian d ' u n thEvt&e
de Hawdot6~,
C. R. Acad. Sc. P a r i s , 277 ( A ) , 269-271. Frafss6,
R.
1973, Course of Mathematical Logic, v o l
.
1974, Course of Mathematical Logic, v o l .
1, R e i d e l , Dordrecht.
2, R e i d e l , D o r d r e c h t .
Laver, R. 1971, On
Ft&sC's
o x d e t ,type canjectuhe, Thesis, B e r k e l e y 1969, and Annals
Math. v o l . 93, 89-111.
Dgpartement de Mathgmat iques U n i v e r s i t g de Provence M a r s e i l l e , France.
Non-Classical Logics, Model Theory and C o m p u t a b i l i t y , A.I. Arruda, N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland P u b l i s h i n g Company, 1977
F'ROJECTIVE LOGICS AND PROJECTIVE BOOLEAN ALGEBRAS (*I by R I C A R D O hlORAIS
I, INTRODUCTION, Lusin and S i e r p i n s k i s t a r t e d , i n 1325, the development of the theory of p r o j e c t i v e sets ( c f . Lusin 1925 and Sierpinski 1925) b u t soon afterwards t h i s research p r a c t i c a l l y ended due t o the complexity of the problems and the lack of b e t t e r t o o l s t o work with. I t was not u n t i l t h e l a t e s i x t i e s t h a t some new r e s u l t s were ( c f . Fenstad 1971, Moschovakis 1973, and Kechris 1973) using t h e proposed, b u t s t i l l questionable, axiom of Addison and Moskovakis 1968). Recently A. Nerode thought of develop
obtained recently
" P r o j e c t i v e Determinancy"
a l o g i c LA t h a t would b e s u i t -
a b l e f o r t h e study o f t h e a n a l y t i c s e t s , o r p r o j e c t i v e s e t s of level L
WlW
(cf.
was t o Bore1 s e t s . This was done by h i s s t u d e n t P . J . Campbell,
1 , as and
f u r t h e r strengthened in E.Ellentuck 1975 u s i n g d i f f e r e n t methods. This paper follows a sugestion of Eilentuck of t r y i n g t h e g e n e r a l i z a t i o n of t h i s approach t o a l l l e v e l s of the p r o j e c t i v e hierarchy. [ * ) The a u t h o r i s i n d e b t e d t o P r o f e s s o r E l l e n t u c k f o r t o Rutgers U n i v e r s i t y (U.S.A.)
and
support.
20 1
h i s o r i e n t a t i o n , and
CAPES ( B r a z i l ) f o r
their financial
202
R I CARD0 MORAl S
11, PROJEC I V E LOGICS, 0, PRELIM NARI ES s
Let [hey]’ denote’the s e t of a l l n - t u p l e s (uZ, ...,w n ) of f i n i t e s e quences of natural numbers satisfying e ( ~ , )= ... = e ( ~ ~where ) , L(u)denotes the length of u . Throughout t h i s t ex t 6 and g will denote elements of %; n ; m, and k will and f o r The subsets
be natural numbers and u will stand both f o r elements of [neqJn”, elements of ney the s e t of f i n i t e sequences of natural numbers. most important tool developped in t h i s work i s a pair of families of of [hey]’, denoted respectively by FLLeen and F u l l ;
.
D E F I N I T I O N 1 . a) F E F L U , ( F 0 a Fd2l-n b e t ) i d and vney ii F c [neq]’ and F 402%6iu t h e 6aUvwing cvndition giuen by a 4.7XLng 0 6 quant.idiehn
w i t h n aetmnntionn: (3
I
d,)(v dz)*..(Q6,)(Q’k)((d,
I
k,**-,&,
k ) E F)
whme Q and 2’ ahe din-tinct yuanLL&iem. S , i m d ! d y we
b)
dedine
G E F u l l ; (G 0 a F L L e e - v ~ - n Z m n e t ) id and vney i d G c [ n e d n und
(v
61)(362 )-.*(Q’6,)(Qk)((d,
I
k 9 * * * r 6 n
I
k) E GI.
F and G will denote elements of F u l l and F u l l * respectively,and the word countable will always mean e i t h e r f i n i t e or denumerable. Finally, i f @ = {@iI i 8 I ) i s a countable s e t of formulas o f a formal I\ $ . or simply I \ ~ $ ~ language, the conjunction A @ will also be written i e 1 4. i f no confusion a r i s es .
1, THE
LANGUAGE L~~
We s t a r t with a f i r s t order logic L with countably many re la tion, funcbe the infinita rylogtion and constant symbols, and w l variables. Let L W1W
by adi c over L as defined in Keisler 1971, p . 6. We obtaion L from L Pn WIW joining two new operators P, and P; i n the following way: F o m a of L ar e those of L with the addition: i f @ i s amap from Pn W 1W [neq]” i n t o formulas of L then Pn(@) and P i ( @ ) are formulas of L Pn‘ Pn Extend the notion of n u t h 6 a o t i o n by defining:
203
PROJECTIVE LOGIC
i f f ( 1 F € F L L W M ) ( w w € F ) CR C$(w) [ h ]
[h]
(1) CR CP,($)
( 3 G € Full;)( W w € G) d ! = $ ( w )
iff
d bpi(@) I h ]
O c c m e ~ c ao f v a r i a b l e s and c o n s t a n t s i n P a ( @ ) and $(u) f o r
be those i n t h e c o l l e c t i o n o f a l l
w € [neq]'
[h!
P i ( @ ) are .
defined t o
D e f i n e d - w a e i d i t y and w a e i d i t y i n t h e usual way. The n o t i o n o f nub&mnula o f K e i s l e r 1971, p. 11 (2)
= {PM($)j
Sub(P,($))
i s extended by
uw S u b ( $ J ( v ) ) .
u
Uw S u b ( $ ( v ) ) *
S u b ( P ; ( @ ) ) = { P l ( @ ) }IJ
Extend t h e n o t i o n o f t,iouing Rhe c z e g d o a i n ~ i d e(cf. K e i s l e r 1971, p.11) as f o l l o w s : (3)
where
(P,J@)) 1
is
Pi(%)
(Pi($))1
is
P,(l@)
I
l$ i s t h e map t h a t takes w i n t o
Axivmn o f L
P"
l$(w).
w i l l c o n s i s t o f t h e n i n e axioms f o r p r o p o s i t i o n a l
as p r e s e n t e d i n B e l l and Slomson 1969, p . 36, p l u s t h e s i x axioms f o r as i n K e i s l e r 1971, p. 15.
logic
Lwlw
The Rda ad Irz@tence w i l l be (Rl)
Modus Ponens
(R2)
i f I-$
(R3)
i f f o r every
$ S @ t- J, +I$
(R4)
i f f o r every
F € FullM
(R5)
i f f o r every
G
P;($)
Vx$
t h e n + $+
+$
€
I-
if
x i s not free i n
t h e n I-$
AF$(w) + $
Full; I- J, + VG$(w)
+
$J
.
AQ.
I - P n ( $ ) + VJ.
then then
I- $ +
P,*($).
ble o n l y have t o w o r r y about i n t r o d u c i n g P,($) because t h e f a c t s a r e e a s i l y g o t t e n f r o m those about P,($) u s i n g t h e f o l l o w i n g
tance o f t h e axiom + $ 1
PROPOSITION 2 .
w
I-Pi(l$)
H
1 $
.
Now b e f o r e we proceed w i t h t h e s t u d i e s o f L
b.tatemk?VLt
abvut w
me Rnue meta
-
ins-
lP,($)
p o r t a n t p r o p e r t i e s o f t h e f a m i l i e s FuUM and
P R O P O S I T I O N 3.
about
l e t us c o l l e c t some
Pn FILee;
.
in-
Suppane t h a t dvti ewmy W E [ n e q ] ' , x(v) o a m e t a l& Q and Q' be bn in U e 6 i n i t i o n I . Then t h e ~vUviu&g
.
htatemenb.
204
RICARDO N O R A I S
a) ( 3 61)...(2dn)(2'k)A(d1
I
k,...,6n
b) ( V ~ , ) . . . ( 2 ' l i , , ) ( 2 k ) A ( d l I k , . . . , c) ( 3 F E F u U n ) ( W v E
F)A(w)
I
k)
dn 1 k )
id6
4 5 6 (3 F i d 6 (36
Ffin)(Vv
E Fuu;)(Wv
E F)A(v) 8
G)X(u)
( W G E Full;) ( 3 w E G ) A(") ( 3 G E FuU;)
d) ( W F E F u l l n ) ( 3 u E F ) A ( u )
E
(Ww 8 G)A(w)
I t i s important here t o note the complete symmetry between t h e s e two f a m i l i e s of sets. I t i s tnis symmetry t h a t w i l l make possible the majority of our proofs, besides e l e g a n t l y reducing a l l the work i n h a l f . we Now, in order t o b e t t e r understand the behavior of t h e s e f a m i l i e s , have:
PROPOSITION 4 .
16
IT
> 1,
and H c : s e q ] "
E
= { ( " p . . . , " , )I
(5 I
[(",),
, Let
"2,..*>"11)
E
HI.
Then
a) H E F u U n
.id6
(36) (Hd E
b) H E Full;
.id6
( v 6 ) (HS
Fa;-,)
E FLLeeYl-,)
F i n a l l y the next proposition shows t h a t the f a m i l i e s F u l l M and n.
a r e well mixed t o g e t h e r . The proof i s by induction on
2,THE CONSISTENCY PROPERTY
F&f;
I
Let C be a countable s e t of constant symbols not appearing i n L . Let c E C t o L , and from bl c o n s t r u c t t h e l o g i c M PM A banic tm i s e i t h e r a constant symbol of bl o r a term of the form P" function d(t t k )where R L,...,t k a r e b a s i c terms and 6 i s a k - a r y symbol o f L . but The d e f i n i t i o n t h a t follows was taken from Keisler 1971, , p . 11, clauses here, besides adapting i t t o the present s i t u a t i o n (namely adding C9 and C9) we a l s o modified, t o simplify the proofs, the notion o f b a s i c bl be t h e f i r s t order l o g i c obtained by adding each
.
,,...,
teroi, and clauses C10 and C11.
DEFINITION 6. n e s:
A CoiuDtency PhUpULty D a be* S batin6y.ing
doh
each
205
PROJECTIVE L O G I C
to
The d e f i n i t i o n o f Consistency P r o p e r t y i s t h i s l o n g because we want have:
THEOREM 7 . and
40 E S ,
PROOF:
(Model E x i s t e n c e Theorem). 16 S 0 a C a ~ n O t e n c y P h u p e h t g
then
Without
han a modet.
40
l o s s o f g e n e r a l i t y we way assume t h a t each subset o f an e l -
elllent o f S i s a g a i n i n S . To c o n s t r u c t t h e model s a t i s f y i n g w i t h t h e s m a l l e s t s e t Y o f f o r m u l a s o f E.4
(i)
no
(ii) Y
Let and
T =
f o r which:
i s c l o s e d under subformulas.
(iv) i f l@ E Y If
c E C
then
a b a s i c term and
1
@(t)E Y then I$(t') E Y
c = t E Y.
be t h e c o u n t a b l y i n f i n i t e s e t o f sentences o f Y ,
{to,tl, . . . I be t h e s e t o f b a s i c terms. S t a r t i n g w i t h
h0
construct
an i n c r e a s i n q sequence o f elements o f S as f o l l o w s . Suppose we have 4nl+l
4,,, :
.
$1 8 Y .
and R i s a b a s i c t e r m t h e n
X = { I $ o , I$,,...
and we b u i l d
start
Y
(iii) I f t i s a t e r m , t '
(v)
Pn
oo we
h,,,,
R I CARD0 MORA I S
206 (1)
i f A,, U
{@,,,Ig
(2)
i f sm U
{a,}
=
A;+,
;
i s V@ then f o r some $ E @,
(2.1)
$ ,,
(2.2)
$m i s 3 x $ then f o r some c E C, A;+,
(2.3)
@m
(2.4)
i s P,($)
IAF$(v)}
U
{$,,,I
U {AG@(v)}
f i n a l l y , s i n c e i n any case Am+,
Next d e f i n e
e sw
c = d
L e t [c]
u
= A;+,
sw
=
.
{c =
um sm
fml
&A+,
s+ ;,
=
E
S,
=
o;+~
=
sm u {$,,,I E S; t h e r e i s c E C such t h a t
8 S,
E S.
and d e f i n e an equivalence r e l a t i o n on C by: c+d
c E C and l e t A =
be the equivalence c l a s s o f
Now f o r each k - a r y r e l a t i o n symbol P,
6,
l
sm u {$m}U { $ ( c ) }ES,
G E F a ; ,
This i s the universe o f the model t h a t w i l l satisf.y symbol
@
E S,
i s any o t h e r formula,
$,
=
E S,
i s P i ( @ ) then f o r some
$,
(2.5)
U
u {$mlu I
= A,,,
F E FU.ee,,s;+,
then f o r some
u {@,,,I
= A,,
(3)
sm
we consider t h e f o l l o h i n g cases:
E S
= A,,
iff
let
S
o f L define a r e l a t i o n
so
I [c] i
.
o f L and each k - a r y
c E C}.
function
Rm on Ah and a f u n c t i o n F,, from
Ak
i n t o A by:
(a) (b)
,..., r e k ] ) E Rm Fm( [c,] ,. ..,[ch]) = [c,] ( [c,]
Note now t h a t i f is A@,
@ E 6,
then
iff
Pm(cl
,..., c),
iff
co =
b,,,(c,,
E
...,ck)
E
.
and
0 E Y
f o r each
(a)
@
(b)
@ i s any o t h e r formula, then
8 E @ ;
$ E Y.
Then use t h i s f a c t t o show t h a t the s t r u c t u r e
a satisfies
=
so
1
m 6 wl, {F,,,
I
m E w}, A >
.
Theorem 7 i s a n i c e t o o l t o use i n the p r o o f o f
THEOREM 8. (The Completeness Theorem f o r
L
)
Pn
16 @ 0 a s e n t e n c e
06
207
LOGIC
PROJECTIVE
To show t h a t e v e r y theorem i s v a l i d we Drove t h a t t h e r u l e s o f i n -
PROOF:
ference (R4) and (R5) p r e s e r v e v a l i d i t y . Rule (R4).
(a)
Suppose
3F E FuU,,
VF
E
F u l l R , 02 t=
U? b AF@(u) A 1J,
A F @ ( w ) + J, t h e n i t i s n o t
t h e case
.
i m p l i e s t h a t 3 F E FuRe,,
B u t by D e f i n i t i o n 1, Ce CP,(@) and t h e r e f o r e i m p 1 i e s n o t UL C P,($)
A 1$ o r e q u i v a l e n t l y ,
that
U? I = A F $ ( w ) ,
CL c P,($)+$.
Rule (R5).
(b)
d C J, * VG $ ( w ) .Then a!= 1ji o r Suppose VG E F u l l ; , ( 3 u E G ) OZ k = $ ( w ) a n d h e n c e b y P r o p o s i t i o n 3, CR
( W G 8 FLU;) 11) or
U? C J ,* P,($).
( I F E FullR)(W w E F) CE i=@(v), which i m p l i e s
Now we have t o show t h a t e v e r y v a l i d sentence i s a theorem. I n o r d e r t o do t h a t we l e t S be t h e s e t o f f i n i t e s e t s o f sentences n o f o n l y f i n i t e l y many
c E C
o c c u r i n n and n o t I-
MA.
M
YJn
such t h a t
We t h e n show S i s a Consistency P r o p e r t y and t h e r e s u l t f o l l o w s hence
lip,
t h e n @ i s n o t a theorem i n
cause i f @ i s n o t a theorem i n L
r7M
{ I $ } E S. By t h e Model E x i s t e n c e Theorem @:
beand
has a model and t h e r e -
fore @ i s not valid. We e x e m p l i f y t h e p r o o f t h a t S i s a Consistency P r o p e r t y be p r o v i n g (C8) and (C9). (C8)
Suppose P,($)
Full,
U {A,@(u)}
WF
(WF E F u l l , )
( I- A F @ ( w )
1An); t h e n by (R4)
since
E n,
P,(@)
P,(@)
P,*(@)
Suppose
I-An-,
( I-
S.
I- P,(@)
+
1A 0
and,
lAn, a contradiction.
E n b u t ( V G E FLU;)
I- l A ( n u { A , @ ( w ) } ) ( W G 6 FILL$)
I-
+
E
e
I- lA(n U { A F @ ( u ) } ) , and so
E n, we have
Since
(C9)
n b u t (WF E FuU,)(n
E
f o r every
(n U { A G @ ( u ) }g S ) ; t h e n again
G E Full;
and
so
An + V G l @ ( w ) ) , which i m p l i e s , by (R5),
P,(l@).
Therefore, by P r o p o s i t i o n 2, a contradiction.
I-
An + l P i ( @ ) o r , e q u i v a l e n t l y , I- l h n ,
206
R I CARDO MORA I C
There i s another p r o j e c t i v e l o g i c of i n t e r e s t t o us, namely: DEFINITION 9 . h ~ n bowm w,
The logic L
P
0 dedined t o be t h e u n i o n oh & L
o h i n othetr ~0oh.d~:
(a)
ln L
(b)
The &en
UA
n
Pn($) 0 a domda doh ewmy n .
P '
06
(R4)
(Vn E
(R5)
( i n E w ) id
id
w)
Pa
indmence (R4) and (R5) now head
( W F E FuUn) I- A F $ ( w )+ $
then
I-
(VG E FLU;)
then
I-$+ P,($).
I-$+ VG@(w)
i s complete s i n c e a l l L are. P Pa There i s one important theorem p a r t i c u l a r t o L
Pn($) + $,
Obviously L
THEOREM 1 0 . (R4')
In L
P
0 a h.u&
t h e @f%LCLing
= 1)
id
V6 I- Ak'$(d
(b) (doh. n > 1)
id
Vd I-
(a) ( d o h n
whehe
$d
(w2,.
..,wn)
= $( 6
:
06 in&?kence: k) --f
I .t(w,),
P
+
$
$
w2,.
then
then
I-
+
PI(@) + $ ;
pn($)+
n-
;
.., w n ) .
We conclude this s e c t i o n with the remark t h a t t h e downward Skolem-Tarski theorem holds f o r both L and L P Pn *
111,
$J
Lowenheim-
PROJECTIVEBOOLEAN ALGEBRAS,
1 I NTRODUCTI O N , I
In t h i s s e c t i o n we d e f i n e a new kind o f Boolean a l g e b r a s , c a l l e d n-proj e c t i v e Boolean a l g e b r a s , which a r e g e n e r a l i z a t i o n s of t h e S u s l i n algebras introduced by L . Rieger in 1955 ( c f . Rieger 1955). Our work, however, i s patterned a f t e r a recent paper by E . E l l e n t u c k (Ellentuck 197+) i n which he s t u d i e s the S - a l g e b r a s o f Rieger based on his previous paper on S u s l i n l o g i c (Ellentuck 1975). R i e g e r ' s idea with t h e S u s l i n algebras was t o provide a s t r u c t u r e i n which one could model nn1 a n a l y s i s .
1. BASIC
nt
a n a l y s i s . Our algebras a r e intended t o help model
RESULTS,
Let B be a Boolean algebra.
209
PROJECTIVE LOGIC
The joim and nieeA of B w i l l be denoted r e s p e c t i v e l y by Sup and 7ng. The iizditzite j o i n of t h e family {bi 1 i E I } i s denoted by Sup bi or simply by Sup bi
i
is7
i f i t c l e a r which s e t 7 i s .
I f Q i s a map from [neq!" i n t o 6 we s h a l l use the n o t a t i o n Pit($) f o r the following element of 73, provided i t e x i s t s : Pi*($)
=
and, s i m i l a r l y ,
SUP
F
Ini( $ ( w ) , wEF
where, a s u s u a l , F runs over The symbols P,(@) and and t h e previously defined confusion.
DEFINITION 1 2 . (W
- PBA
ijoh n h h t )
F u l l n and G over
FU.eek
.
P;($) w i l l be used both f o r t h e above suprema formulas o f Lpn, b u t t h i s should l e a d t o no
A u - B C J C J ~d~ gU e~ b~ t ~ ~ U M w - I 3 4 O j ! L d W t 600tea~d g c b h n id Lt 0 it - PBA dot C V U i ~ i E W.
Formula ( 4 ) i s a very powerful d i s t r i b u t i v e law and not a l l algebras closed under Pit and P i s a t i s f y i t . In f a c t , t h e r e a r e complete B o o l e a n algebras i n which ( 4 ) f a i l s . In our work, however, we need t h i s d i s t r i b u t i v i t y t o t i e t h i n g s up ( s e e f o r example condition ( 6 ) below), and we a r e thus forced t o introduce i t a s p a r t of t h e d e f i n i t i o n . To g e t an example i n which ( 4 ) f s i l s s e e Morais 1976. Another way t o see the importance of ( 4 ) i s t h e next proposition which presents t h r e e e q u i v a l e n t formulations of ( 4 ) .
PROPOSITION 13. 7 6 B 0 c( Bootenti d g e b h a i n t o B , tt,t -I$ be t h e tNnp deijined by (-
whehe - 0 t h e nytnbat eqUiWdent:
604
Q)( w )
=
-$ (w)
c a t i i p L ~ i e n t d L oi ~n ~ B
aid
Q 0 a riiap
. Theit tlze
6hotti
[Aeq]'
60ttCJdt7g
ah&
210
RICARDO
MORAIS
Now, using these equivalences, we can get several properties of projective Boolean Algebras, namely:
PROPOSITION 1 4 . A u - B o o L ~ ~dMg e b h a B 0 n-PBA i6 m d Only i6 d a s e d u n d e h the P i a p e h a t a h and (4) holds. PROPOSITION 1 5 .
16
M > 1
PROPOSITION 17. Evehg
whehe
w
PROPOSITION 1 8 . PROOF:
Now
M-
PBA, then B 0 ( n - 1) - PBA.
an example.
The cornple*e B a o l e a ~d g c b m 2
= {O
,I} 0 w - PBA.
Since 2 i s complete we have just t o show ( 4 ) holds i n 2 .
P,($)
=
0
i f f sup In6 - $ ( w ) . = G uEG
.
B 0
- PBA ~ a t i n 6 i e A :
iotoak-n add
iff
i f f (by Proposition 3 )
holds
M
aMd
8 0
Sup In6 $(u) = 0
F uEF ( 3 G E FuRe;)(Wu
1 iff Pi( - $ ) = 1
i f f (WF E FuRen)(3w E F)(@(u)=O) E
iff
G)($(u) = 0 ) i f f In6 Sup $(u) = O G uEG
- P i ( - $ ) = 0 and therefore (6)
211
PROJECTIVE L O G I C
The most i m p o r t a n t example o f an w - PBA however i s g i v e n by t h e
fol
-
lowing: The Lindenbawl dgebaa L
THEOREM 1 9 . w
- PBA.
PROOF:
Let
1
@
I
06
P
,the w - pfihujedue Logic
LP
denote t h e e q u i v a l e n c e c l a s s o f t h e f o r m u l a @ i n L
P
.
i n t o L and d e f i n e a map $J from [bey]" P by choosing f o r each W E [hey]' a r e p r e s e n t a t i v e f o r -
L e t @ be a map f r o m [hey]" i n t o f o r m u l a s of L
P
mula @(u) o f t h e e q u i v a l e n c e c l a s s
(8)
P,(@) = !pn(@)l and hence
T,
We t h e n show
(li(w).
,
F i r s t we have t o p r o v e t h a t t h e f o r m u l a choice o f t h e map
Pn o p e r a t o r .
i s c l o s e d under t h e
P
Pn(@) does n o t depend on t h e
@.
I t s enough t o show t h a t f o r any o t h e r map
'Ju E [ ~ e y ] " I- @ ( w )
+
$(u) t h e n
I-
JJ
:
if
Pyz(@)+ P , ( ~ J ) .
By (R4) t h i s f o l l o w s f r o m (9)
VF E F a n I-
AF@(u)
+
pn($)9
which i n t u r n f o l l o w s from, (WG E F u R e V : ) ( V F E
Fan)
(by R5), I-
AF@(w)
+
VG$(u).
B u t t h i s i s t r i v i a l s i n c e by P r o p o s i t i o n 5, g i v e n any F and G , F n G # @ . T h e r e f o r e (9) h o l d s . Now t o f i n i s h t h e proof of Theorem 19 we have t o show t h a t t h e d i s t r i b u t i v e law (4) holds i n L
P'
We s h a l l need,
(10) P i ( @ ) = I P p 4 which i s e q u i v a l e n t t o ,
S U P I AG @ ( u ) I = I P i ( $ J ) I G and so we have t o prove: (i) ( W G E F f i i )
I
I- h G @ ( V )+
Pi(@)
and
(ii) I f (WG E Fufl;)
I- h G $ J ( u + )
11 t h e n
I-
P i ( @ )+ $
.
212
RICARDO MORAIS
PROOF of (i):
From p r o p o s i t i o n 5 g e t
(WG E FU.eei) (WF E F d n )
+
AF l$(v)
+
VG l $ ( v )
now a p p l y (R4) and use P r o p o s i t i o n 2. S t a r t w i t h the hypothesis
PROOF of ( i i ): (WG € FULL;)
I-
1
$J
+ VG 1 $(v),
t h e n a p p l y (R5), and use P r o p o s i t i o n 2. F i n a l l y ( 8
,
(10) and P r o p o s i t i o n 2
give
Pn(@)
=
-
P,*(
- 0)
and t h e r e f o r e ( 6 ) h o l d s , which i s e q u i v a l e n t t o ( 4 NOTE: L
PM
E v i d e n t l y e x a c t l y t h e same p r o o f shows t h a t t h e Lindenbaum a l g e b r a of
(denoted L
PM
) i s n - PBA.
3, FREE n - PROJECTIVE BOOLEAN ALGEBRAS, DEFINITION 2 0 .
An nP
-
BooLean d g e b t a 0 a a - ho-
Izornornotpkm b-een
momohpkinm t h a t pk,hedmve~t h e Pn opehatoh. An W P - homomotpkinm 0 a u - honiomohpkintn
&at p u e h v e n Pn d o t evehy n
DEFINITION 2 1 .
L e t B be m
- genehaten B i d
(a)
G nP
(b)
G dheely
E w.
n - PBA and G
c
B. Tken:
B 0 t h e nm&ent
n - PBA containing G
.
nP- genmaten B i d G nP-genehaten B UJ~C! in a d d i t i o n given m y o t h m n - PBA B' and m y map h : G + B' t h e h e i b an nP - homomohpkidm H : E + B' w h i c h extend6 h
.
- net
06
gen -
An n - PBA 0 a dhee nP- d g e b h a i d contaia n P - n e t ad g e n ma to a . S . i m . 2 d y , dedine a 6hee ~ P - u Q e b h a .
a
dhee
(c)
S .in i. 2 dy dedine W P - neA 06 genehatom and 6hee
WP
ehato4,5.
DEFINITION 2 2 .
I f i s a common p r a c t i c e . i n any t e x t about " f r e e " s t r u c t u r e s t o
first
t a l k about i t s uniqueness and a f t e r w a r d s t o prove i t s e x i s t e n c e .
The
l o w i n g two p r o p o s i t i o n s a r e proven i n t h e same way i t i s u s u a l l y
done f o r
general Boolean a l g e b r a s . See f o r example Halmos 1963, p. 42.
fol-
213
PROJECTIVE L O G I C
PROPOSITION 23. 76 B 0 u dhee n P - d g e b h u , G t h e he,t 06 6hee nP-genefu7Xoh.S and h .the given map 6honi G into .the n - PBA B',then t h e nP - hamomahpkinm H : B + B' t h a t extendh h 0 unique. PROPOSITION 24. Any &oo 6hee n P - d g e b h a whohe .the hame catr&&y atre nP-0oma5pkic.
h d
0 6 genmatom
have
Now t o p r e s e n t an example o f a f r e e U P - a l g e b r a ( t h e e x i s t e n c e o f a f r e e n P - a l g e b r a i s proved s i m i l a r l y ) we proceed as f o l l o w s . F i r s t d e f i n e a phOpOh.iJ%onCdl o g i c LK f o r each c a r d i n a l
K
and t h e n
show t h a t t h e Lindenbaum a l g e b r a L~ o f LK i s a f r e e U P - a l g e b r a
we
with
K
generators. LK i s g o i n g t o have a s e t o f
{Pa j
c1
< Kl
K
variables
,
and t h e p r o p o s i t i o n a l c o n n e c t i v e s 1 and A o p e r a t o r s P,? and P;
. As
in L
PM '
we i n t r o d u c e
and l e t t h e s e t o f f o r m u l a s be t h e l e a s t s e t such t h a t
.
(a)
pa
(b)
if @
(c)
i f 0 i s a c o u n t a b l e s e t o f f o r m u l a s t h e n A @ i s a formula.
(a)
i s a f o r m u l a f o r each o r d i n a l
c1
<
K
i s a f o r m u l a t h e n so i s l @
i f @ i s a map from
[heq]' i n t o f o r m u l a s t h e n P,(@) and Pi(@) a r e
formulas ( f o r every
n E w).
Define "riaving t h e negation h i d e " f o r formulas o f LK as we d i d
LPn
the
for
with the addition:
For axioms t a k e t h e n i n e axioms o f p r o p o s i t i o n a l l o g i c as i n B e l l and I- @l*l@ and I- A @ + @, where @ i s a c o u n t a b l e
Slomson 1969, p. 36, p l u s s e t o f f o r m u l a s and @ 8
@.
For r u l e s o f i n f e r e n c e t a k e those o f L p n w i t h t h e e x c e p t i o n o f (R2). A r e a l i z a t i o n o f LK i s a map 2 =
6
f r o m t h e s e t o f v a r i a b l e s i n t o t h e w-PBA
{ o , 1 1 , which i s i n d u c t i v e l y extended t o a l l f o r m u l a s as f o l l o w s :
(4 d(l@)
=
- 6(@),
(b)
d ( A 0 ) = In6 d ( @ ) ,
(c)
6 ( P n ( @ ) ) = P n ( 6 ( @ ) ) and
$80
b(P;(@))
= P;(d(@))(
214
RICARDO M O R A I S
6(@) i s
where
the map defined by
6($)(u)
=
6 ( @ ( u ) ) for u
€ [neq]"
. 6.
We say t h a t a formula @ i s valid i f d ( @ ) = 1 in a l l realizations Now, before we prove t h a t LK i s an UP-algebra on K generators, need: 8 be an w P - d g e b h a and
PROPOSITION 2 5 . L e t ablu
06
LK & t o
B . EXtend
6
.to a l l d
o
6
we
any map dhom .the u a h i
m by~ trdu (a) X h h o u q h
-
(c).
Then
imfiu
I-dl
A($)
= 1.
In pa)Lticdah, by PhopohLi5on 18, eue-hy theohem oh LK 0 v a l i d . F i r s t note t h a t because of properties (a) and ( b )
PROOF:
(11)
d ( @ + $)
= 1
i f and only i f
d satisfies:
d(@) 5 6 ( $ ) .
I t i s routine t o show t h a t the axioms are mapped into 1 , b u t we check, as an example, t h a t the axiom $1 -l@ i s mapped i n t o 1 f o r the case @ i s Pn($). By (11) we have t o show, B(PYl(VJ)1 ) = 6 ( 1 P n ( $ ) )
.
But 6(Pn($)1 1 = P;(
-6($))
6 ( P,*(l$))
= -Pn(6($)) =
= p;(6(1$)
-
=
6(Pn($)) = 6 ( 1 P n ( $ ) ) *
where the fourth equality follows from ( 6 ) . Similarly, using ( 1 1 ) i t i s easy t o prove t h a t the rules of inference preserve the property of being mapped into 1 . As an example we check f o r (R4). Suppose W F € FU.een,
6 ( h F @ ( u *) $ ) = 1
and
we
have
to
show
d ( P n ( @ ) * IrJ) = 1 Sy (11) and property ( b ) we have ( V F E F a Y l ) In6 Therefore
UEF
b(@(U))
5
A($)
*
d(@(u)) 5 d($). F u€F B u t by definition t h i s i s P n ( 6 ( $ ) ) 5 6 ( $ ) , and hence 6 ( P n ( @ ) )5 ~ ( J J ) T h u s by ( 1 1 ) , 6 ( P n ( @ ) * $1 = 1 . We therefore conclude t h a t every theorem of LK i s mapped into 1. SUP ,In6
.
PROJECTIVE L O G I C
215
We a r e now i n o o s i t i o n t o show THEOREM 2 6 .
LK 0 a
64ee wP - d g e b t u an exac.tQ
K
genmcLtau.
F i r s t i t i s c l e a r t h a t t h e same p r o o f used t o show t h a t
PROOF:
L
w-PBA (Theorem 19) can be r e p e a t e d h e r e t o show LK i s w-PBA. Next l e t G = { ! p a l la gebra B t o g e t h e r w i t h a map
h : G + B.
6
=
~ ( u J , )
6
and l e t t h e r e be g i v e n an a r b i t r a r y wP-al-
K)
Now u s i n g h d e f i n e a nap
and e x t e n d
f r o m t h e v a r i a b l e s o f LK i n t o B by
h ( / p a1
ly
i n d u c t i v e l y t o a l l f o r m u l a s o f LK
.
By P r o p o s i t i o n 25 and (11) i t i s easy t o show t h a t every equivalence class
was
P
I @ 1 , and
is c o n s t a n t
so t h e f o l l o w i n g i s a w e l l
in
defined
map
f r o m LK i n t o 8 :
H(i@ This
I1
d(@).
=
H i s t h e d e s i r e d U P - homomorphism e x t e n d i n g h , and hence i t o n l y
remains t o show t h a t t h e c a r d i n a l i t y o f G i s given
a ,B <
K
with a # 6
i s n o t a theorem and hence
4, A
,
K
.
But t h i s i s e a s y ,
P r o p o s i t i o n 25 can h e l p t o show t h a t pa
I pa I
#
for H
I pB 1 .
REPRESENTATION THEOREM FOR FREE nP-BOOLEAN ALGEBRAS
pB
I
We s t a r t t h i s s e c t i o n w i t h a completeness theorem f o r L K . T h i s i s done t h e same way we d i d f o r L
Pn
and so we o m i t t h e p r o o f , a l t h o u g h we p o i n t o u t
t h e b a s i c p o i n t s . F i r s t we d e f i n e :
216
RICARDO M O R A I S
ththetle
1 5 S 0 a K - CoMnOtency P h O p M y and oo E S t h e n a h e ~ z c L t i o n 6 0 6 LK doh rukich d ( $ ) = 1 doh & @ E no
PROOF:
T h i s p r o o f i s p a t t e r n e d a f t e r t h e one f o r t h e Model E x i s t e n c e The-
PROPOSITION 2 8 .
.
orem (Theorem 7 ) . We s t a r t no and c o n s t r u c t a sequence (A,) o f S w i t h t h e d e s i r e d c l o s u r e p r o p e r t i e s . Then l e t map f r o m t h e v a r i a b l e s o f LK
d(PJ
d
Then e x t e n d quence (A,)
no =
i n t o 2 by
iff
= 1
o f elements o f
u nm
m
and d e f i n e a
Pa e
i n d u c t i v e l y t o a l l f o r m u l a s and because o f t h e way t h e
se-
was c o n s t r u c t e d we have
A($)
= 1
+ E nu .
for all
F i n a l l y , we have :
PROPOSITION 2 9 .
16 $ 0 not a
6(@) 0.
theatem 06 LK then doh
bOMe
tluLizaLLon,
J u s t l i k e we d i d f o r 1 we show t h a t t h e s e t o f a l l f i n i t e s e t s Pfl o f f o r m u l a s o f LK f o r which n o t I-1 h b i s a K - Consistency Property.Then
PROOF:
use P r o p o s i t i o n 28 t o g e t t h e r e s u l t .
An n P - d i d d 0 6 A& 0 a 0 - 6 i & l 06 A & 16 a 0- d i d d 06 b& 0 cloned undm Pn w e c a t t it an w P - 6ieLd 0 6 ~ t . t b .
DEFINITION 30.
cloned u n -
dm t h e opetlatoh Pn.
doh
n 8 o
Notice
evmy
t h a t we d i d n o t m e n t i o n any d i s t r i b u t i v e l a w here. T h i s however
i s no s u r p r i s e because we have:
PROPOSITION 31. Eumy nP THEOREM 3 2 .
- @Ld 06
n&
0 n - PBA.
(heSpecFOX each cahd+u?l K t h m e 0 an nP - 6.ietd 06 A & 06 b d ) that 0 n P - g e n m d e d [ ~ P - g e n e h a t e d ) by K 06
L L v d y U P - d.ieXd ia%
dements.
PROOF:
Let
X = ZK be t h e s e t o f maps f r o m
K
into
2 =
l o , 11 and d e f i n e
PROJECTIVE L O G I C
a <
f o r each
Next, l e t let
BKn
taining
217
K
Q
9, =
r6 e
= {g,
I
(respectively
ZK
I 6(.)
=
11
a<
K}
BK)
be t h e s m a l l e s t n P - a l g e b r a ( U P - a l g e b r a ) con-
which i s a s u b s e t o f t h e power s e t o f X,and
2. BKM and BK a r e
Since t h e power s e t o f X i s a complete f i e l d o f s e t s , well defined.
Q i s K t a k e a # B and choose 6 ( a ) # tj(0). Hence i f , say, d ( a ) = 1 t h e n 6 E g,
F i n a l l y , t o show t h a t t h e c a r d i n a l i t y o f any map but
6B
6 E
2K
gB,
f o r which and t h e r e f o r e
g,
'go.
Now copying what we d i d f o r LK we c o n s t r u c t a p r o p o s i t i o n a l l o g i c LKn f o r each n E w i n such a way t h a t t h e i r c o r r e s p o n d i n g Lindenbaum algebras
LKn a r e f r e e n P - a l g e b r a s .
Our r e p r e s e n t a t i o n theorem f o r
M P - a l g e b r a s i s an immediate consequence
o f the next very importdnt proposition.
PROPOSITION 3 3 . ~ ~ P - i A o m o t ~ p kt ioc BK L~ 0 iA MP - iAomohipkic t o BKn. PROOF:
; and 6 o t ~e v w y
We p r o v e o n l y t h a t L~ i s WP - isomorphic t o BK
The w P - i s o m o r p h i s m H : L~
--f
n E w, LKn
.
BK we a r e l o o k i n g f o r i s d e f i n e d i n d u c -
t i v e l y by: (a)
For every o r d i n a l
(b)
H ( I I@ 1 ) = H (
(c)
H(!AQl) =
(a
I
fl
@
c1 < K , H ( I I ) ' , where A'
p a / ) = 9., denotes t h e complement o f A .
H(l@:).
@ E@
H(IPI1(@)l) =
uF v EnF
H(I@(U)I).
T h i s d e f i n i t i o n makes H an U P - homomorphism, and we have t o show i t i s o n e - t o - o n e and o n t o . of
To show H i s o n e - t o - o n e we d e f i n e f o r each LK by
d'(P,) ( o f course e x t e n d i n g
6'
=
6
E ZK a r e a l i z a t i o n
6'
5(.)
i n d u c t i v e l y t o a l l formulas).
Next, by i n d u c t i o n on t h e c o m p l e x i t y o f @,we show
H(l@l) = { 6 € ZK1f(@)=1).
218
R I C A R 0 0 MORA I S.
F i n a l l y , we have t o prove t h a t i f H( t h e 1 o f LK But i f
.
H( 101 ) =
2K t h e n f o r e v e r y
e v e r y r e a l i z a t i o n o f L, t i o n 29, and hence
]@I
satisfies
101) i s t h e
6
€ 2K
1 of
, 6' (0) =
@ T. h e r e f o r e
.
BK
then
1 $1
1 which means
is that
0 i s a theorem b y Proposi-
i s t h e 1 o f LK L a s t l y , s i n c e t h e image o f L , under H i s an U P - a l g e b r a which c o n t a i n s
2 , the U P - s e t o f generators o f
BK
,
Now g i v e n any U P - a l g e b r a 8 , l e t
we have t h a t H i s onto. K
be t h e c a r d i n a l i t y o f t h e s e t 8 .
Since LK i s a f r e e U P - a l g e b r a we can g e t an wP- homomorphism f r o m L, o n t o
B. Therefore the previons p r o p o s i t i o n gives: THEOREM 3 4 . (a)
(The R e p r e s e n t a t i o n Theorem f o r P r o j e c t i v e A l g e b r a s ) .
Any nP - dyebha h an nP - homomohpkic h a y e
n&. (b)
Any P - dgebha 0 an
UP - kotnomohphic
huge
06 an nP - 5 i e L d
06
an UP - 6 i e L d
o6
06
beh.
IV, CONCLUSION, Our r e p r e s e n t a t i o n theorem f o r f r e e p r o j e c t i v e Boolean a l g e b r a s p r o v i d e d US
w i t h a " b r i d g e " f r o m l o g i c t o s e t t h e o r y , b u t so f a r n o t h i n g was s p e c i
f i c a l l y shown so as t o g i v e a r e l a t i o n s h i p between t h e p r o j e c t i v e f i e l d
-
of
s e t s and t h e p r o j e c t i v e s e t s o f L u s i n and S i e r p i n s k i . Our t e r m i n o l o g y t h e r e f o r e l a c k s some j u s t i f i c a t i o n , which i s however g i v e n b y t h e f o l l o w i n g
and
l a s t theorem: THEOREM 3 5 .
Foh n > 0 ,
1
.i~ an n - phojeotiwe 6 i e l d
06 b&,
whehe
A,
6Zand6 doh "boLd6ace A". PROOF:
( f o r a d e t a i l e d p r o o f p l e a s e see M o r a i s 1976).
We w i l l show t h a t
i s c l o s e d under t h e Pn,
b u t t h i s i s not enough,
however, t o p r o v e t h a t . i t i s n - p r o j e c t i v e because t h e d e f i n i t i o n n - p r o j e c t i v e algebra s t a r t s w i t h a o - a l g e b r a .
B u t i t i s easy t o see
t h e same argument used below can be r e p e a t e d t o show t h a t L I ~ + ~i s
o f an that closed
under t h e P1 o p e r a t o r , and t h i s i n t u r n i s a g e n e r a l i z a t i o n o f c o u n t a b l e
219
PROJECTIVE L O G I C
unions and i n t e r s e c t i o n s ( c f . Kuratowski and Mostowski 1968, p. 341). L e t now @ be any map f r o m
x
E
1
i n t o Qn+l,
[AQQ;'
1
P,($) can be g i v e n b o t h by a Jn+l and a By P r o p o s i t i o n 3
x
P,($)
E
JA+l
a n d we
show
that
predicate.
has two e q u i v a l e n t f o r m u l a t i o n s , namely:
(a)
( 3 F E F U . e e n ) ( W w E F)(x E $ ( w ) )
(b)
(WG E FU.eei)(3v E G ) ( x E @ ( w ) ) .
and
We a r e g o i n g t o use (a) ( r e s p e c t i v e l y (b)) t o show t h a t g i v e n by a
+
F i r s t , since w onto
i s countable, there i s a
[bey]"
F and
s t i t u t e the sets
1- 1 r e c u r s i v e map
by means o f X
{O
G by t h e i r r e s p e c t i v e
tw
E [bey]"
, 1 1 , and as we d i d f o r
I g(w)
= 1) E
The e x p r e s s i o n range
(Wm
.
I n addition,
= 0
{ w 8 [neq]'
ik
.
)'i+lp r e d i c a t e .
@(k) E
[(tlange g
=
{O, 11
.
g(n1) = 1).
otl
I g(w)
= 1) E
1 4n+1 c $ + ~f o r
T h e r e f o r e i f we w r i t e W1 and
sub-
11 i s w r i t t e n Fulln
( ~ 6 1 ) ( ~ 5 * ) " ' ( Q 6 , ) ~ Q ' ~ ) ( 9 ( 6 1 1 m* ,* .
F i n a l l y since
we
we t h i n k t h e domain o f g as w .
@
F a n and g ( k ) = 1) + x E @ ( k ) ]
g = {O,
E w)(g(m)
The e x p r e s s i o n
which i s
X from
c h a r a c t e r i s t i c functions
T h e r e f o r e (a) i s e q u i v a l e n t t o ( 3 9 E "u) (Wk E w )
and
is
P,*($)
E
and so we can t h i n k t h a t t h e domain o f $ i s w . L e t _v be t h e
[bey:"
i n t e g e r a s s o c i a t e d w i t h w C [beq]" g : [hey;"+
x
1 ( r e s p e c t i v e l y iln+l)p r e d i c a t e .
1
i s equivalent t o
A,i)4
= 1)
a l l k we can make
x
E $(k)
a
31 f o r q u a n t i f i c a t i o n o v e r r e a l s and WO
and 30 f o r q u a n t i f i c a t i o n o v e r numbers, t h e statement ( a ) now reads:
31 WO [ ( W O A,31
... "
Q1 Q'O) j L 3 1
- Kuratowski
we s i m p l i f y t h e above t o 31 3 1 VO [ ( A
...
v n+1
) +
v
Q'l,QO]
.
n+1
n Then u s i n g t h e T a r s k i
...
a l g o r i t h m s ( c f . Rogers 1967,
1
p.
307)
R I CARD0 NORA I S
220
which i s a
zi+l predicate.
Now using (b), since
G €
FuRel
is
we, s t a r t w i t h
[
W130
... 2'1 20) +
(WOA W 1
u n
and end up w i t h a $+1
predicate.
JLi and s i n c e @ ( k )
V1
E
1 4n+l
1
... 21 2 ' 0 1
n+ 1
came
I t i s c l e a r by now t h a t one o f t h e most i n t e r e s t i n g notions t h a t
up along t h i s work was t h a t o f
Full;
and i t s counterpart FLLeen
symmetry between these two classes o f
. The
,
c ;n+l
generalizing Ellentuck's F u l l s e t s , sets
n o t o n l y helped c u t t i n g a l l our proofs i n h a l f b u t also, and more s i g n i f i c a n t l y , w i t h o u t t h i s symmetry most o f our p r o o f s
-
c o u l d n o t have
come
through, s p e c i a l l y our l a s t theorem i n which the simultaneous use o f
FuUn
FuRe;
and
was fundamental.
For these reasons we foresee an i n c r e a s i n g use o f these n o t i o n s i n
the
f u t u r e s t u d i e s o f p r o j e c t i v e sets.
To conclude t h i s work, among several i n t e r e s t i n g q u e s t i o n s f o r which a l l t h i s machinery i s applicable, we s e l e c t e d two t h a t we are p a r t i c u l a r l y i n t e r e s t e d i n i n v e s t i g a t i n g , namely: (1) (2)
L or L i f any? Pn P ' If. M. i s 3 universe o f s e t s and B i s an n - p r o j e c t i v e Boolean a l gebra, what can be accomplished i n s i d e t h e Boolean valued m o d e l
What k i n d o f i n t e r p o l a t i o n theorem holds i n
MB ?
REFERENCES Addison, J. A. and Y . Moskovakis 1968,
Some comequenca Nat. Acad. Sci.,
06
t h t axiom o d de6&abLe d e L m i n a t e n a s ,
B e l l , J. L. and A. B. Slomson 1969,
Proc.
Vol. 59, 708- 712.
Models and Ultraprqducts, North
- Holland,
Amsterdam.
Ellentuck, E. 1975,
The ~owzdatiom06 S w f i n Logic, The Journal o f symbolic Logic, v o l . 40, 567-575
PROJECTIVE
197+,
22 1
LOGIC
Fhee SwL& d g e b m , S u b m i t t e d t o Czech. Wath. J o u r n a l .
Fenstad, F. 1971,
The a x i a m
0 6 deLetuninateflcbb,
Proceedings o f t h e Second Scandinavian
L o g i c Symposium, Ed., J . E. Fenstad, N o r t h - H o l l a n d , Amsterdam,41-61. Halmos, P. 1963,
Lectures on Boolean Algebras, Van Nostrand K e i n h o l d Company, London.
Kechris, A. 1973,
i'leubwre and categohy i n eddeotiwe denchipLLwe
bet
theahy, Annals
of
Math. L o g i c , V o l . 5, 3 3 7 - 384. K e i s l e r , H. J. 1971,
Model Theory for Infinitary Logic, N o r t h - H o l l a n d , Amsterdam.
Kuratowski, K. and A. Mostowski 1968,
Set Theory, N o r t h - H o l l a n d , Amsterdam.
L u s i n , N. 1925,
S w l Lcb emembLcn p o j e o t i w e h de M. H e i d Lebcngue, C. R. Acad.
Sci.
de P a r i s . Morais, R. 1976,
Projective Logic, Ph. D. T h e s i s , Eutgers U n i v e r s i t y , U.S.A.
Moschovakis, 1970,
Y.
DeLmtirzaflcy und
pmLJ&Otld&Mgb
ad t h e co~dizuum,in Mathematical
Logic and Foundations of Set Theory, E d . Y . B a r - H i l l e l , H o l l a n d , Amsterdm, 24 - 62.
North-
Rieger, L . 1955,
Cancmning Suofin d g e b m (S - k e g e b m ) and t l z e i n (Russian), Czech. I l a t h . J o u r n a l , Vol. 5. 99
htphtbtntathion
- 142.
Rogers, H. 1967,
Theory of Recursive Functions and Effective Computability, MacGraw H i 11. Sierpinski, W. 1925, S w l une &abbe d'emembLcb, Fund. Math. Vol. 7, 2 3 7 - 2 4 3 .
-
l n s t i t u t o de Matematica U n i v e r s i d a d e F e d e r a l do R i o de J a n e i r o R i o de J a n e i r o , RJ.,
Brazil
Non-Classical Logics, Model Theory and Computability, A.I. Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
TYPES, WITH MODEL COMPLETENESS, AND RELATED PROPERTI ES ,
SOME THEOREMS ON OMITTING
APPLICATIONS AMALGAMATI
TO
ON,
by C H A R L E S C .
PINTER
1, INTRODUCTION, Many t o p i c s of c u r r e n t i n t e r e s t i n model theory involve m o d e l s which omit designated s e t s of types. For example, t h e e x i s t e n t i a l l y closed models of a theory T a r e p r e c i s e l y those which omit a c e r t a i n s e t of types. The Same i s true f o r t h e g e n e r i c models of T f o r f i n i t e f o r c i n g , t h e completing models of T ; t h e models which a r e analganation bases f o r T , and so on. I n f a c t , i n p o r t a n t p r o p e r t i e s of a theory T hold i f f a l l the models of T omit c e r t a i n given types: t h i s i s t h e case of a theory being model complete, f i ni t e l y f o r c i n g complete, having the amalgamation property, the congruence extension property: and many o t h e r s .
O n e o f t h e o b j e c t s of this paper i s t o show t h a t many s u p e r f i c i a l l y u n r e l a t e d r e s u l t s of model theory a r e , i n f a c t , consequences of the same simple theorems on omitting types. By s y s t e m a t i c a l l y developing c e r t a i n r a t h e r elementary observations on omitting types, we can r e c a p t u r e - and unify-many known r e s u l t s , f i n d a number of new ones, and almost t r i v i a l i z e some c l a s s i c a l r e s u l t s . As an example of t h e l a t t e r , we show t h a t Lindstroai's t h e o rem on model completeness is a consequence of a very simple observation on omitting types i n t h e o r i e s which a r e c a t e g o r i c a l i n some i n f i n i t e power. In Section 2 of this paper we develop s e v e r a l p r o p e r t i e s o f omitting types, and i n Section 3 we i l l u s t r a t e t h e i r uses with a v a r i e t y of examples. Our standard reference t o model theory w i l l be Chang and Keisler 1973. 223
224
C. C. PINTER
We assume throughout t h a t L i s a countable, f i n i t a r y , f i r s t - o r d e r guage; a .themy T i s a c k d u c t i v e l y closed s e t o f sentences o f L . p l i f y n o t a t i o n , we w i l l l e t 7 designate any f i n i t e sequence (u1, v a r i a b l e s , and, i f quence (al, symbols
a
...,a n ) o f
@(V)
lan-
To sim-
...,w n )
of
i s a s t r u c t u r e , we w i l l l e t Cr d e s i y a t e any f i n i t e semembers o f U l . For any formula @
.1 and @ j
, we
will
the
use
w i t h t h e i r obvious meanings, provided t h a t t h e f r e e
v
v a r i a b l e s o f $ occur among
and t h a t t h e sequence
a
matches t h e sequence
lJ.
2, SOME L e t u(; theory o f L
or4
THEOREMS
{on(;)
=
, we
:
n
OMITTING I?
be a sequence o f formulas o f L . I f T i s a
say t h a t T Lac&y
which i s c o n s i s t e n t w i t h T ,
n
U}
TYPES,
o m i h u i f f f o r every formula
I$(;)A T u n ( ; )
i s consistent w i t h T
6 w. L e t C bc a countablc s c t o f sequences
u o f formulas; i f
T
(V)
@
for
some
locally
u 8 C, wc w i l l say t h a t T i s Z - compLeAe. A model aI= T w i l l be s a i d t o V M LC ~ i f f a omits cach u I? C . The h i t t i n g - types Theorem
ornits every asserts :
.
C 76 (2.1) 16 T h C - compL&te, t h e n T h a a m d e L which ofii& compLeAe Rheafiy: T h C - campLett id5 T h a a tiiodel wkich o n i i h
Ile say t h a t T i s 1 - c o r z n h t e n t i f f T C_ T',bfhere
DEFINITION.
c o n s i s t e n t and C
- complete.
C l e a r l y , T has a nodel vl;tittiw C more, i f T i s
, that
.id6 T
0 a
T'
is
0 Z - carznhtent. FurtherT 0 C - cmpLeAe.
complete t h e o r y T 0 C - c a ~ b h t e n t.id:
-- -
PROOF: (iii) (i) T
B
T
.
(ii) (i) i s immediate. (iii): Every model o f T i s a model o f a complete extension
of
i s , o f a'complete, C - complete theory. Thus, f o r eachformula $(;)I
SOME THEOREMS O N OMITTING TYPES
every model Ol t==
225
T s a t i s f i e s the sentence
(ji)+(i)
* V i e W ( 3 G ) [ + (;)
A l~i(;)J
By compactness, we have (iii). 0 EXAMPLE, Let T be a consistenttheory in the language L" of w-logic.Then, every extension of T has an w - model i f f every extension of T i s w-complete i f f f o r each formula @(;) there are integers il, i such t h a t
..., Q
The next two theorems have a great many applications, which will be developed in Section 3 of t h i s paper.
PROOF : (i) i s immediate, using the contrapositive of the definition of"T 10cally omits u " . (ii) : He will show t h a t i f T i s C - complete and @, i s any sentence which i s consistent with T , then T U { @ I i s Z - complete; (ii) will follow immediately from t h i s . Hell, suppose T i s C - complete and q(;) is consistent with T U { @ } . Then I)(;) A @ i s consistent with T I so f o r some n 8 w ,$(;) A @ A l u n ( ; ) i s consistent with T . Thus $(;) A l u n ( ; ) i s
consistent with T U {@I. 0 ( 2 . 4 ) THEOREM
The doaRoLuing me equivalent:
(i) T h C - cornpdete (ii) Fa& & nentencen tokich or;l.LtA c .
@
,Tt
@
idd
@
0 h u e i n e v # y model
06
T
We will show t h a t (ii) holds i f f T i s the intersection of a l l i t s complete, c - complete extensions; our r e s u l t n i l 1 then follow by (2.3). Suppose (ii) holds, and l e t { T :~ i € I} be the s e t of a l l the c o m p l e t e ,
PROOF:
C
- cornplete
theories containing T
. Plow,
T
5
2 T ; furthermore,if
@€
?Ti,
226
C. C. PINTER
, hence @ E T . Conversely, T = T. ; c e r t a i n l y @ € T i m p l i e s t h a t @ i s t r u e i n e v e r y A. model o f T o m i t t i n g C . On t h e o t h e r hand, i f @ i s t r u e i n every model o f T o m i t t i n g Z , then € Ti f o r each i € I, so @ Ti 0 t h e n @ i s t r u e i n e v e r y model o f T o m i t t i n g C
suppose
€a
.
D E F I N I T I O N , L e t T be C - c o n s i s t e n t . The i n t e r s e c t i o n o f a l l t h e complete, C
- complete
n o t e d by
T
e x t e n s i o n s o f T i s c a l l e d t h e C - comp.&,t,Lano f T , and i s
C
.
TC i s t h e s m a l l e s t C - c o m p l e t e t h e o r y c o n t a i n i n g T . We w i l l
see
dein
t h e n e x t S e c t i o n t h a t f o r c i n g - companions, m o d e l - companions, and o t h e r f a m i l i a r c o n s t r u c t i o n s o f model t h e o r y a r e s p e c i f i c examples o f 1-completions. I t i s obvious t h a t any m u d e l
06
T wkich v m h C hub .to be a modeL
v6
Tz. From t h i s f a c t , we i m m e d i a t e l y deduce: (2.5)
THEOREM, T C I- @
c.
4 5
@
A h e i n ewehy made[ a6 T which o m h
I n many cases which we a r e l e d t o c o n s i d e r , t h e c l a s s J o f a l l t h e mode l s o f T which o m i t C t u r n s o u t t o be an elementary c l a s s . F o r such cases, the following i s useful: (2.6)
THEOREM, SUppVbe Rhe d k b b 3 of, ClRe Rhe mod&
06
T which o m d C
0 an elementaht; d k b b . Then TC = Th (31, and Tc A a d o m d z e d b y
whete PROOF:
each
vi
T *U { ( . W ; ) v i i b
IDi(;)
:
0
E C)
a 6ini.te d i b j u n c L i o n .
T C = Th(7) b y ( 2 . 5 ) . Thus, each model o f T C o m i t s C , and o u r
r e s u l t f o l l o w s f r o m a s i m p l e a p p l i c a t i o n o f compactness. 0 I n t h e sequel, i t w i l l be u s e f u l t o n o t e t h a t , i f
T i s C-consistent,
t h e r e i s an obvious i n d u c t i v e procedure f o r c o n s t r u c t i n g t h e C - c o m p l e t i o n TC O f T. o f consistent To c o n s t r u c t T C , we d e f i n e a sequence U o E UIc t h e o r i e s as f o l 1ows : (2.7) (i) U o = T. (ii) Given UK, d e f i n e UK+l by: f o r each u E C and each f o r m u l a a ( ; ) , i f UK +I)(;) + o n ( ; ) f o r each n E w , t h e n 1(3;)$(;) E UK+l
...
.
(iii) I f
c1
i s a l i m i t o r d i n a l , U, =
F i n a l l y , l e t TC be t h e d e d u c t i v e c l o s u r e o f
u
U Y. UK<W U ~ Our assumption t h a t Y
.
227
SOME T H E O R E i l S O N O M I T T I N G T Y P E S
T i s C - c o n s i s t e n t guarantees t h a t each UK i s c o n s i t e n t , and c l e a r l y , f o r some countable ordinal 1~ , u 2 1~ implies (Iy = UU . The preceding construction may be i n t e r p r e t e d as a completeness theorem f o r truth in t h e c l a s s of models of T which omit C , w i t h t h e closurecond i t i o n (ii) taken as an i n f i n i t a r y r u l e of proof. S p e c i f i c a l l y , we have t h e following g e n e r a l i z a t i o n of t h e w - r u l e , which we s h a l l c a l l t h e C-&e:For each a E C , from I. (;) + un(;) f o r each n E o i n f e r 1 ( 3 ; ) I ) ( ; ) ,where I)(;) i s any formula of L . C - l o g i c i s formed by adding t h e C - r u l e t o t h e axioms and r u l e s of inference of t h e f i r s t - order l o g i c L and allowing i n f i n i t e l y long proofs. Ik have the following completeness theorem f o r C-logic:
A Rh~.clqT
(2.8) t i i q ill
C
iri
1 i4 c o r t 5 i ~ t t r ~ ,it n C - Logic
+
idA
T h a a madel o m 2 -
. Fuhthetuliohc, i.6 i n abiq seaterice od L , $ can b e deduced ((ham T h j i c .id6 .in thul? i l l e w u y model 06 T Which om& C Ile say t h a t T ~~dii.~d',kj om& C i f f every infini-Le model of T can be I:
.
-
extended t:, a model of T which omits C . The following simple observations have many a p p l i c a t i o n s : I f every denumerable model of T oinits C then every model of T omits C . Indeed, i f OZ I= T , Cc i s i n a , and 5 r e a l i z e s a then by t h e downward LBwenheim- Skolem Theorem, T has a denumerable model d ' + a such t h a t i s i n OZ'; and c l e a r l y 2 r e a l i z e s u in OZ'. From t h i s observation and t h e upward Lanenheim- Skolem Theorem, we i n f e r t h e following f o r every i n f i n i t e cardinal a :
a
(2.9)"
16 CVehy model!
06
T
05
p c w m a om&
C
, then
e v m y model a6 T om&
C .
Furthermore, the downward Lowenheim- Skolem Theorem y i e l d s : (2.10)
14 T
t o a model $
c o ~ i n & y vmLi% x , t h c n cumq model UZ C T can be e x t e n d e d T nuch t h a t $ om& I: avid cahd 01= cahd &
.
k=
(2.9)" has manifold a p p l i c a t i o n s , among which a r e t h e following: a theory T has t h e amalgamation property i f f every model of T of a given i n f i n i t e power i s an amalgamation base of T . (This improves Yasuhara 1974, Theorem 1.13.1). T i s model- complete i f f every model of T of a given i n f i n i t e power a i s e x i s t e n t i a l l y closed over T Analogous statements may be i n f e r r e d f o r t h e strong amalgamation property, t h e congruence extension Thus, to property ( s e e Bacsich and Rowlands Hughes 1974),and many o t h e r s . know i f a theory T has one of t h e above- named p r o p e r t i e s , i t s u f f i c e s t o watch only the models of a given i n f i n i t e power a .
.
228
C.
PlNTER
C.
We end t h i s s e c t i o n w i t h some o b s e r v a t i o n s on categohicity
iiz
pvweh and
omitting types. Our f i r s t p r o p o s i t i o n g e n e r a l i z e s (and, L i n d s t r o m ' s Theorem on model
- completeness.
in
a sense,
trivializes)
( 2 . 1 1 ) PROPOSITION L&t T be a theohy idzLch 0 a - categohical doh dome 1 w. L e i C be any countabde n e i 06 types nuch t h a t T c o ~ i n & y o m i t s Z Then ewmy model 06 T am& C u
.
.
PROOF: Take any u E C , and suppose u i s c u n s i s t e n t w i t h T . Then T has a model o f power c1 w h i c h r e a l i z e s u Eecause T i s a - C a t e g o r i c a l , no model o f T o f power u or,iits u : i n view of (2.10), t h i s c o n t r a d i c t s our assumpt i o n t h a t T c o f i n a l l y o m i t s u . !le must conclude t h a t u i s i n c o n s i s t e n t w i t h T , hence e v e r y model o f T o m i t s C .O
.
(To o b t a i n L i n d s t r o m ' s Theorem, we n o t e : T i s m o d e l - c o m p l e t e i f f e v e r y model o f T i s e x i s t e n t i a l l y c l o s e d o v e r T ; OZ i s e x i s t e n t i a l l y c l o s e d o v e r T iff
o m i t s a c e r t a i n s e t o f types C ; i f T i s i n d u c t i v e ,
t h e n every
node1 o f T can be extended t o an e x i s t e n t i a l l y c l o s e d model o f T.) Now, l e t T b e a t h e o r y which i s u l - c a t e g o r i c a l
b u t n o t u - c a t e g o r i c a l .If
a t y p e u i s c o n s i s t e n t w i t h T and T l o c a l l y o m i t s u , t h e n u i s o m i t t e d i n some c o u n t a b l e models o f T , b u t r e a l i z e d i n a l l i t s uncountable models. I f e v e r y c o u n t a b l e model o f T o m i t t i n g u were t o a d m i t a c o u n t a b l e p r o p e r
elementary e x t e n s i o n o m i t t i n g
u , an w l - c h a i n o f such m o d e l s c o u l d
be
formed, and t h e i r u n i o n w o u l d be an uncountable model o f T o m i t t i n g u , w h i c h
i s i m p o s s i b l e . Thus, t h e r e a r e maximal c o u n t a b l e models o f T o m i t t i n g 0 , ( c ~ u x h a li n t h e sense t h a t t h e y a d m i t no p r o p e r elementary e x t e n s i o n o m i t t i n g u ). The same argument h o l d s f o r a c o u n t a b l e s e t o f t y p e s I:. Lle conc l ude :
Lei T be w l - c a t e g g ~ h i c dbLL.t W - Categohicad. 06 typen which m e c o n ~ D t e i z *i n T and duchRlzat T 0 C-coMhZent. Then thehe m e mod& ad T which m e niaxhicd i,&h h e d p e c t t o omLtZing C These tnvdeh me cvuntabde: evmy &enlentmy chain 06 countabLe mod& 06 T omLtZing C m u t end &h one 06 t h a t maximal mod(2.12)
LeX
C
PROPOSITION.
be a countabde b e 2
.
&.
A g r e a t many consequences may be drawn f r o m t h i s s i m p l e o b s e r v a t i o n .
A
c a n o n i c a l a p p l i c a t i o n i s t o atomic models: every t h e o r y T w h i c h i s wl-categ o r i c a l b u t n o t w - c a t e g o r i c a l has a c o u n t a b l e maximal a t o m i c model,
which
229
SOME THEOREMS O N O M I T T I N G T Y P E S
i s therefore a l s o m i n i n i a l .
Applying t h i s r e s u l t
as t o t h e t h e o r i e s T h m A w h e r e
to
T ,
as w e l l
01 i s a n y c o u n t a b l e m o d e l o f T
,
we
r e c o v e r many o f t h e r e s u l t s o f K o r l e y 1967.
3, SOME APPLICATIONS, I n t h i s s e c t i o n , we s e l e c t a few t y p i c a l a p p l i c a t i o n s o f t h e r e s u l t s d e veloped i n S e c t i o n 2. notion o f
I n p a r t i c u l a r , we w i l l i l l u s t r a t e t h e u s e s
o f the
C-completion.
( I ) EXISTENTIALLY
CLOSED MODELS,
L e t T be a t h e o r y ; we r e c a l l t h a t i f 01 k= TV , 02 i s s a i d t o be e x DA.ed&y c l o s e d o v e r T i f f f o r any $ b T, d = & i n p l i e s UZ + I & , We
w i l l abbreviate " e x i s t e n t i a l l y closed over T For each f o r m u l a and T
+
E
-f
a E
a l l t h e formulas o f V l T iff
'I
by e.c. o v e r T .
,
l e t oa d e s i g n a t e t h e t y p e { a , 1 E : E 8 31 a}. L e t Z be t h e s e t o f a l l t h e types oa as CY ranges o v e r
OZ o m i t s Z
.
V1
.
I t i s w e l l known t h a t i f
I t i s i m p o r t a n t t o n o t e t h a t 01 i s e.c.
(v;)
indeed, t h e sentences
LE
each t y p e ua a r e sentences o f T
+
m
b T
V'
m
i s e.c.over
o v e r T i f f OZ i s e.c.
over T . W '
a] which determine t h e c o m p o s i t i o n o f
w.
By Theorems 2.4 and 2.5, we immediately conclude:
-
oen( 3 . 1 ) P R O P O S I T I O N , T 0 Z-complcde id5 T D t h e net a6 & . t h e t w c u w!uch me h u e i i z CVUIJ C . C . model ad T . F w d h m o h e , T 0 the V t h e u t y ud & t h e mode-& which me e.c. U V C R T . @ D h e i i z every model iulzich 0 e.c.auen ( i n o t h e r words, I$ E T: TI.
I t i s w o r t h remarking t h a t
T w z n a y be c o n s t r u c t e d f r o m T
by
procedure o u t l i n e d i n ( 2 . 7 ) , and i t s theorems may be o b t a i n e d f r o m T deduction i n C
-
logic.
w
the by
Me n o t e a l s o t h a t ( 3.2 )
?
C
, a mLLtu&y
model - c o a A t e n t w L c h T
Indeed, every model o f TV i s , t o a model o f
c .
TV
,a
hence d t h T
.
c a n b e e x t e n d e d t o an e.c. model o f T , t h a t W
Now, suppose t h a t t h e c l a s s o f a l l t h e models which a r e e.c. o v e r T i s elementary. By Theorem 2.6,
t h i s c l a s s i s a x i o m a t i z e d by T V c and
230
C. C. PINTER
T wz = T v U { a where,
f o r each
a EW,,
Obvioulsy TwE i s model and t h e r e f o r e a l s o t h e
E
E
Y1)
i s an e x i s t e n t i a l formula such t h a t T I-E+a.
T
E W
o f Tw,
has a mod&- companion, Rhe lmXm 0 axioma-
Twt = T w U { a a
:
E
- complete, hence T i i s t h e model - companion model - companion o f T . We, conclude:
( 3 . 3 ) PROPOSITION, 1 6 .tized by
whehe, doh e a c h
-+
,,
E
-+
: a E Wl},
E
0 an e x i s t e n t i a l d o t m u l a
s u c h Rhat
T +€+a. I t i s e a s i l y v e r i f i e d from t h e r e l e v a n t d e f i n i t i o n s (see, e.g. ,Robinson
1971) t h a t the i n f i n i t e l y T - g e n e r i c models a r e T w - g e n e r i c models. Thus, by
exactly the i n f i n i t e l y
.
Fisher and Robinson 1972, Theorem 3.4
PROPOSITION. Foh any Rheohy T, Rhe 6oUoLuing ahe eqLLivdent: (a) The i n 6 i n i t e R y T -'gm&c mod& 60m an eRementatLy c h ~ h . (b) The mod& wkich ahe e.c. o v a T dohm an U.kmentatLq! c h 6 . (c) T has a mod& companion. 7 6 one 06 Mese condition6 hold, t h e n T ~ ' 0 t h e mod& companion o d T and axiomatized Rhe ceads 06 i n 6 i n i t c L y T - gen&c mod&. (This i s a v a r i a n t of Fisher and Robinson 1972, Theorem 3.4. I n p a r t i c (3.4)
u l a r , t h e l a t t e r a p p l i e s t o i n d u c t i v e t h e o r i e s only, whereas t h e p r e s e n t (3.6) holds f o r a r b i t r a r y t h e o r i e s ) . I n o n e o f t h e m a j o r r e s u l t s o f t h e i r paper,Fisherand Robinson demonstrate t h a t t h e f o r c i n g companion of any i n a u c t i v e theory T ( t h a t i s , t h e theoryof the generic models o f T ) can be decomposed i n t o t h e f o r c i n g companions t h e components o f T w
. ( A component o f a u n i v e r s a l
theory
u
is a
of
minimal
i r r e d u c i b l e extension o f U ) . We show now t h a t an analogous r e s u l t h o l d s f o r T i
,
t h e theory o f t h e models which a r e e.c.
over T .
PROPOSITION, L e t { T ~: i E I } be the seA 06 t h e components The d a s h 06 mode,& wkich ahe e.c. ovm T 0 Rhe union 06 t h e TW.
(3.5) 06
PROOF: then $I=
I f a i s e.c. over T v , hence
versely, i f OL i s e.c.
T!,
a I=
T~
f o r some i e l . t i o w i f m s $ i T L ,
t h i s shows t h a t
a is
e.c. over
Ti.
Con-
over T~ f o r some i € 7 , then 02 can be embedded i n
23 1
SOME THEOREMS ON OMITTING TYPES
a model o f T I = Ti U {I$ E 3 1 : 0 E T ~ } ,hence 01 I= T I . Thus;OZ cannot over be embedded i n a model o f any T . , j # i. So c l e a r l y , a i s e . c . J
Tw
.0
Tc =
(Note t h a t i f T i s i n d u c t i v e , t h e n
(11) THE
*
ie1 T -1i
AMALGAPIATION PROPERTY I
A model 02 t T~ i f each diagram
i s c a l l e d T- mndgmcLtive ( o r a T- amulgamation b a t ) ,
--.* \ t..-*-.* YJ
m /&-..
(3.6)
can be completed. (See; e.?.,
&,y,D!= T
B a c s i c h and Rowlands Hughes 1974,and Yasuhara
1974). F o r e v e r y p a i r ( a ,B ) o f u n i v e r s a l formulas, l e t
u ( ~ ” ) designate
the
{ 1 ~ , 1 6 : ~ , 6 € 3T ~t ,- ~ + a and T + 6 + B } .
type
L e t Z be t h e s e t o f a l l t h e types
u ( ~ ’ ~ as ) , (a , B ) ranges o v e r
p a i r s o f u n i v e r s a l f o r m u l a s such t h a t
a1 1 the
+ a v B. I t i s known (Bacsich and Rowlands Hughes 1974, Theorem 2.4) t h a t i f a t T w , t h e n a i s T - amalgam a t i v e i f f OZ o m i t s C . We n o t e , as i n t h e p r e c e d i n g example, t h a t a model i s T - amalgamative i f f i t i s Ti- amalgamative. T
By Theorems 2.4 and 2.5, we i m m e d i a t e l y conclude:
( 3 . 7 ) PROPOSITION, T 0 Z-compLete
t e n c u which
aht
id6
T 0
the net
-0uLe i n evehy T - a m d g a m a t i v e model
06 aU .the T -
06
06 T
.
t h e benFwLthehmote,
amdgamahive mod&. 4I .iO -0uLt i n evehy T - a m d g a m d v e model1 We remark t h a t T t may be c o n s t r u c t e d f r o m T w by t h e p r o c e d u r e o f (2.7), and i t s theorems may be o b t a i n e d f r o m T w by d e d u c t i o n i n Z - l o g i c .
T$
0 Rhe Rheony
c
.
[That 0 , T V I- 41
:T A m w y model- conb0tent ulLith T w , hence with T . (3.8) c t T w can be extended t o a T-amalgamative m o d e l b k T w F o r e v e r y model
I f t h e c l a s s o f a l l t h e T - amalgamative models i s elementary,
Theorem 2.6, (3.9)
i t i s a x i o m a t i z e d by T t Tic =
Tw U
then
.
by
and
v 6z : z E
w h e m Z i n t h e b e t 0 6 all p a i h n ( a , B )
Z}, 06
uncvetldd ~omunLLea nuch
232
C.
M T I - a v f 3 ,
and
T
+ gZ
+
cZ,6,€
C . PINTER
31,anddaheach z = ( a , B ) € Z ,
TI-c,+a
8.
By compactness, T
has the amalgamation p r o p e r t y (AP) i f f T includes a
s e t o f formulas T U { c Z v 6, : z E Z} as described under (3.9). But then W the theory T~ U { E , v 6, : z E Z} already has t h e AP. Thus, we a r e l e d t o the f o l l o w i n g scheme f o r c l a s s i f y i n g t h e o r i e s having t h e AP:
Evehy theoty T V U{E, v 6 z : z € Z } which h m W y modeL-conM e AP. Among Rhe Rheohien which m e mu&&y modeL c o r n h t e n t uLth T , a Rheohy U h a Rhe AP i d 6 U cuntainh a b e 2 06 bentence) T U { E ~v 6, : z 8 Z}. (3.10)
b h t e n t luith T h a
W
A minor g e n e r a l i z a t i o n o f the AP i s very u s e f u l and occurs increasingly i n the l i t e r a t u r e : I f S and T a r e t h e o r i e s , we say t h a t S h a ,the T-am&
gamation phopetLty (S h a t h e T-AP) i f f the diagram (3.6) can be completed f o r every model Ol C S ; t h a t i s , i f f every model Ol I= S i s T-amalgamative. S ha amafgamation p h o p e n t y id6 S I h a Rhe T - amdgamation p h o p e h t y W Thus, t h e r e i s no loss o f g e n e r a l i t y i f we now c o n f i n e our a t t e n t i o n t o the case where S i s an i n d u c t i v e theory and T i s a u n i v e r s a l theory. We p r o -
One v e r i f i e s , by an easy argument such as (Yasuhara 1974, 1.9.3),that
fhe
.
T-
ceed now t o g i v e a decomposition r e s u l t f o r t h e AP i n t h e s p i r i t o f Propos i t i o n (3.5); we w i l l show t h a t t h e r e l a t i o n "S has t h e T-AP" mined by r e l a t i o n s T'
O f
"S' has t h e T'-AP"
among t h e components
s'
i s deterof
S
and
T.
(i) 16 T 0 a u n i v m d RhtOhy, Zuo n.trutcXLULen Ol and & 06 ,the name component 06 T id6 Rhey have a joint externion ukich 0 a mod& ad T. (ii) 16 S 0 an inductive Rheofcy, Zuo nmcXLULen OL and me mod& 06 Rhe name component 06 S id6 Mme 0 a mod& f 06 S nuch M 0 1 4 1 9 and & + 1 f.
( 3 . 1 1 ) LEMMA,
me mod&
( P a r t (i) i s well-known.
P a r t (ii) may be e s t a b l i s h e d by using P a r t
(i)
a f t e r expanding t h e language so as t o make e x i s t e n t i a l formulas atomic).
( 3.12 ) COROLLARY L e t T . be a u n i v m d Rheohy, and upp pone T 0 mod&c o r n 0 f e n t ULith Ol Then & .the mod& 06 T which m e u t e r n i o r n 06 OL ate mod& ob Rhe n a m ~component T, 06 T. I
.
( 3.1 3 ) COROLLARY
L e t T be a u n i v m d RheOhy, S an inductive Rheoty,
SOME THEOREMS O N O M I T T I N G T Y P E S
233
.
and nuppone T 0 model- c o m D t e n t w i t h S 16 S b h e d u c i b l e , .then exa c t e y one component TI ad T 0 c_ S, and evetry o M m component 06 T 0 incombintent LtLith S (Te_S w , so i f O%,g I= S, then by 3.11 (ii), t h e r e i s a f b S such t h a t
.
P and &
01 +
of T
, and
+
P i by 3.11 (i), 02 and
are models o f the same component
a l l the o t h e r components o f T are i n c o n s i s t e n t w i t h
S).
( 3 . 1 4 ) THEOREM, Let T be a u d v m d fitheohy, S a n i n d u c t i v e C h e o h y , and suppone T 0 model - comhintent l o a h S . Then (i) S h a the T - AP evetry component 0 6 S h a .the T - AP. Nex-t, M h W e fithat S 0 i m e d u c i b l e : (ii) S h a ,the T - AP id6 S h a .the T ' - AP doh bVme component T ' 06 T.
((i) i s immediate and (ii! follows from C o r o l l a r i e s 3.12 and 3.13). I n p a r t i c u l a r , S h a Rhe T - AP Ll;d evetry component S' a6 S h a the TI-AP 6 0 1 ~nome
compvneflt T ' o d T .
(111) COUNTABLE
FRAGMENTS OF L~~~
.
Throughout t h i s discussion, we l e t LA be an a r b i t r a r y b u t f i x e d countable fragment o f L and we take
WlW
.
The l o g i c a l symbols o f LA are 1 ,
t o be defined by
A@=
v,
!,
3 , and=,
1V { 1 0 : @ E 01. L e t L be t h e f i
n i t a r y language whose r e l a t i o n , f u n c t i o n and constant symbols are those
of
L A . We form a new f i r s t - order language L' by adding t o L a new r e l a t i o n f o r every i n f i n i t e d i s j u n c t i o n VQ i n L A . Every f o r m u l a @ i n symbol R
V@
LA i s associated w i t h a formula
0' i n L ' ,
obtained from
0 by
r e p l a c i n g each
occurrence o f any i n f i n i t e d i s j u n c t i o n V @ by R V t J . L e t V ' b e the theory o f L ' whose axioms are a l l t h e formulas where
0e
@ and VQ ranges over a l l the
$
+
RV0,
i n f i n i t e d i s j u n c t i o n s i n L A . We
c a l l 01 a n a n d a t d model o f L ' i f 02 I= V ' and, f o r each i n f i n i t e d i s j u n c t i o n V @ i n L A , 01omits t h e type
{RV@
, I@'
: @
E t ~ }L.e t 1 denote t h e
s e t o f these types, w i t h VQ ranging over a l l t h e i n f i n i t e d i s j u n c t i o n s i n L A . Then a model 02. o f L ' i s standard i f f 01 b V ' and Ol omits C . The r e l a t i o n s h i p between standard models o f L ' and models o f L A i s g i v en by: (3.15)
(i)1601 0 a n t a n d n n d r n o d e l o d L ' ,
RhenUlha ahduOt01'wkich
234
C. C. PlNTER
0 a m o d e l 0 6 L A , and 01 != @ ++ $ ' doh evmy 60hmuf.a @ i n LA. (ii) 16 3 D a m o d e l 06 L A , & h a a unique expwuion t o a ntundahd m o d e l $' 06 1' A U C ~t h a t +' b $ .- 9 ' doh evmy 60hmuf.a @ i n 'A ' (This r e s u l t i s e s s e n t i a l l y known, and r o u t i n e t o v e r i f y ) . L e t V A designate the 1
- completion
o f V'
.
Obviously, every s t a n d a r d
model o f 1' i s a model o f V A . I f @ i s any sentence o f LA ,then @ i s t r u e i n every model o f LA i f f @ ' Thus,
.
VAI-
0' i s t r u e i n every standard model o f
1'
i d 6 V A I- @ ' . A The f o l l o w i n g i s a c h a r a c t e r i z a t i o n o f E-complete t h e o r i e s i n
(3.16)
(3.17)
Foh
evmy
60hmu.h
i n LA
@
,
iff
I-L 0
PROPOSITION, Let T b e a f i e o h y i n 1' nu&.thcLt
L'.
VA5,T. T D
z - complete id6 T natibdies f i e 6oUmLng condition: doh each indinite d h junction VO i n L A , id TI- ( W v ) [ @ + ' J,] doh evehy $ € O , t h e n T + b v ) inv0 $1. +
PROOF: Suppose T i s E - complete; t h i s means t h a t i f T I- 5 + l$' f o r every @ E O and T I- 5 - + R then 5 i s i n c o n s i s t e n t w i t h T . Now supV@ ' pose t h a t TI-@' J , , t h a t i s , T I - l $ + I@' f o r every @ € 0. We c l a i m t h a t T I-1 ii, + l R v O ; otherwise, 1 ii, I\ PyO i s c o n s i s t e n t w i t h T, y e t -+
T
+ 1 J, A
RVO
-+
1 4 ' f o r each
possible. Thus, T I-
-+
J,
.
@ € @
and
+
1 J, A RVO
+
,
RVO
which i s i m -
The converse i s imnediate. 0
I f T i s any s e t o f sentences i n LA, l e t T 1 = V A U
:@ €
{@I
TI. A
set
o f sentences T i n LA i s s a i d t o be deduotivdy cloned & LA i f f T includes the axioms o f LA and i s closed under the r u l e s o f i n f e r e n c e
for
LA.
By
(3.16) and (3.17) we conclude t h a t f o r any s e t T o f sentences i n LA, (3.18)
T 0 deductively d o n e d i n LA id6 T' 0 Z - compLde.
I f T i s a s e t o f sentences o f LA , t h e deductive d o n u h e o f T i s t h e s m a l l e s t d e d u c t i v e l y closed s e t o f sentences c o n t a i n i n g T. One e a s i l y ver-
if i es t h a t , (3.19)
U 0 f i e deductive d o n u h e
06
T id6 U' 0 .the I: - campLeA*ion
T'. We say t h a t T i s a compLete s e t o f sentences i n LA i f f f o r every
06
sentence
235
SOME THEOREMS ON OMITTING TYPES
€ T ( b u t not b o t h ) . T i s con,5Dtent i n LA i f f the in LA , @ E T or i@ deductive c l o s u r e of T i s a proper subset of LA T h u s , a complete s e t of sentences of LA i s c o n s i s t e n t i f f i t i s deductively closed. Combining ( 3 . 1 8 ) - (3.19) with ( 2 . 3 ) - ( 2 . 5 ) , we g e t t h e following, where T i s any set of sentences of L A :
.
(3.20) T 0 deductively cloned id6 T 0 t h e inte,uecZioian compleRe, c o n 5 D t e n t extenhion5 . (3.21 )
T
(3.22)
16 T 0 coii5inZcni in
0 c o n n h t e n t i n LA
06 T .
456
06
all
.i&
T h a a model.
LA , T
A
@
id6 $I 0 h u e i n evehy m o d e l
The l a s t statement immediately yields the completenun Thevmm doh L
WlW
.
The O n i i t t i n g - Rypeb ThcV4CJii 304 LA follows e a s i l y from (3.18) and the observation t h a t i f T ' i s C-complete and 0 - complete, i t i s a l s o ( C U @)-corn plete. Although ( 2 . 1 ) , t h e Omitting- types Theorem, does not hold f o r uncountable languages, t h e construction of L ' o u t l i n e d above i s p o s s i b l e even when LA i s an uncountable fragment and when L has uncountably many r e l a t i o n , WlW function o r constant symbols; (3.15) remains v a l i d i n t h i s case. From t h e ( i n L ' ) 0 6 a ntandmd model observation t h a t e v u y elerneit,t,tahy nu6n.t~uct~ute 0 a Atandad model?, t o g e t h e r with t h e downward Lowenheim - Skolem Theorem f o r f i n i t a r y languages, we g e t t h e following s t r o n g form of the Lowenheim: ( I f LA i s any fragment of L , we l e t Skolem Theorem f o r L #LA WlO
W1W
designate the cardinal 11 L 11 + the cardinal number of t h e s e t of sentences of LA which a r e i n f i n i t e d i s j u n c t i o n s VO ; we note t h a t # L A = 11 L ' I I ) ;
(3.23) L e t LA 6e a dhagment v 6 L (uhehe L may have u n c o u n t a b k j 1W W lW many nvn- logic& nym6vD). L e t 02 be a model 0 6 LA nuch 2Ita-t catld m=a and
.
* OZ a L B 2 = LA Given any b& X c_ A 0 6 polom 5 6 , thehe 0 a 06 LA poweh 6 containing X . Many o t h e r p r o p e r t i e s of L follow s i m i l a r l y from t h e i r f i n i t a r y W1W counterparts and ( 3 . 1 5 ) - i n p a r t i c u l a r , known statements r e l a t i n g t o element a r y c h a i n s , homogeneous models and i n d i s c e r n i b l e s . Related l o g i c a l systems, such a s weak second order l o g i c and l o g i c w i t h t h e added q u a n t i f i e r " t h e r e e x i s t i n f i n i t e l y many", may be t r e a t e d i n a s i m i l a r manner.
-
236
C. PINTER
C.
4, FINAL
REMARKS,
In Section 2 we gave the definition: T c o d i n a l l y om& X i f f every inf i n i t e model of T can be extended t o a model of T which omits Z . I t i s shown in Bacsich and Rowlands Hughes 1974 t h a t t h i s notion has several imp o r t a n t applications in the special case where T i s an inductive theory and each u in C consists of universal formulas. A semantic characterization of "T cofinally omits u " i s given in Bacsich and Rowlands Hughes 1974 for t h i s special case, b u t i t has the disadvantage of involving a r a t h e r u n wieldy sentence of L which seriously r e s t r i c t s i t s usefulness. The f o l WlW lowing i s a simple characterizarion of the same notion: ( 4 . 1 ) THEOREM, L e t T be an i n d u ~ 2 v etheohy, and Z a couiztabLe b e t 06 u conb0LLng a d u n i v t & d doRmuLa. T c o ~ i n & y o m h 2 c id6 doh each a i n Z , t h e ~ a U v w i n g condition h o l h : .@pa
(4.2) T k E~
+
Foh any c h o i c e ufl dot each n
: M E w
,
E w} :
0 6 ~Xistetentidd o t m u t a b buch t h a t n E w } 0 incvnbAtenten,t t k t h T.
( 4 . 2 ) i s equivalent to: u 0 n o t ~ J tTh e deduotive d V b w l e v d a n y conbhtent exhtebu%d .type, w a h h a p e c t t o T
REP,IARK:
.
o f t h e Theorem: Our condition i s necessary: f o r i f t h e r e i s a u E Z and {E,? : n E w } as in (2.9) such t h a t { E ~: ti E w } i s canbibtent with T , then there i s a model 02 L T and an a in OZ such t h a t Ul k E n i Z l ] f o r each n E w ; c l e a r l y , no extension of a t o a model of T can omit u . To prove the sufficiency of our condition, we will construct an extensionof 02 != T which will omit z . Let 0 be a s e t of sentences of L(02) which i s maximal with respect t o the two conditions: (i) D ( a ) 0 j1 (a), and (ii)T U @ i s consistent. Let a' b T U 0 ; by thc m a x i m a l i t y o f 0 , i f OL b ' @[Z: f o r any $ E W, and a in dl, then T U 0 F $(;),hence T k E + $ f o r some E E 3I and a'I= E[?L]. Thus, f o r each u E C and a in Ol , i f 5 s a t i s f i e s u in a: then a s a t i s f i e s {cfl : n E UJ} inOZ',where T k E~ + un f o r each n E w . By assumption, t h i s i s impossible. I t follows t h a t foreach 5 in a, a does not s a t i s f y any u in a'. Using t h i s l a s t observation w times, ( n E w ) of models of T , we construct a chain a = a,, gal 5 ...5 a, 5 with an+1 = 0 2 ; , and l e t & Clearly i$ i= T and & omits each
PROOF
=idwan.
...
u e z.0 One e a s i l y v e r i f i e s d i r e c t l y t h a t i f T and Z are as i n ( 4 . 1 ) , then T
237
SOME THEOREMS O N OMITTING TYPES
CO&&ai&
om&
C
4 6
C h omiZted i n euehy e x i n t e n t i d y d o b e d mod&
06
T.
A f i n a l observation r e l a t i n g t o c a t e g o r i c i t y i s o f some interest. We will say t h a t T i s a - categohicd w&h h u p e c t t o I: i f f T i s 1 - c o n s i s t e n t and a l l t h e models o f T of power a which omit X a r e isomorphic. PJow suppose T i s w - c a t e g o r i c a l -;!ith r e s p e c t t o Z ; models of T which o m i t E a r e models o f Tz, and TZ i s t h e i n t e r s e c t i o n o f a l l i t s complete e x t e n sions having countable models omitting I: ; we deduce t h a t TC 0 compleke. Let T be any type such t h a t TC l o c a l l y omits T ; i f a model of T omit t i n g Z were t o r e a l i z e T , so would a countable model omitting C t. b u t ano t h e r countable model omitting Z would omit T , which i s impossible. T h u s , every model of T omitting X i s atomic: T A o - categohicd uLith hunpect t o Z i d 6 t h e mod& 06 T which (4.3) OWZ d ~ e exaotey . t h e aiomic mode& 0 4 TC . 1n patLticLLeah, T z h a compKeke, atomic theohy. The notion o f a - c a t e g o r i c i t y w i t h respect t o c y i e l d s t o t h e same methods, roughly, a s a r e used i n L , and s i m i l a r r e s u l t s can be obtained. W1W
REFERENCES, Bacsich, P. and D. Rowlands Hughes 1974, Syntactic chahactehizatioM ad onidgumation, convexity and hehted p ~ t o p e ~ ~ Y uThe , Journal of Symbolic Logic: vol. 39, 433-451. Chang, C. C. and H. J . Keisler 1973, " M o d e l T h e o r y , N o r t h Fisher, E . and A . Robinson 1972, Inductive. t h e o ~ L uand 12, 95 - 107. Morley, M. 1967, CountabLe mode& 06 5 , 6 5 - 72.
-
H o l l a n d , Amsterdam.
theh dohung companionh, I s r a e l J . Math., vol.
zI - categohicd
theo&iu,
I s r a e l J . Ffath., vol.
Robinson, A. 1971 l n ~ i d eB O h u M g i n modeL theohb!, P r o c e e d i n g s of t h e S e c o n d Scandinavian Symposium in Logic ( O s l o 1971 ) , North-Holland, Amsterdam, 317-340.
238
C . C . PINTER
Yasuhara, M. 1974, The am@amaLion genehic mod&,
ptopehty, -the u n i u m a l - homogeneow madeh,and -the Math. Scand., v o l . 34, 5 - 36.
Department o f Mathematics Buckne 1 1 Uni v e r s i t y Lewisburg, P e n n s y l v a n i a , U . S . A .
Non-Classical Logics, Model Theory and Computabi lit y , A . I . Arruda, N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
SEMI-FORMALBETH TABLEAUX by
ANDRES R . RAGGIO
Beth Tableaux a r e one of t h e most e l e g a n t systematizations of quantif i c a t i o n theory. B u t they have a s e r i o u s drawback: formulas w i t h functions cannot be analyzed d i r e c t l y b u t only i n d i r e c t l y eliminating t h e f u n c t i o n s w i t h t h e help of new p r e d i c a t e s . Hilbert-type axiomatizations a r e i n t h i s respect simpler; i n order t o handle functions only t h e morphology must be changed extending t h e term-definition. The reason f o r t h i s drawback of Beth tableaux i s t h e following: a n universal a t e x i s t e n t i a l q u a n t i f i c a t i o n a t the r i g h t of t h e tableaux ( a the l e f t ) must be r e f u t e d ( v e r i f i e d ) by a l l i n s t a n t i a t i o n s using t h e terms t h a t have occurred so f a r i n the tableaux. If there a r e no f u n c t i o n s , these terms a r e f i n i t e in number; but with f u n c t i o n s the possible i n s t a n t i a t i o n s a r e denumerable. And t h i s case cannot be handled because of t h e f i n i t a r y character of t h e Beth tableaux rules. If we drop t h i s f i n i t a r y r e s t r i c t i o n , wo can analyse f o r m u l a s w i t h e must change o n l y two functions i n a very e l e g a n t and perspicuous way. W r u l e s : e x i s t e n c i a l q u a n t i f i c a t i o n t o t h e r i g h t , and universal q u a n t i f i c a t i o n t o t h e l e f t . In both cases we p r e s c r i b e t h a t t h e next l i n e of t h e t a bleau must contain a l l i n s t a n t i a t i o n s of the q u a n t i f i c a t i o n u s i n g the terms - functions and v a r i a b l e s - already employed i n t h e t a b l e a u . In t h i s way we g e t eventually a new l i n e formed by a d e n u m e r a b l e l i s t offormul a s having a q u a n t i f i e r l e s s . The l i s t must be handled a s a s i n g l e node of
.
the tab1 eau A semi-formal Beth tableau - e i t h e r a closed o r an open one - i s no longer always a f i n i t e o b j e c t . B u t i t i s s t i l l a c o n s t r u c t i v e l y g e n e r a t e d
239
240
ANDRES R . R A G G I O
o b j e c t , and t h i s i s t h e o n l y i m p o r t a n t p r o p e r t y i n p r o v i n g t h e f o u r fundamental metatheorems o f q u a n t i f i c a t i o n t h e o r y ; namely, completeness, c u t e l i m i n a t i o n , i n t e r p o l a t i o n , and Herbrand. By t h e way, s y n t a c t i c compactness i s a l s o v a l i d because o f t h e d e f i n i t i o n o f a c l o s e d t a b l e a u as a f i n i t e development
-
denumerable nodes c o u n t as a s i n g l e e n t i t y - w i t h
the
same
p r i m e f o r m u l a s a t t h e l e f t and r i g h t i n e v e r y branch. I n a c l o s e d t a b l e a u we d e l e t e a l l f o r m u l a s which a r e n o t used and we o b t a i n i n
this
way
a
s t r i c t l y f i n i t a r y proof. We show n e x t how t o p r o v e t h o s e f o u r metatheorems. The p r o o f s
are not
more c o m p l i c a t e t h a n t h e usual ones and i n one case, Herbrand, d e f i n i t e l y s h o r t e r and e a s i e r .
Completeness: we f o l l o w t h e u s u a l p a t t e r n . To be s u r e t h a t , i f necassign t o each f o r m u l a a l e f t i n d e x e x p r e s s i n g i t s d e p t h i n t h e t a b l e a u , and a r i g h t essary, e v e r y f o r m u l a i n a t a b l e a u i s g o i n g t o be analysed, we
i n d e x e x p r e s s i n g t h e p l a c e o f t h e f o r m u l a i n i t s node f r o m l e f t t o
right.
We add t h e r u l e t h a t f o r m u l a s s h o u l d be handled f o l l o w i n g t h e o r d e r o f
the
sum o f t h e i r two i n d i c e s .
Cut-elimination: we f o l l o w t h e u s u a l p a t t e r n . As L o r e n z e n and S c h t h t e have skown, t h e t r e a t m e n t o f t h e two i n f i n i t e r u l e s p r e s e n t no d i f fi c u lties. Interpolation: ,we f o l l o w t h e u s u a l p a t t e r n . As i t i s w e l l known t h i s p r o o f i s v e r y cumbersone when f u n c t i o n s a r e n o t e l i m i n a t e d i n t r o d u c i n g p r e d icates.
Herbrand: i n t h e semi-formal B e t h t a b l e a u x we do n o t need t o u s e t h e ext r e m e l y c o m p l i c a t e d p r o o f v i a Gentzen's extended Haupstsatz;wecan use H i l b e r t and B e r n a y ' s e l e g a n t p r o o f , b u t w i t h t h e f o l l o w i n g e s s e n t i a l s i m p l i f i c a t i o n : instead o f using t h e i r
€-theorem
we argue as f o l l o w s . Suppose t h a t a f o r -
mula i n prenex normal f o r m i s d e d u c i b l e ; u s i n g c u t - e l i m i n a t i o n w e p r o v e t h a t t h e f o r m u l a o b t a i n e d by d e l e t i n g i n t h e p r e f i x a l l u n i v e r s a l q u a n t i f i e r s a n d s u b s t i t u t i n g i n t h e nucleus a l l v a r i a b l e s bound by t h o s e u n i v e r s a l q u a n t i f i e r s by new f u n c t i o n s whose arguments a r e t h e f r e e v a r i a b l e s o f t h e o r i g i n a l f o r m u l a and t h e v a r i a b l e s bound by a n e x i s t e n t i a l q u a n t i f i e r p r e c e d i n g t h e p r e f i x t h e c o r r e s p o n d i n g q u a n t i f i e r , i s a l s o d e d u c i b l e . F o r example:
in
SEMI-FORMAL BETH TABLEAUX
24 1
But t h e n t h e n t h l i n e o f t h e c l o s e d t a b l e a u s t a r t i n g w i t h t h i s f o r m u l a ( n = number o f e x i s t e n t i a l q u a n t i f i e r s i n t h e p r e f i x ) i s a denumerable l i s t o f q u a n t i f i e r f r e e f o r m u l a s o b t a i n e d from t h e new nucleus by s u b s t i t u t i o n o f terms b u i l t up f r o m f u n c t i o n s i g n s and v a r i a b l e s . Because o f t h e c h a r a c t e r o f t h i s c l o s e d t a b l e a u o n l y a f i n i t e number o f f o r m u l a s n t h l i n e have been used. We b u i l d t h e d i s j u n t i o n t h e r e o f ; t h i s
finite in
the
disjunction
i s a l s o d e d u c i b l e . We c o n t i n u e as i n H i l b e r t and Bernays. Semi-formal Beth t a b l e a u x seem t o be t h e most s i m p l e and e l e g a n t
sys-
tematization o f q u a n t i f i c a t i o n theory.
REFERENCES
I
H i l b e r t , D. and P. Bernays 1939, Grundlagen der Mathematik, Bd. 1 1 , S p r i g e r - V e r l a g , B e r l i n . Lorenzen, P. 1951, Atgeb-che
und L v g D t h c h e u n t w u c h u n g e n Ubeh 6heie. wehbtlnde,
The
J o u r n a l o f Symbolic L o g i c , v o l . 16, 81-106. Raggio, A. R. 1974, A nimpLe phvv6 a6 Hehbhnud'n t h e o h m , N o t r e Dame J o u r n a l o f Formal L o g i c , v o l . XV, nO 3 , 4 8 7 - 4 8 8 . SchUtte, K. 1951, Bw&thevh&thche
theahit,
eh6anung deh u n c n ~ c h e ni n d u k x i v n i n dm zahten-
Mathemdtische Annalen, Bd. 122, 369-389.
Departamento d e Matematica and Centro d e Logica e Epistemologia Universidade Estadual d e Campinas Campinas, Sao Paulo, Brazil.
Non-Classical Logics, Model Theory and Computability. A.I. Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company. 1977
QUANTIFIER
by
ELIMINATIONI N FIELDS J. R. S H O E N F I E L V
Among t h e most i m p o r t a n t c o n t r i b u t i o n s o f Model Theory t o Algebra have been t h e r e s u l t s o f T a r s k i on a l g e b r a i c a l l y c l o s e d f i e l d s
a n d r e a l closed
f i e l d s , t h e r e s u l t s o f Robinson on d i f f e r e n t i a l l y c l o s e d f i e l d s ,
and
the
r e s u l t s o f Ax and Kochen on c e r t a i n v a l u e d f i e l d s . Even a p r e l i m i n a r y e x a m i n a t i o n o f t h e s e r e s u l t s show t h a t t h e y have much i n common.
The ( b y no means o r i g i n a l ) t h e s i s o f t h i s a r t i c l e i s
t h e b a s i c u n i f y i n g concept i s t h a t o f q u a n t i f i e r e l i m i n a t i o n . t h a t t h e c e n t r a l theorem o f each s e t o f r e s u l t s says t h a t
that
We w i l l show
a c e r t a i n theory
admits q u a n t i f i e r e l i m i n a t i o n , and t h a t a l l o t h e r r e s u l t s f o l l o w
easily.
B e c a u s e o f t h e l i m i t e d space, we o n l y t r e a t a l g e b r a i c a l l y c l o s e d f i e l d s and r e a l c l o s e d f i e l d s i n d e t a i l .
1
I
Q U A N T I F I E R E L I M IN A T I O N ,
We s h a l l suppose t h a t a l l ( f i r s t - o r d e r ) languages c o n t a i n o n l y c o u n t a b l y many n o n - l o g i c a l symbols and c o n t a i n a t l e a s t one c o n s t a n t . T h e s e
assump-
t i o n s a r e n o t e s s e n t i a l ; b u t t h e y s i m p l i f y t h e statements and p r o o f s o f t h e r e s u l t s , and a r e s a t i s f i e d i n a l l a p p l i c a t i o n s . A t h e o r y T a d m h quantidieh e l h i f l a t i o n
( a b b r e v i a t e d QE)
if
for
e v e r y f o r m u l a $ o f T t h e r e i s a q u a n t i f i e r f r e e f o r m u l a J, o f T such t h a t $<->J,
i s a theorem o f
T.
There a r e a number o f methods f o r p r o v i n g t h a t s p e c i f i c t h e o r i e s a d m i t
QE. We g i v e one which i s p a r t i c u l a r l y s u i t e d t o t h e cases
243
w h i c h we con-
244
J.
R. SHDENFIELD
s i d e r . We f i r s t need some d e f i n i t i o n s . A type i s a s e t @ variable.
o f formulas, each o f which c o n t a i n s e x a c t l y o n e f r e e
These f o r m u l a s may c o n t a i n , i n a d d i t i o n t o t h e symbols
language being considered,
names o f i n d i v i d u a l s o f
of
the
a structure
A. We t h e n say @ i s h e d i z e d i n A i f t h e r e i s an i n d i v i d u a l a i n A such t h a t $ ( a ) i s t r u e f o r e v e r y f o r m u l a $ ( x ) i n @ W e s a y @ i s 6irzitely he&zed i n A i f e v e r y f i n i t e subset o f @ i s r e a l i z e d i n A .
.
A structure A i s
cvuntabl?y
oatuhated
i f every type
0
containing
names o f o n l y c o u n t a b l y many i n d i v i d u a l s i n A which i s f i n i t e l y r e a l i z e d i n A i s r e a l i z e d i n A.
An isomorphism behueen A and B i s an isomorphism o f a o f A and a s u b s t r u c t u r e o f
substructure
B . Such a n i s o m o r p h i s m i s maximdif
i t has no p r o p e r e x t e n s i o n which i s an
isomorphism between A and B .
The f o l l o w i n g r e s u l t i s proved i n S h o e n f i e l d 1970.
Q U A N T I F I E R E L I M I N A T I O N THEOREM. A theory T a d m QE i66 u ~ h ~ e w h LA a couv~&~bCe mcdee o { T, B 0 n countably oatwrated model 06 T and F i b a maximal? homohpkinm between A and B, then t h e domain a d F LA
A
A. The p r i n c i p a l uses o f QE
a r e i n e s t a b l i s h i n g completeness and model
completeness. R e c a l l t h a t T i s complete i f e v e r y sentence o f T i s
decid-
a b l e i n T , where $ i s decidable i n T i f e i t h e r $ o r l$ i s a theorem o f
T
. The
theory T i s
model compLeLe i f whenever A and B a r e models o f T
such t h a t A i s a s u b s t r u c t u r e o f B , t h e n A i s an elementary s u b s t r u c t u r e of B.
PROPOSITION 1 . L e t T a d m i t QE . Then T i 0 model c o m p l e t e . 16 euehy yuantidieh-6fiee bentence 06 T decidabLe i n T , t h e n T i b complete.
&o
PROOF:
L e t A and B be models o f T such t h a t A i s a s u b s t r u c t u r e o f B,
and l e t
$(a,,
...
, an)
be a sentence c o n t a i n i n g names a,,
i n d i v i d u a l s o f A . We must show t h a t i t i s t r u e i n B . Because T admits
fier-free.
$(a,,
.. . ,
. .. ,
a
of
an) i s true i n A iff
QE, we may suppose t h a t $ i s q u a n t i -
B u t i n t h i s case t h e r e s u l t i s c l e a r .
Now assume t h e a d d i t i o n a l h y p o t h e s i s and l e t $ be a s e n t e n c e
of T
.
QUANTIFIER ELIMINATION I N FIELDS
245
Choose a q u a n t i f i e r - f r e e J, such t h a t $I<->* i s a theorem of T . Replacing each f r e e v a r i a b l e of J, by some c o n s t a n t , we may suppose t h a t J, i s a sentence. Then J, i s decidable in T ; so @ i s a l s o . Q.E.D. We s h a l l now give some a p p l i c a t i o n s of completeness.
An axiomu,x%z&on of a s t r u c t u r e A i s a theory T such a sentence o f T i s a theorem of T i f f i t i s v a l i d in A . We can always obtain an axiom-
a t i z a t i o n of A by taking a s axioms of T a l l sentences t r u e in A. However, we a r e g e n e r a l l y i n t e r e s t e d in obtaining a simpler axiomatization. For t h i s purpose, we s e l e c t some sentences we know a r e t r u e in A and adopt them a s axioms. This gives a theory T having A a s a model. To t e s t whether T i s an axiomatization, we use t h e following t r i v i a l r e s u l t .
PROPOSITION 2. 7 6 A 0 a model u6 T, then T 0 an axiomu,x%zu,x%on 06 A id6 T 0 comptktc. The second a p p l i c a t i o n of c o m p l e t e t h e o r i e s i s given by an equally trivial result.
PROPOSITION 3. 16 T 0 complete, then eve49 nentence lokich . it h~ ue i n one model 06 T 0 h u e i n evehy model 06 T . For our t h i r d a p p l i c a t i o n , we r e c a l l two d e f i n i t i o n s . A t h e o r y T i s axiornatized (decidable) i f t h e r e i s an algorithm f o r deciding whether or not a given formula i s an axiom (theorem) of T . To m a k e t h e n o t i o n o f algorithm p r e c i s e here, we need the theory of r e c u r s i v e functions. A basic r e s u l t i n t h a t theory then leads t o t h e following.
PROPOSITION 4. 16 T 0 axiomu,x%zed and complcte, then T 0 decidable. W e conclude t h i s s e c t i o n with a method f o r e s t a b l i s h i n g the additional hypothesis in Proposition 1 . W e say A i s a p h c s t r u c t u r e f o r T i f every model of T has a subs t r u c t u r e isomorphic t o A . (Note t h a t A need not be a model of T.)
PROPOSITION 5. 16 T has a phime n.~~thuctw~e, then e v m y quanti6ie4-@~ee
246
J . R . SHOENF I ELD
bentence
06
PROOF:
L e t A be t h e p r i m e s t r u c t u r e and l e t @ be a q u a n t i f i e r - f r e e sen-
T 0 decidubLe i n T.
tence. I f @ i s t r u e i n A , t h e n i t i s t r u e i n e v e r y model o f T a n d hence i s a theorem o f T . I f 41 i s i s a theorem o f T . Q.E.D.
2, ALGEBRAICALLY
f a l s e i n A , t h e same argument shows t h a t l @
CLOSED FIELDS,
We b e g i n w i t h t h e f o l l o w i n g problem: f i n d a s i m p l e a x i o m a t i z a t i o n o f t h e complex f i e l d C
.
F i r s t we must s p e c i f y t h e language o f C . (Warning: Theorems o n
QE
a r e v e r y s e n s i t i v e t o small changes i n t h e language . ) As n o n - l o g i c a l symbols o f t h e language o f f i e l d s , and
-
and t h e c o n s t a n t s
we t a k e t h e b i n a r y o p e r a t i o n s y m b o l s
0,
1, and
-1. I n t h i s language we can
+
easily down
w r i t e down t h e f i e l d axioms. F o r each p o s i t i v e i n t e g e r n we can w r i t e
a sentence which says t h a t e v e r y p o l y n o m i a l o f degree n has a root.We t h e n have t h e axioms o f t h e t h e o r y
ACF o f a l g e b r a i c a l l y c l o s e d f i e l d s . admitb
THEOREM 1 ( T a r s k i ) . ACF
QE.
PROOF: L e t A be a c o u n t a b l e model o f ACF; B a c o u n t a b l y s a t u r a t e d model o f ACF; F a maximal isomorphism between A and B We m u s t s h o w t h e domain D o f F i s A. S i n c e D i s a s u b s t r u c t u r e o f A , i t i s a subring containing
.
1.
We know e v e r y isomorphism d e f i n e d on such a s u b r i n g can be extended t o
t h e s m a l l e s t s u b f i e l d i n c l u d i n g i t . Hence D i s a s u b f i e l d o f A. By r e p l a c i n g B by an i s o m o r p h i c f i e l d , we may suppose t h a t
i d e n t i t y on D . I t i s t h e n there i s a which i s the
b & B and an isomorphism o f t h e f i e l d s D(n) i d e n t i t y on
t e l l s us what c o n d i t i o n s
F
is
the
c l e a r l y enough t o show t h a t f o r e v e r y
D and t a k e s a i n t o b
b must s a t i s f y :
.
a &A, a n d D(b)
The t h e o r y o f
fields
i f a i s algebraic o v e r
D, b
m u s t b e a l g e b r a i c w i t h t h e same minimal p o l y n o m i a l ; i f a i s transcendent a l o v e r D , b must be t r a n s c e n d e n t a l . I n t h e f i r s t case, such a b e x i s t s because B i s a l g e b r a i c a l l y c l o s e d . Thus i t s u f f i c e s t o show t h a t some menber o f B i s transcendental over D .
If
d ( x ) i s a non-constant
p o l y n o m i a l w i t h c o e f i c i e n t s i n D,
247
QUANTIFIER ELIMINATION I N FIELDS
6 ( x ) # 0 may be c o n s i d e r e d as a f o r m u l a w i t h one f r e e v a r i a b l e c o n t a i n i n g names o f i n d i v i d u a l s i n D . formulas i s r e a l i z e d i n B.
We want t o show t h a t t h e s e t @ o f t h e s e
Since D i s c o u n t a b l e
i s c o u n t a b l y s a t u r a t e d , we need o n l y show t h a t
a.
A )
(as a subset o f i s finitely
realized
B in
B . Thus we must show t h a t , g i v e n f i n i t e l y many non-constant p o l y n o m i a l s , t h e r e i s an e l e m e n t o f B w h i c h i s n o t a r o o t o f any o f them. F o r t h i s , i t s u f f i c e s t o see t h a t B i s i n f i n i t e .
B u t i f b,,
...
, bn
were a l l t h e members o f B , t h e n
- ... -
(x-bl)
(x-b,
+ 1
w o u l d h a v e n o r o o t i n B . Q.E D. We c a n now s e e why t h e
Quantifier E l i m i n a t i o n Theorem i s so
s u i t a b l e f o r o u r purposes. I t d e a l s w i t h e x t e n s i o n s o f isomorphisms, t h i s i s a q u e s t i o n a b o u t which f i e l d t h e o r y
and
t e l l s us a g r e a t d e a l . I n more
c o m p l i c a t e d cases, we s h a l l be d e a l i n g w i t h f i e l d s w i t h a d d i t i o n a l
struc-
t u r e . I n such cases, t h e c r u c i a l new p o i n t i s t o f i n d c o n d i t i o n s
under
which t h e extended isomorphisms a r e isomorphisms f o r t h e a d d i t i o n a l
struc-
ture.
COROLLARY 1 .
ACF 0 model compLete.
PROOF: By Theorem 1 and P r o p o s i t i o n 2 .
Q.E.D.
We g i v e an a p p l i c a t i o n o f C o r o l l a r y 1. The f o l l o w i n g r e s u l t i m p o r t a n t i n a l g e b r a i c geometry.
i s
very
An a l g e b r a i c p r o o f o f i t , w h i l e n o t d i f -
f i c u l t , i s not entirely t r i v i a l .
PRINCIPLE OF ALGEBRAIC ZEROES. L e t 6 , ( x 1 , ... , x n ) ,..., dr( xl, . . . , x n ) be pOeyMOmiaeh uLith c o e 6 6 i c i e ~i n a n d g e b h a i d y cloned 6 i e l d F . 1 6 t h e y h a v e a camman z a o i n home e x t e n b i o n G t h e y have a cammon zeha i n F . PROOF:
06
R e p l a c i n g G by a l a r g e r f i e l d , we may suppose t h a t G i s
b r a i c a l l y c l o s e d . There i s a sentence 0 , u s i n g names o f m e m b e r s which says t h a t G . By C o r o l l a r y 1 ,
6,
,..., 6,
h a v e a common z e r o .
F,then algeof F
,
Then 0 i s true i n
F i s an elementary s u b s t r u c t u r e o f G . Thus $ i s t r u e i n
248
J. R.
SHOENFIELD
Q.E.D.
F.
We s t i l l do n o t have our axiomatization of C , since ACF i s n o t complete. We therefore add axioms saying t h a t the c h a r a c t e r i s t i c i s 0 , t h u s obtaining the theory ACF( 0 ) . COROLLARY 2 .
ACF(0) i n complete.
PROOF: By t h e Theorem, A C F ( 0 ) admits QE. Moreover the rational f i e l d i s a prime s t r u c t u r e f o r A C F ( 0 ) . Now apply Proposition 5 and 1 . Q.E.D. COROLLARY 3 .
By Corollary 2
PROOF:
.L5 a n a x i a m d z d o n
ACF( 0 )
06
C
and Proposition 2. Q.E.D.
COROLLARY 4 . 16 a s e n t e n c e @ ( v 6 t h e l a n g u a g e 06 6iddh) .L5 a2ue i n C , t h e n it i n t h u e i n eumq a l g e b n a i c a l t q d o b e d d i d d 06 c h M a c t a D l z C
0.
PROOF: By Corollary 2
and
Proposition 3.
Q.E.D.
Corollary 4 i s a special case of a vague empirical principle Lefschetz's Priciple: any a l g e b h a i c 6act &ue i n C 0 &ue i n dgeb&c&q d o b e d d i d d 06 c h m c t e h i b f i c 0 . COROLLARY 5 . ACF ( 0 )
PROOF: By C o r o l l a r y 2
3, REAL
called euekq
decidabee. and
P r o p o s i t i o n 3.Q.E.D.
CLOSED FIELDS,
Now l e t u s a t t e m p t t o f i n d a n a x i o m a t i z a t i o n o f the real f i e l d To apply our method, we must find a theory which admits QE and has R as a model. However, there i s n o s u c h theory in the language o f f i e l d s . For i f @ ( x ) i s a quantifier-free formula, then the s e t x in R such R.
t h a t @ ( x ) i s true
i s e i t h e r f i n i t e or c o f i n i t e . (This i s e a s i l y verified
249
QUANTIFIER E L I M I N A T I O N IN F I E L D S by i n d u c t i o n on t h e l e n g t h o f
3q(x = q
R such t h a t
.q )
On t h e o t h e r hand, t h e s e t o f x i n
$(x).)
i s t r u e i s neither f i n i t e nor c o f i n i t e .
.
What we must do i s i n t r o d u c e a new symbol so t h a t 3 y ( x = y q ) w i 1 1 We i n t r o d u c e t h e b i n a r y r e l a -
be equivalent t o a q u a n t i f i e r - f r e e formula. t i o n symbol
5 ; then
3q( x = y
- y)
i s equivalent t o
We now have t h e language o f o r d e r e d f i e l d s .
can
Now R , i n a d d i t i o n t o b e i n g an
w r i t e down t h e axioms f o r an o r d e r e d f i e l d . ordered f i e l d ,
0sx.
I n t h i s language, we
i s a real closed f i e l d ; t h a t i s ,
e v e r y polynomial
which
has b o t h p o s i t i v e and n e g a t i v e v a l u e s has a r o o t . I t i s easy t o w r i t e down an i n f i n i t e s e t o f axioms which express t h i s ; we need one axiom f o r degree of a p o l y n o m i a l .
We t h u s o b t a i n t h e t h e o r y RCF
of
real
each
closed
fields.
RCF a d r n i t b QE.
THEOREM 2 ( T a r s k i ) . PROOF:
L e t A be a c o u n t a b l e model o f R C F ;
model o f R C F ; F
a maximal isomorphism
B
a countable
between A and B .
saturated We must
show
t h a t t h e domain D o f F i s A . As i n t h e p r o o f o f Theorem l , D
i s a s u b f i e l d o f A . (We need here t h e
f a c t t h a t t h e e x t e n s i o n o f a n isomorphism f r o m a r i n g t o t h e s m a l l e s t f i e l d
mdemd fields.)
including i t i s v a l i d f o r suppose t h a t
F i s t h e i d e n t i t y on
A l s o as i n t h a t p r o o f ,
we may
D.
By t h e p r o p e r t i e s o f r e a l c l o s e d f i e l d s ,
F can be extended t o an i s o -
morphism o f t h e a l g e b r a i c c l o s u r e o f D i n A and t h e a l g e b r a i c c l o s u r e D i n B . Hence by t h e m a x i m a l i t y o f F ,
A and B . Now l e t
a E A.
A g a i n we must f i n d
morphism o f t h e o r d e r e d f i e l d D
and takes a i n t o b Let
@
D i s a l g e b r a i c a l l y closed i n
.
D(a)
and
of both
b E B so t h a t t h e r e i s an i s o D ( b ) which i s t h e i d e n t i t y o n
a I D. x < d , where d E D and a < d , d E D and d < a. We show t h a t 0 i s r e a l -
We may suppose
consist o f a l l formulas
d < x, where i z e d i n B . Because B i s c o u n t a b l y s a t u r a t e d and D i s countable, i t i s enough t o r e a l i z e a f i n i t e subset o f @ . F o r t h i s , i t i s enough t o see t h a t and a l l f o r m u l a s
if
dl < a < d2 w i t h
d,
< b < d,
.
Let b E B
d,,
d,
E D,
thenthere i s a b E B
B u t we c a h s i m p l y t a k e realize 8
(1) a < d - > b < d
.
Then f o r a l l
such b = l2( dl + d 2 ) .
d E D,
that
250
J. R. SHOENFIELD
and
d < a ->
(2)
d < b.
a E D, a
Since
i s transcendental over D .
b I D;
From (1) and (2),
so b i s t r a n s c e n d e n t a l o v e r D . Hence we have a unique f i e l d isomorphism of
D(a)
on D and takes a i n t o b.
D(b) which i s the i d e n t i t y
and
We m u s t s h o w t h a t t h i s i s o m o r p h i s m preserves o r d e r . An element o f D ( a ) i s o f t h e f o r m polynomials into
I t w i l l t h e r e f o r e be s u f f i c i e n t
6(b)/g(b).
(3)
0
c
6(a)->
0
6
f o r every polynomial If
d ( a ) / g ( a ) w, h e r e
s
6,
i t holds f o r a l l
i t holds
for
6,6,.
6
If
6
i s constant, then
pose
6
f o r some
has degree
t o prove
d.6
d € D.
H e n c e we
(1) h o l d s f o r
6
H e n c e we may s u p p o s e t h a t i s clear. If
(3)
d E D; so
2 2 . Since
with
i s 1. I f
reducible.
6(x) = x-d
are this
with coefficients i n D .
(1) h o l d s f o r
6,.
g
and
6(b)
may suppose t h a t t h e l e a d i n g c o e f f i c i e n t o f and
6
w i t h c o e f f i c i e n t s i n D ; and t h e isomorphism c a r r i e s
6
(3)
6
i s o f degree
f o l l o w s from
(1) and
1
i s ir-
,
then
(Z).Now
i s i r r e d u c i b l e , i t has no r o o t s
sup-
in
D
Then i t has no r o o t s i n A o r B ; f o r t h e s e r o o t s would be a l g e b r a i c D and hence i n D . Since t h e l e a d i n g c o e f f i c i e n t o f
l a r g e values o f
x in D
. Combining
these f a c t s
6
6,
i s 1, 6 ( x ) > 0
for
with the f a c t t h a t A and
B a r e r e a l closed, we see t h a t 6 assumes o n l y p o s i t i v e v a l u e s f o r o r B . Thus (3) h o l d s . Q.E.D.
argu-
ments i n A
The f o l l o w i n g c o r o l l a r i e s o f Theorem 2 a r e proved l i k e t h e
corre-
sponding c o r o l l a r i e s o f Theorem 1.
COROLLARY 1 .
RCF i b model? c o m p l e t e .
COROLLARY 2 .
RCF 0 campLett.
COROLLARY 3 .
RCF 0 an ,axiomdzaLLon
R
.
1 6 a sentence $ (06 t h e language 06 o h d u e d t h e n it i b t h u e i n e v e h y cloned 6 i d d .
COROLLARY 4 .
thue i n R ,
06
.
over
i b
251
QUANTIFIER ELIMINATION I N FIELDS COROLLARY 5 .
RCF 0 decidable.
We g i v e an a p p l i c a t i o n of Corollary 1 . In o r d e r t o s o l v e H i l b e r t ' s 17th problem, Artin proved the following r e s u l t . Let d(xl , . .,x,) be a p o l y n o m i a l with r a t i o n a l c o e f f i c i e n t s which assume only non-negat i v e values when i t s arguments a r e r e a l . Then 6 i s a sum of squares of rational functions w i t h rational coefficients.
.
To prove t h i s , A r t i n considered t h e f i e l d G of r a t i o n a l functions o f with rational c o e f f i c i e n t s . If the conclusion i s f a l s e , 6 i s not a sum of squares in G . A t h e o r e m o f Artin and Schreier then t e l l s us t h a t t h e f i e l d G may be ordered so t h a t 6 < 0. A r t i n t h e n used t h i s t o show t h a t 6 must assume a n e g a t i v e v a l u e f o r some real arguments. This p a r t of t h e proof follows r e a d i l y from C o r o l l a r y l . F i r s t , the theory
x1
, . . . ,x,
of real closed f i e l d s shows t h a t G can be extended t o a real closed f i e l d and R i s a s u b f i e l d of H. There i s a sentence I$ w h i c h s a y s t h a t 6 a s s u m e s a negative value; and t h i s sentence i s t r u e i n H, s i n c e 6 < 0 . I t follows t h a t 4 i s t r u e i n R. F i n a l l y , we can o b t a i n an axiomatization f o r R i n t h e l a n g u a g e o f fields. W e take t h e axiomatization in the language of o r d e r e d f i e l d s , and r e p l a c e each formula o 5 Z by 3 x ( o + x 2 = Z ) .
H;
4, OTHER
CASES,
We s h a l l not t r e a t t h e o t h e r two cases i n d e t a i l , s i n c e t h e additional material i s mostly a l g e b r a i c . This material i s discussed i n Robinson 1 9 5 6 f o r d i f f e r e n t i a l l y c l o s e d f i e l d s and i n Kochen 1 9 7 4 f o r v a l u e d e only make a few remarks which w i l l enable t h e reader t o connect fields. W t h e material i n Kochen 1974 w i t h our methods. The language of valued f i e l d s can be formulated i n several ways. However i t i s done, i t i s possible t o make a s s e r t i o n s in t h i s language a b o u t the r e s i d u e c l a s s f i e l d and t h e value group. For example, l e t $(xl, x,) b e a f o r m u l a o f t h e f i e l d language. Then there i s a formula
...,
..
I$*(xl,. ,xn) of the valued f i e l d language with t h e following p r o p e r t y . Let F be a valued f i e l d ; u l ,..., a i n t e g e r s o f F ; F t h e r e s i d u e ,..., t h e c o s e t s o f a l , . . . , a i n F. Then class field;
an
252
R. SHOENFIELD
J.
+(a,,. . .,a
i s t r u e i n F i f f +*(al
I n t h e v a l u e d f i e l d language
,...,an) i s
t r u e i n F.
we can f o r m t h e t h e o r y
of
T
valued
f i e l d s which s a t i s f y H e n s e l ' s Lemma a n d h a v e r e s i d u e c l a s s f i e l d s c h a r a c t e r i s t i c 0. To make t h i s i n t o a t h e o r y a d m i t t i n g QE,
we m u s t
of add
new s y m b o l s f o r e a c h f o r m u l a a b o u t t h e r e s i d u e c l a s s f i e l d o r t h e v a l u e group.
Thus i f
I$(
xl,.
.., x n )
i s as above, we add a new
relation
symbol
P and a new axiom P(X1'
..., xn)-
+*( X1,
...' X n ) .
We c a n now p r o v e t h a t t h e r e s u l t i n g t h e o r y T* admits
QE, and
u s e ' t h i s t o o b t a i n , f o r example, t h e Ax-Kochen r e s u l t s on A r t i n ' s c o n j e c t u r e . Making more use o f t h e r e s u l t s t i z a t i o n o f the f i e l d o f
i n Kochen 1974, we can g e t an axioma-
p - a d i c numbers. However, t h e problem o f
finding
an a x i o m a t i z a t i o n o f t h e f i e l d o f meromorphic s e r i e s w i t h c o e f f i c i e n t s i n a f i n i t e f i e l d i s s t i l l open.
REFERENCES, Kochen, 1974,
S. The
modeL t h e o h g
06
l o c u e d i e l d b , L o g i c Conference K i e l 1974,
L e c t u r e Notes i n Mathematics, S p r i n g e r - V e r l a g , 384-425. Robinson, A. 1956,
Complete Theories, N o r t h - H o l l a n d P u b l i s h i n g Co., Amsterdam.
Shoenfield, 3 . R. 1970,
A themem on quunti6ieh e l h i u z a t i o n , Symposia Mathematica, v o l . V,
173-1 76.
Department of Mathematics Duke U n i v e r s i t y Durham, N o r t h C a r o l i n a , U.S.A.
PART I11 COMPUTABILITY
Non-Classical Logics, Model Theory and Computability, A . I . Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
THE DECISION PROBLEM OF THE CONGRUENCE LATTICES OF PSEUDOCOMPLEMENTEDSEMILATTICES(*)
ON
by
H.
P . SANKAPPANAVAR
1, INTRODUCTION, The s t u d y o f t h e d e c i s i o n problems f o r v a r i o u s c l a s s e s o f ( b o t h a l g e b r a s and r e l a t i o n a l s t r u c t u r e s )
-
which began i n t h e 1930's
-
Church gave t h e f i r s t u n d e c i d a b i l i t y r e s u l t (see Church 1936) c r e d i t a vast
c l a s s e s o f s t r u c t u r e s which a r e , classes o f structures
-
with o t h e r
congruence l a t t i c e s , subalgebra
t i c e s , l a t t i c e s o f subspaces o f geometries, except f o r
lat-
l a t t i c e s o f v a r i e t i e s o f alge-
b r a s , automorphism groups, endomorphism groups, e t c . i n t h e e a r l y 1960's,
until
t h e d e c i s i o n problems f o r
i n a n a t u r a l way, a s s o c i a t e d
f o r example,
when
has f o r i t s
l i t e r a t u r e , and an e x c e l l e n t survey o f t h e work done
1965 has appeared i n Ershov e t a1 1965. However,
much l a t e r ,
structures
-
were c o n s i d e r e d o n l y
r e s u l t s o f Tarski a n d
Grzegorczyk which were p u b l i s h e d i n 1949 and i n 1951
respectively
of (see
T a r s k i 1949, and Grzegorczyk 1951). Tarski proved t h a t the ( f i r s t - o r d e r ) theory o f the spaces o f two
- dimensional
p r o j e c t i v e geometries
lattices
of
sub-
( w i t h p o i n t s h a v i n g homo-
geneous r a t i o n a l c o o r d i n a t e s ) i s undecidable ( T a r s k i 1949).
The problem o f
whether t h i s t h e o r y i s r e c u r s i v e l y i n s e p a r a b l e f r o m t h e s e t o f i t s f i n i t e l y r e f u t a b l e sentences appears t o be s t i l l open. ~~
~~~
~~
Grzegorczyk
(in
the
above
~~~
(*) This work was essentially done in 1973 at the University o f Waterloo, Waterloo, Ontario, Canada a n d was supported by a Province o f Ontario Graduate Fellowship. 255
256
H. P. SANKAPPANAVAR
mentioned paper) ccnsidered the d e c i s i o n t a i n classes o f algebras which are
problems f o r the t h e o r i e s o f cer-
associated w i t h t o p o l o g i c a l spaces:
he
showed the e s s e n t i a l u n d e c i d a b i l i t y o f t h e t h e o r i e s o f closure algebras, o f Brouwerian algebras, o f t h e algebras o f bodies, o f t h e algebras o f convexity and o f t h e semi - p r o j e c t i v e algebra. He a l s o
deduced Tarski's
mentioned above from one o f h i s r e s u l t s . I n 1962 Kargapolov study o f d e c i s i o n problems f o r l a t t i c e s
of
subgroups by
result
initiated
theory o f subgroup l a t t i c e s , and hence a l s o o f t h e congruence l a t t i c e s , Abelian t o r s i o n
- free
reduced groups i s undecidable
the
showing t h a t t h e of
( c f . Kargapolov 1962).
As c o r o l l a r i e s he mentions t h e u n d e c i d a b i l i t y o f t h e theory o f subgroup l a t t i c e s o f groups and o f Abelian groups. In 1970 Kozlov proved t h a t t h e theory o f l a t t i c e s o f subgroups o f f i n i t e Abelian p - g r o u p s
separable ( c f .
Kozlov 1970). As consequences Kozlov
i s recursively i n -
l i s t s a l s o t h e un-
d e c i d a b i l i t y o f t h e t h e o r i e s o f l a t t i c e s o f subgroups o f Abelian p - groups, o f f i n i t e Abelian groups, o f t o r s i o n l i a n groups.
- free
Abelian groups and o f f r e e Abe-
He asks whether t h e u n i v e r s a l theory o f l a t t i c e s o f subgroups
o f Abelian groups i s decidable.
Also i n T a i t s l i n 1970 i s e s t a b l i s h e d
h e r e d i t a r y u n d e c i d a b i l i t y o f t h e theory o f the l a t t i c e o f subgroups d i r e c t sum o f two i n f i n i t e c y c l i c groups and hence t h a t o f
the
the
o f the
theory of
subgroup l a t t i c e s o f any c l a s s o f groups which contains such a d i r e c t sum; he a l s o gives some p o s i t i v e r e s u l t s ,
such as the d e c i d a b i l i t y o f t h e theo-
r i e s c f subgroup l a t t i c e s o f f i n i t e Abe:ian
groups w i t h
f i n i t e Abelian p - g r o u p s w i t h
)L
d i r e c t sums o f
f i n i t e c y c l i c groups.
h
isomorphic
T a i t s l i n proved i n 1968 ( c f .
h
generators,
of
generaLors and o f Abelian groups which are Moving on t o
rings,
T a i t s l i n 1968a) t h a t the theory o f t h e
lat-
t i c e o f i d e a l s of a polynomial r i n g over a f i e l d w i t h a t l e a s t two unknowns i s h e r e d i t a r i l y undecidable, w h i l e t h a t o f a polynomial r i n g i n one unknown i s decidable; i n f a c t , he observed the d e c i d a b i l i t y o f the i d e a l l a t t i c e o f a Dedekind domain.
I n the same year T a i t s l i n published a n o t h e r paper
( T a i t s l i n 1968b) i n which be proved t h a t the theory o f the p a r t i a l l y dered s e t o f simple i d e a l s o f a polynomial r i n g i n a t l e a s t t h r e e
or-
unknowns
over a f i e l d i s h e r e d i t a r i l y undecidable and he mentions the c a s e o f t w o
un-
knowns as an open problem.It i s a consequence o f the r e s u l t s p r o v e d i n R a b i n 1964 t h a t the theory o f congruence l a t t i c e s o f countable Boolean
algebras
i s decidable. More r e c e n t l y , B u r r i s and Sankappanavar have examined t h e dec i s i o n problems o f the t h e o r i e s o f l a t t i c e s o f subrings o f r i n g s w i t h u n i t y ( t h e case o f Boolean algebras i s a p a r t i c u l a r case), o f congruence l a t t i c e s o f s e m i l a t t i c e s , semigroups and unary algebra, and o f l a t t i c e s o f v a r i e t i e s
257
PSEUDOCOMPLEMENTED S E M I L A T T I C E S
( cf.
B u r r i s and Sankappanavar 1975). Boolean algebras are the o n l y non- t r i v i a l proper s u b v a r i e t y (see
e.g.
Sankappanavar 1974) o f t h e v a r i e t y o f pseudocomplemented s e m i l a t t i c e s ,
and
the class o f Boolean algebras i s a subclass o f t h e c o n g r u e n c e - d i s t r i b u t i v e pseudocomplemented s e m i l a t t i c e s . Since t h e theory o f congruence l a t t i c e s o f countable Boolean algebras i s decidable as mentioned above, a question n a t u r a l l y arose whether the theory o f ( a s l i g h t l y l a r g e r class o f )congruence l a t t i c e s o f congruence- d i s t r i b u t i v e pseudocomplemented s e m i l a t t i c e s i s a l s o decidable. I n t h i s note we show t h a t t h i s theory i s r e c u r s i v e l y inseparable (and hence undecidable).
From t h i s we deduce an u n d e c i d a b i l i t y
result of
Ershov and T a i t s l i n 1963 which i n t u r n i s an improvement on an undecidabili t y r e s u l t due t o Grzegorczyk ( c . f .
Grzegorczyk 1951). I t i s a l s o
observed
t h a t the theory o f Heyting l a t t i c e s and t h a t o f f i l t e r l a t t i c e s o f pseudocomplemented s e m i l a t t i c e s are r e c u r s i v e l y inseparable.
2, PSEUDOCOMPLEMENTEDSEMI
LATTICES
AND THEIR
CONGRUENCE LAT-
TICES,
, 0 > i s a A - n PmieaU;ice Luith zeLo i f f A i s a b i L , and 0 i s a d i s t i n g u i s h e d element o f L , s a t i s f y i n g x A y = y A x, x A(y A 2) = ( x A y) A 2, x A x = x , and
An algebra f =
x A 0
L e t f be a A - s e m i l a t t i c e w i t h zero and l e t
= 0.
ordering o f f
.
we d e f i n e L a ,
I f a, 6 € L,
element x i n L i s a peudocomptemerzt g r e a t e s t element such t h a t
8 = $
<S,A,
*,
- semilattice
O>
x A a = 0
61
=
{x
5. be
€ L: a
the
o f an element a i n L i f f x i s the and i s denoted by a*.
An
i s a peudacamptemented nem.Lld-tice i f f < S , A , O >
w i t h zero and
*
partial An
5 x 5 b}.
algebra is
a
i s a unary operatior! on S such t h a t a* i s
the pseudocomplement o f a i n S . The c l a s s o f a l l psecdocomplemented semil a t t i c e s i s a v a r i e t y and i s denoted by PCS. I f S € PCS, we w r i t e S i s a PCS, and i t s congruence l a t t i c e i s denoted by Can S. study o f t h e congruence l a t t i c e s o f
For
a
systematic
pcs's one should r e f e r t o Sankappanavar
1974. We mention here only those concepts and r e s u l t s needed f o r our
pur-
pose. I n t h e sequel S denotes an a r b i t r a r y PCS. An element a i n S i s cloned i f f a** = a
, and
t h e s e t of suchelements i s denoted by B(S),while i t s com-
plement i n S by N ( S ) . The elements o f N(S) are s a i d t o be non - d a h e d , o r e q u i v a l e n t l y , an element a i s non - closed i f f a < a** R ( S ) i s a sub-
.
algebra o f S
and a l s o forms a Boolean algebra i n t h e usual sense.
It
is
258
H. P . SAEIKAPPANAVAR
well-known t h a t t h e c l a s s o f Boolean algebras can be d e f i n e d by a s e t i d e n t i t i e s involving only A
,*
and 0
, hence
of
as a s u b v a r i e t y o f t h e v a r i e t y
PCS. For c € B(S) d e f i n e D C ( S ) = { x € S : x** = c } . I t i s known t h a t Dc(S) i s a subalgebra o f S. For 0 i n Con S, ( e ) , denotes t h e r e s t r i c t i o n o f 0 t o B ( S ) . I t should be noted t h a t (e)B can be regarded e i t h e r as a PCS- congruence o r as a B A - congruence on B(S) considering B(S) as a PCS o r as a Boolean algebra ( i n t h e usual sense) r e s p e c t i v e l y . The mapping ** : S S, b b b * * , i s Q ( S ) o r simply Q , i te. a PCS - homomorphism whose kernel i s denoted O ( S ) = { < x , p: x , q € S and x** = q**}. For a € S we d e f i n e t h e congruence li ( o r ( a ) - )on s as f o l l o w s : +
<x,q>€
LEMMA
1.
ii
iff
Foh a E S,
x , y e S.
x I\ a = q A a,
(a),
= ((a**)- ) B
2 . 16 a , B 8 Con S me huch ,then a = B . LEMMA
.t(2dt
CY
Q, B 2 Q a d ( a ) , = (B),
The proofs o f t h e above lemmas a r e n o t hard and hence omitted. We
also
need t h e f o l l o w i n g p r o p o s i t i o n which i s proved i n Sankappanavar 1974. PROPOSITION
Con S (0) W X W Y . ( x ' < q**
Let
.iA -+
d i n - t % i b m v e i6d S b c l t i b d i t b oh q 5 x ) .
the
condition
x 5 q
We say t h a t S i s congmence- dintzibu-tiwe i f f Con S i s d i s t r i b u t i v e . K be a c a r d i n a l . A PCS S i s a K - Boolean G e b m i f f S i s congru-
e n c e - d i s t r i b u t i v e and lUc(S)l 2 K f o r every c € B(S). I t f o l l o w s t h a t a PCS i s a Boolean algebra i f f i t i s a 1 - Boolean algebra. As and V denote t h e e q u a l i t y r e l a t i o n on S and S x S r e s p e c t i v e l y . e(a,b) denotes t h e congruence generated by < a , b>
3, LOGICAL TERMINOLOGY
AND THE BASIC
THEOREM,
By a lungage we mean a f i r s t - order language w i t h e q u a l i t y ( f o r
these
and o t h e r r e l a t e d n o t i o n s see Shoenfield 1967) which has only a f i n i t e nunber o f non- l o g i c a l symbols. I f L i s a language, a theohy T i n L i s a
set
o f sentences o f L which i s closed under l o g i c a l deduction. A sentence u i n L i s &bLt@ k e d h b l e i n a theory T i f f t h e r e e x i s t s a f i n i t e model o f T i n which l u
i s t r u e . We denote by
t h e s e t o f a l l sentences
which are t r u e o f a l l t h e f i n i t e models o f T , and by T
f
in
L
the set o f all
259
PSEUDOCOMPLEMENTED S E M I L A T T I C E S
f i n i t e l y r e f u t a b l e sentences i n L . lie say t h a t T i s xecwmiveLy inbepaha-
bee i f f t h e r e e x i s t s n o r e c u r s i v e s e t o f sentences A i n L such t h a t T and A i s d i s j o i n t f r o m T r a b l e t h e n T and T f i ,
5
A
It i s clear t h a t i f T i s recursively insepa-
f'
a r e undecidable.
L e t L be a language w i t h one b i n a r y p r e d i c a t e symbol P language(not n e c e s s a r i l y d i f f e r e n t f r o m L ). L e t
,
another
LI
,y)
6 ( x ) and P(X
be f o r L J w i t h one and two f r e e v a r i a b l e s r e s p e c t i v e l y . F o r e v e r y s t r u c 6 t u r e M~ o f L I w i t h u n i v e r s e M I , we d e f i n e a s t r u c t u r e o f L induced by and p - which i s denoted by M1(6 ,p ) - as f o l l o w s :
mulas o f
M1(6
.p)
=
where
0
= {A E
Al : M I k 6 ( ( n ) l
R
= {
t> E M I
and : 6
, t E 0 and MI
b p ( A ,Z)j
The f o l l o w i n g theorem- w h i c h i s t a k e n f r o m B u r r i s a n d 1975- i s t h e b a s i c t o o l i n t h e n e x t s e c t i o n . THEOREM 3 . 1 .
L e X T be a theoxy .&I a Language L ulith t h e popeL@ that T
0 kecwmively h e p a h a b L e . LeA T I be a theahy in 1, and p ( x , y) m e ma 6ohmLLean i n L, nuch that (1)
Sankappanavar
. k n w n e that
6(x)
doh evehy 6inite model N 06 T thetle e x h b a &hLte model MI induced nLtuc2uhe MI (6 p ) N , and
06
T~ nuch that t h e
(2)
60'1- wetly model M1 a6 T I t h e induced nLtuctwle M 1 ( 6 model 06 T .
Then T I 0 heCWL4iV@
4,
in
a
LmepahabLe.
ELEMENTARY THEORY
Let LI
,p )
OF CONGRUENCE L A T T I C E S I
denote t h e language o f l a t t i c e s , i . e .
L I has two
non- l o g i c a l which
symbols A and V . One can w r i t e i n L I formulas Caatom(x), I N LA ( x ) say r e s p e c t i v e l y t h a t "x 0 a c o a t o m : ' , " x 0 A-.ihtleducibLe". kle denote by
DCON t h e c l a s s o f a l l ( d i s t r i b u t i v e ) congruence l a t t i c e s
o f congruence- d i s t r i b u t i v e PCS'S and ~h (DCON) denotes DCON
the theory
of
i n L I , i . e . t h e s e t o f a l l sentences i n L I t h a t a r e t r u e o f !ICON.
260
H . P . SANKAPPANAVAR
THEOREM 4 . 1
Th(DC0N) -in trecwlniu&y imepatrable.
PROOF: L e t T f denote t h e t h e o r y o f an i r r e f l e x i v e , symmetric b i n a r y r e l a t i o n R . I t i s shown i n Ershov 1965 t h a t T f and T' a r e r e c u r s i v e l y i n f separable. L e t 5 be t h e sentence 3X3Y3Z(X # Y &
X
# z
Y #
Z)
and l e t T be t h e t h e o r y a x i o m a t i z e d by T' u e x t e n s i o n o f T', T and T
f
{c l
. Then
since T i s a f i n i t e
a r e r e c u r s i v e l y i n s e p a r a b l e and so
T qualifies
t o be t h e t h e o r y T o f Theorem 3.1. L e t M = < A , R> be a f i n i t e model o f T a, b E A
such t h a t € R (and hence
, so
/A1 2 3.
a # b
gebra <2A
,n , ' , O>
regarded as a PCS
#
denotes:
t h e power s e t
(where 0 stands A
a
A : S
S i n t o t h e u n i v e r s e o f a PCS. D e f i n e an o p e r a t i o n
al-
f o r the
U Al
A). Letting S = 2
s e t and we use 1 f o r 0' which i s
choose
Al = {tab :
new symbol tuband r e q u i r e t h a t tab = tba. Let and l e t 2A denote t h e power s e t o f A and
For every pair
necessarily),
x
we S
+
empty w i l l make S as f o l -
lows:
A
,
(i) i f
b , t E 2
(ii) i f
a , b E A with
(iii)i f
a,b , c
b A t = b
E A with
A RaC = Xac
nt;
< a , b> E R , .tabA tab =
< a , b> E R
'ab ; and < a , c> E R t h e n
A tab = {a) ;
(iv) i f a , b , c , d E A s u c h t h a t { a , b }
r
and
if
a, b E A
tab A
x
=
n
{ c ,d} = B
,
w i t h < a , b> E R
x A tab=
and x E 2A
then
{ a }
if
x={al,
{ b l
if
x = { b } ,
tab i f
x
x
= { a ,b l ,
x # {al,{bl,{a,bl.
A {a,b} if
We t a k e t h e 0 E 2 A as a d i s t i n q u i s h e d element i n S and d e f i n e f o l 1ows : (i) i f
b E
zA , b*
(ii) i f
b E
A,
, b*
= b'
*:
S
+
S
as
;
= { a , 6)" where
Then i t i s easy t o v e r i f y t h a t
< a , b> E R
t u b A t c d = 0;
$ = <S, A ,
b = Lab
* , O>
.
i s i n d e e d a PCS
with
PSEUDOCOMPLEMENTED S E M I L A T T I C E S
B ( S ) = 2A and N ( S ) = t L a 6 : < u , b > € I?). Me a l s o n o t e t h a t {<.tub,{ u , b } > : < a , b > 6 R } . I t i s c l e a r t h a t g s a t i s f i e s condition
26 1 0 =
(D)
and hence i s congruence- d i s t r i b u t i v e . Since A i s f i n i t e , S i s also f i n i t e . The above construction i s i l l u s t r a t e d in Figure 1 , where A = { a , 6 , c , d } and R = { < a , b > , < a , c > , < b , d > , < c , d > , < b , a > , < c , a > , < d , b > , < d , c > l . Figure 1 1
26 2
H. P . SANKAPPANAVAR
L e t us choose Con 8 f o r t h e M~ o f Theorem 3.1 and c o n s i d e r t h e f o l
-
, where x q y i s an a b b r e v i a t i o n f o r t h e f o r m u l a # q &Wz((x = X A z & z = z A q ) * ( z = x oh Z = q)).
l o w i n g formulas i n L J
x
=xA q & X
6 ( x ) -+d e f
Coatom ( x )
-
and
def
P(x , g )
For a € A
6(X) &
6 ( q ) & 3 Z ( l m ~ ( z &)
4 X
2
g).
i t i s c l e a r t h a t {a}" i s a coatom i n Con
since i t
has
two congruence c l a s s e s , namely [ { a } , I ] and L U , { a l * ] . We c l a i m t h a t 6 ( x ) p i c k s o u t p r e c i s e l y t h e congruences o f t h e f o r m { a } " w i t h CLAIM 1.
Con 8 I=
claim, i f and so
a
E A
Con 8
+
a E A.
{a}" f o r some a E A. T o p r o v e t h i s {a}" i s a coatom i n C U M 8 6 ( { a } " ) . Conversely, suppose Con 8 k= S ( Y ) ; then Y i s iff
S(Y)
Y =
then i t i s already noted t h a t
a coatom i n Con 8
. Since
2 i s t h e o n l y s i m p l e PCS, Y
congruence c l a s s e s , n a n e l y [ I ]
and YO]
has c x a c t l y two
(2 i s t h e PCS whose u n i v e r s e
is
{O, J } ) . I t f o l l o w s t h a t [ I ] = [ R , J ] f o r some t E S s i n c e S i s f i n i t e and so Y = 2 . If h E S and tr < .C t h e n h = 0, f o r , if JI # C t h e n d < E < V since < h , . t > E X but <,t,.t>@ .t and < O , r , > E 8 ; b u t < O , r , > @ E , t h i s i s i m p o s s i b l e s i n c e 2 i s c o a t o m . Thus i t f o l l o w s t h a t t i s an cLtom i n S. Since, by t h e c o n s t r u c t i o n o f S , atoms i n S a r e p r e c i s e l y o f t h e f o r m { a } w i t h a € A we have .t = {a} f o r some a E A, which proves Claim 1. CLAIM 2.
For
a,b
E A, < a ,b > E R
iff
p r o v e t h i s , f i r s t suppose < a ,b > E P,
{dA A ibl" 2
@;
=
f o r e by
Lemma 2 we g e t
{ a , b}"
in
CVJZ S
{a,bl"
I t i s c l e a r t h a t (tab)" 2 @
and
{a , b l A by Lemma 1 and ( { a ) " A { b ) n ) B =
, we
have ( ( t a b )) B = ( { a } " i\ {S1")B.ThereA
(,tab)" = {a}" A { b } A . Now
c o n s i d e r t h e congruence
,tab5 {a,b } , we have
{a,b}" 5 (tab)" ;since
. Since
{a,b } > E (,tab)"and < t a b{a, , b) >
.
{ a , b}" < (tab)" I n f a c t , s i n c e and
To
=
since
( { a }A ) 8 A ( { b l " )
.
Con S I== p ( { a } " , { b } " ) .
( { < t a {ba,, b } > } U A g )
{a,b l " +
k? {a,b}"
(Rub)" = { a , b}"
i s an atom i n
.
(tab)* Next we c l a i m t h d t
s i n c e t h e o n l y congruences g r e a t e r t h a n
we i n d e e d
have
v ( I < t a b , { a , b l > }UA,)
C O J Z ~ , we c o n c l u d e
that
- irreducible ;
for,
{a,b}" i s A { a , b}"
A
"
a r e (tab)" , a ,b
and
8,
263
PSEUDOCOMPLEMEIITED S E M I LATT I CES
i t i s clear that
c
Can $
p
Y <
{a,b}" i m p l i e s I ? ( t a b ) " , p r o v i n g t h e c l a i m . Thus
IdA> { b l " ) . To p r o v e t h e converse, suppose
c l e a r t h a t t h e congruence c l a s s e s o f
[{a),
17 n
[{b}, I ] ,
n
and ( S - [ { a } , J ] ) [{a},J]
*],
Let
Y
(S
(S - [ { b } , J ] ) .
(S- [ { a },
(S - I { b } , J : )
have
J] n
-
[{b},
J ] ) , (S -
Since < a , b >
n [ { b } , J ] = r { a , b } , J ] . A l s o w e have
[ { a } ,{b}
n
[{a},
[O, { a , b } * ]
=
a {a}" A { b } "
" 3{a,b}"
i s i m p o s s i b l e ; hence we c l a i m t h a t
Y
2
.
B R,
we
[{a},J] {a}*],
n
see
that
(S - [ { b l , J j ) =
and ( S - [ { a } ,
J
1)
Then i t f o l l o w s t h a t {a}" A { b } " = { a , b } " .
and we w i s h t o show t h a t Y i s n o t A - i r r e d u c i b l e . W e I A l 2 3, {a,b l # 7 . I f Y B = ( { a , b } " ) B t h e n
and s i n c e
<{a, b}, J > E yB 5 Y
u , w E B( $), u #
J ] )n [ { b } , J ] = [ I b } ,
,J ]
[ { a } , J ] ) nL i b }
and so
e ( { a , b}, I ) 5
Y
whence
{ a , b}" = Y
which
W,
YB < ({a, 6})B and consequently t h e r e exist such t h a t < u , W > E ( { a , b l " ) B and < u , w > f Y B . Plow
0
F o r if Y $ @ t h e n f o r some non-closed n,
.
B = {
YJ
.
t h i s i t f o l l o w s t h a t Y i s t h e meet o f maximal elements s i n c e [ a , V] Con ( B ( $ ) ) , i m p l y i n g t h a t 'Y i s n o t A-irreducib1e;hence Con$ #; p ( I a } " , { b } " ) T h i s proves C l a i m 2. From c l a i m s 1 and 2 i t f o l l o w s t h a t i f
+
U
= {{a}" : a E A}
P
and
=
Con $ p({a}", {b}" ) } then < U , P >
{<{a}",{ b } " > :
f i e d . Therefor? Theorem 4.1 i s proved. I n f a c t , we have proved t h e f o l l o w i n g s t r o n g e r r e s u l t once i t i s
-
noted
t h a t t h e PCS S c o n s t r u c t e d i n t h e above p r o o f i s a 2 - Boolean a l g e b r a .
THEOREM 4.2. T h e t h e o h y 0 6 t h e C l a n 4 0 6 c o n g h u e n c e l a t t i c e 4 2 - Boolean a l g e b h a b i b a e c u a n i w e l y i n n e p a a a b l e . COROLLARY 4 . 3 . 1 w i t h any ( o h &)
(1)
06
L e t K be any c h b 0 6 ~@JX&XA which containb l.~~%ceA 0 6 t h e 6oMoWing pope,kCien:
LO d g e b h a i c , dintkLbLLtiwe and atatnic and hub 0 and J ,
264
H . P . SANKAPPAIIAVAR
e u v l y ,Lnte,kud in L i r p b ~ U d O C O m p ~ e m e ~ Cand d, L h.4 ail &anertt buch ,#i& [o, 6; h a cotrip&te atotriic E o o ~ e a n d g c b h a , 6 0 t h e me& 06 aeR coatomn i n 1 and [d , I ] 0 a n dgebhaic n u b l a t t i c e iuhobe cornpad elemem2 60hm a complemented bub4kdXLce. Then T h ( K ) 0 h e c m i v & y imeparrable. (2) (3)
The above c o r o l l a r y i s immediate from t h e f a c t t h a t t h e congruence l a t tices o f 2
- Boolean algebras
have these p r o p e r t i e s (see Sankappanavar 1974).
From t h i s c o r o l l a r y i t f o l l o w s immediately t h a t t h e theory o f Heyting t i c e s i s r e c u r s i v e l y inseparable, which i s an improvement on
lat-
a result of
Ershov and T a i t s l i n 1963 t h a t t h e theory o f d i s t r i b u t i v e l a t t i c e s i s r e c u r s i v e l y inseparable. Ne a l s o o b t a i n t h e h e r e d i t a r y u n d e c i d a b i l i t y o f t h e theory
of
r e s t r i c t e d c l a s s o f f i n i t e d i s t r i b u t i v e l a t t i c c s as given i n the
a
very
following
corollary. COROLLARY 4 . 4
d t h 0 and 1
Let
bUC!t
06 a i d e dina%ibLLt.iue
F be the &n
L
eatticen
ZhaX
L contaim a n dement 6 nuch Rlzat 10, 61 and [d, 1: me Boolean nub&zt.ticw, and (ii) t h e numben. 06 V - h e d u c i b l e elmew2 $ 6 i n L 0 e y u d to .the numbeh 0 6 coatom i n L . Then rh(F1 0 hetleilitatLiey undecidable. (i)
Using t h e same c o n s t r u c t i o n and t h e same p r o o f o f Theorem 4.1, tice of filters of S THEOREM
hence
,in
p(x
,q)
as i n the
F(S),the l a t -
place o f Con S and thus we o b t a i n t h e f o l l o w i n g .
The t h e o h y
4.5
06 a l l
6 ( x ) and
Claims 1 and 2 can a l s o be proved w i t h
06 6 i L t e ~ t&zt.ticw 06 2 - Boolean
d g e h
(and
PCS'b) i n h e C U h b i V d y i n n e p a h a b l e .
5. CONCLUDING
REMARKS
We observe t h a t t h e PCS's S can have d i s t i n c t non
constructed i n t h e p r o o f o f Theorem
- closed elements
Boolean algebre as a neo
- Boolean
which are n o t d i s j o i n t . C a l l
dgebha i f f any two d i s t i n c t
elements i n i t are d i s j o i n t . The above observation leads t o t h e problem which we suspect has an a f f i r m a t i v e s o l u t i o n .
a
4.1 2-
non- closed following
265
PSEUDOCOMPLEME?ITED SEH I L A T T I C E S
PROBLEM. Is t h e theory of the c l a s s o f congruence l a t t i c e s of neo- Boolean algebras d e c i d a b l e ? I t is alsoof interest t o finda lattice-theoretic characterizationofthe congruence l e t t i c e s o f 2 - Boolean algebras.
The author would l i k e t o express h i s g r a t i t u d e t o S. Burris f o r t h e encouragement.
REFERENCES, Burris, S . and H. P . Sankappanavar 1375, L a t t i c e - t h e v h e t i c decinivn p v b l e n i n i~ U n i v e n n d ALgebtran, Algebra Univ. 5, 1 6 3 - 177. Church, A. 1936, A nvte vn t h e e I ~ c ~ I ~ d u i z g n j ~ h vThe b l ~Journal i, of Symbolic Logic 1 , 40 - 41. Ershov, Y . L . , J . A. Lavrov, A. D. Taimanov, and M. A. T a i t s l i n 1965, Ela:iei&mq T / z e v > L u , Russian Math. Surveys, 20, 35 - 105. Ershov, Y. L . , and M. A. T a i t s l i n 1963, Svi;ie uiv,vlvab&e t h e o h i u . ( i n Russian), Algebra i Logi ka, 2 , 37 41.
-
Gratzer, 6 . 1971, Lattice Theory, W . H. Freeman and Company, San Francisco. Grzegorczyk, G . 1951, Undecidability 06 nofile t v p v l v g i c d Rhevhiu, Fund. Math. ,38,137-152. Kargapolov, M. I . 1962, U M -tlze eL!w~eiuW~y,#tevhq a 6 .P&Licu 1 , 4 6 - 53.
v6
nubgfivup,Algebra i Logika,
Kozlov, G . T. 1970, The cindecidabXity a6 t h e Rlzevhy 04 &attice, 06 nubghvupn ad A b m n p - g h v u p , Algebra i Logika, 9 , 167 - 171.
dinite
Rabin, W. 0. 1964, VecidabiLiZg 06 necvnd - v t d a tlzevnieb and au*vr;iata v n i n d i n i t e & e u , Trans. h e r . Math. S O C . , 141, 1 - 34. 1965, k bblp!A r;i&:vd I v h u ~ u f e & d a m phvv&5 and bVme appfiC&OvLJ,kgic, Methodology and Philosophy of Science, Proceedings
of
266
tl. P SANKAPPANAVAR
the 1964 I n t e r n a t i o n a l Congress,Ear H i l l e l e d . , Amsterdam (1965) ,5868. Sankappanavar, H. P. 1974, A study of congruence lattices of pseudocomplemented semilattices, Ph. D. T h e s i s , U n i v e r s i t y o f Waterloo, O n t a r i o , Canada. S h o e n f i e l d , J . R. 1967, Mathematical Logic, Addison Wesley, Reading. T a i t s l i n , M. A. 1968a, Elementahy lattice theohies
dolr
ideaeb i n p a l u r z o t i u d h i y o ,
A1 gebra
and Logic, 7, 1 2 7 - 129. 1968b, U n o h p l e d e & i n p a l y n v n i ~h i n g b , Algebra and Logic, 7 , 394-395. 1970, 012 dmenta/ry theohies 0 6 l a t t i c e s 06 oubghoupn, Algebra and Logic, 9 , 285- 290. T a r s k i , A. 1949, UndecidabLLLty 06 t h e theohie, ad .&uXLces and p o j e c t i u e gevmu7uk.4, The Journal o f Symbolic Logic, 14, 7 7 - 78. T a r s k i , A., A. Mostowski and R. M. Robinson 1953, Undecidable Theories, North - Holland, Amsterdam.
l n s t i t u t o de M a t e m i t i c a Universidade Federal da B a h i a Salvador, Bahia, B r a z i l .
Non-Classical Logics, Model Theory and Computability, A . I . Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
POLYNOMIALLY BOUNDED Q U A N T I F I C A T I O N O V E R HIGHER TYPES A N D A NEW HIERARCHY OF THE ELEMENTARYSETS ( * I b y JANOS S I M O N
I t i s w e l l known t h a t nondeterminism and e x i s t e n t i a l q u a n t i f i c a t i o n a r e
r e l a t e d . I n p a r t i c u l a r NP can be o b t a i n e d by p o l y n o m i a l l y bounded q u a n t i f i c a t i o n o v e r p r e d i c a t e s on s t r i n g s , were t h e p r e d i c a t e s a r e i n P. Meyer and Stockmeyersuggested c o n s i d e r i n g t h e analogue o f t h e a r i t h m e t i c h i e r a r c h y , where t h e a l t e r n a t i n g q u a n t i f i e r s a r e a l l p o l y n o m i a l l y bounded. I t i s n o t known whether t h e t h e r e s u l t i n g h i e r a r c h y i s p r o p e r . In t h i s paper we cons i d e r p o l y n o m i a l l y bounded q u a n t i f i c a t i o n o v e r s e t s and h i g h e r t y p e s , show t h a t one o b t a i n s a p r o p e r h i e r a r c h y o f t h e elementary r e c u r s i v e
and lan-
guages. I n p a r t i c u l a r , w i t h a s i n g l e e x i s t e n t i a l s e t q u a n t i f i e r (and p r e d i c a t e s i n P) one o b t a i n s e x a c t l y t h e n o n d e t e r m i n i s t i c e x p o n e n t i a l t i m e
rec-
o g n i z a b l e languages. E x i s t e n t i a l q u a n t i f i c a t i o n o v e r t y p e i corresponds t o
i levels
0s
e x p o n e n t i a t i o n o f t h e t i m e r e q u i r e d t o a c c e p t t h e s e t by a non-
d e t e r m i n i s t i c T u r i n g machine (Tm). The r e s u l t s may be c o n s i d e r e d as a c h a r a c t e r i z a t i o n o f t h e computational power o f t h e p r e d i c a t e ' € ' .
1, INTRODUCTION
I
The c o n n e c t i o n between n o n d e t e r m i n i s t i c computations and e x i s t e n t i a l
( * ) T h i s r e s e a r c h was s u p p o r t e d i n p a r t by g r a n t 70/755 from de Amparo
2
P e s q u i s a do Estado d e Sao P a u l o
267
(FAPESP) , B r a z i 1 .
Fundasao
268
JANOS SIMON
q u a n t i f i c a t i o n i s w e l l known ( S c o t t 1968). Given a n o n d e t e r m i n i s t i c T ~ T and ' an i n p u t x we add, as a s e p a r a t e i n p u t , t h e s t r i n g o f c h o i c e s y t h a t T ' makes i n an a c c e p t i n g computation. We may t h e n e a s i l y b u i l d a T m T w h i c h o p e r a t e s d e t e r m i n i s t i c a l l y and accepts e x a c t l y t h e same s e t as T ' , i n
t h e same
number o f moves, whenever y i s (an encoding o f ) t h e s e r i e s o f c h o i c e s
that
T I uses. T r e j e c t s i f q does n o t r e p r e s e n t an a c c e p t i n g sequence o f c h o i c e s o f moves. Thus t h e s e t accepted by t h e n o n d e t e r m i n i s t i c machine i s L(T') =
{XI
( 3 y ) T ' ( x , y ) halts and accepts]
T h i s c o n n e c t i o n was e x p l o i t e d i n Meyer and Stockmeyer 1972 and 1973 t o e x h i b i t an analogue o f t h e Kleene h i e r a r c h y f o r P, t h e s e t o f p o l y n o m i a l t i m e r e c o g n i z a b l e languages, by d e f i n i n g
Cp
i+l
= c l a s s o f languages d e f i n a b l e as P
nil
{ y 1 l P x R ( x , Y ) , R in
nPi + l =. c l a s s
o f languages d e f i n a b l e as P
{ y lWPx S ( x , q ) ,
s in
C.
1
where l P x ( W p x ) means t h a t t h e r e i s a polynomial p ( ) such t h a t t h e quant i f i c a t i o n ranges o n l y o v e r s t r i n g s x w i t h 1x1 5 p ( ] y l ) (1x1 d e n o t e s t h e l e n g t h o f x ) . The p o l y n o m i a l i s f i x e d f o r a g i v e n s e t . P
'i+l
= c l a s s o f s e t s d e f i n a b l e by {yl y
is accepted by polynomially ' Tm with a C p oracle)
bounded
i
Then C
P
0
s e t s a r e languages i n P, C
P
1
a r e languages i n NP,
and
Itp
1
s e t s a r e languages whose complement i s i n NP. Many o f t h e p r o p e r t i e s o f t h e Kleene h i e r a r c h y h o l d f o r t h e s e c l a s s e s . I t i s n o t known, however, whether t h e h i e r a r c h y i s proper, s i n c e i t i s n o t
known whether P = NP ; see Stockmeyer 1975 f o r f u r t h e r d e t a i l s . We p r e s e n t i n t h i s paper a p r o p e r h i e r a r c h y o f t h e elementary
recur-
s i v e languages, by a n e x t e n s i o n o f t h e s e d e f i n i t i o n s . T h i s w i l l be done
by
a g e n e r a l i z a t i o n o f t h e p o l y n o m i a l h i e r a r c h y by u s i n g q u a n t i f i e r s o v e r p o l y -
269
P O L Y N O M I A L L Y BOUNDED Q U A N T I F I C A T I O N
n o m i a l l y bounded o b j e c t s o f h i g h e r type, i n a manner analogous t o t h e d e f i t i o n o f the analytical hierarchy generalizing the arithmetical hierarchy. We w i l l q u a n t i f y o v e r s e t s , c l a s s e s o f s e t s , e t c . , where t h e
elements
o f t h e s e t s a r e p o l y n o m i a l l y bounded. E x i s t e n t i a l q u a n t i f i c a t i o n o v e r
type
i w i l l correspond t o i l e v e l s o f n o n d e t e r m i n i s t i c time. We develop now some n o t a t i o n i n o r d e r t o p r e s e n t these r e s u l t s .
A s t r i n g i s an o b j e c t o f t y p e 0 . An o b j e c t o f t y p e i + l i s a c o l l e c t i o n
i > 1 we say t h a t an o b j e c t o f t y p e i i s poLyno-
o f o b j e c t s o f t y p e i. F o r
bounded
nii&y
by i t s parameter y i f f t h e o b j e c t s t h a t i t i s a
collection
o f a r e p o l y n o m i a l l y bounded. The f u n c t i o n s t i ( n )
a r e d e f i n e d by
t i ( n ) stands f o r i l e v e l s o f e x p o n e n t i a t i o n .
Thus
[d(x) ]
NTIME
nondeterministic
denotes t h e language accepted by
T u r i n g machines w i t h i n t i m e
B ( x ) and DTIME [ d ( x ) ] t h e languages accepted
d ( x ) by d e t e r m i n i s t i c T u r i n g machines. L e t
w i t h i n time
NEXPTIME
=
U
p a poLynomiae
NTIME [ Z p ( n ) ]
;
and, i n general
I t i s w e l l known t h a t t h e elementary languages a r e e x a c t l y t h e l a n g u a g e s i n m
U
i=l
DTIME [ t i ( n )
and i t i s easy t o show t h a t
since
3
2 70
JANOS S I M O N
[Zi(n) ] 5
NTIME
Also f o r a l l polynomials
DTIME
[ ti+l( M ) 3 ,
p ( 1, f o r s u f f i c i e n t l y l a r g e n,
We saw t h a t NP ( i . e . U
p a paLynamiae
"TIME [Z,
(ph)] )
can be c h a r a c t e r i z e d as t h e c l a s s o f languages L which c o u l d b e e x p r e s s e d a s
f o r some p r e d i c a t e R L € P . Our
r e s u l t i s an extension o f t h i s characterization t o t h e classes
s. =
'
U NTIME [Zi (p ( M I p a pot ynam i d
namely, t h a t any language L E Si
)
]
may be o b t a i n e d by a s i n g l e
polynomially
bounded e x i s t e n t i a l q u a n t i f i c a t i o n o f a s i m p l e p r e d i c a t e . I n t h e case i = O , t h e q u a n t i f i e r r a n g e s o v e r p o l y n o m i a l l y bounded o b j e c t s o f t y p e 0 and p r e d i i s computable i n d e t e r m i n i s t i c p o l y n o m i a l time, w h i l e i n t h e
cate
case, t o d e s c r i b e a language
general
L € Si, we q u a n t i f y o v e r p o l y n o m i a l l y bounded
o b j e c t s o f t y p e i, and a l l o w a c o n s t a n t number o f polynomiallyboundedquant i f i e r s over objects
o f t y p e a t most i-1. I n a d d i t i o n , t h e m a t r i x ( t h e
q u a n t i f i e r - f r e e p o r t i o n o f t h e f o r m u l a ) becomes a f i x e d Boolean c o m b i n a t i o n o f d e t e r m i n i s t i c p o l y n o m i a l t i m e computable p r e d i c a t e s and membership predicates
-
t h e form t'c
o r , i n o t h e r words, i f we make t h e c o n v e n t i o n t h a t p r e d i c a t e s o f "u € Y"
may be e v a l u a t e d i n u n i t time, t h e m a t r i x i s d e t e r m i n i s -
polynomial t i m e computable i n t h i s new sense. P
3. x t o denote " t h e r e i s a p o l y n o m i a l l y bounded o b j e c t x o f
L e t us use P
t y p e i", and W . x 1
t o d e n o t e u n i v e r s a l q u a n t i f i c a t i o n . We s h a l l w r i t e O_
f o r a n u n s p e c i f i e d q u a n t i f i c a t i o n (i. e. Say t h a t a language L i s
Qy means W
P
i
P
orji).
C p if i t can be expressed as i 1
P
i
271
POLYNOMIALLY BOUNDED QUANTIFICATION
where Qp
Ji
j , < i, t h e q u a n t i f i e r s a r e p o l y n o m i a l l y bounded by 1 x 1 ,
11
and t h e
a r e f i x e d , and R L i s a p r e d i c a t e i n P, except f o r o c c u r r e n c e s o f 'El
T h i s i s t h e o b v i o u s g e n e r a l i z a t i o n o f t h e a n a l y t i c a l and h i g h e r h i e r a r c h i e s f o r polynomial t i m e bounded computations. We c o u l d define .IIP
c l a s s e s and i A
P
j
type
similarly
classes.
Using t h e n o t a t i o n , o u r main theorem i s
P
I n o t h e r words: a language may be d e s c r i b e d by a f o r m u l a o f t y p e
1 i l ob-
( i . e. a s i n g l e e x i s t e n t i a l q u a n t i f i c a t i o n o v e r p o l y n o m i a l l y b o u n d e d j e c t s o f t y p e i, f o l l o w e d by a f i x e d number o f p o l y n o m i a l l y
bounded
t i f i e r s o f l o w e r t y p e s and a d e t e r m i n i s t i c p o l y n o m i a l t i m e matrix) i f
quan-
computable
and o n l y i f i t can be r e c o g n i z e d by a n o n d e t e r m i n i s t i c
Turing
machine i n t i m e bounded by i l e v e l s o f e x p o n e n t i a t i o n o f a p o l y n o m i a l .
I n p a r t i c u l a r we have: LEMMA 1 .
NEXPTIME =
P C . 1 1
I n o r d e r t o p r o v e o u r r e s u l t , we w i l l proceed by i n d u c t i o n . The case
i = 1, t h e s t a r t o f t h e i n d u c t i o n , i s t h e Lemma a b o v e , w h i c h we p r o v e i n the next section.
2, EXPONENTIAL PROOF OF L E M M A 1 : us assume t h a t
TIME Let
p ( ~= )
M
SI
A SINGLE
L E NTIME[Z~(')
-
SET QUANTIFIER, 1,i.e.
L E S ~ .F o r c l a r i t y ,
use t h e f a c t t h a t we use t h i s p a r t i c u l a r p o l y n o m i a l , b u t t h e somewhat c l e a n e r . F o r e v e r y tion o f the
let
i t w i l l become c l e a r t h a t o u r argument does n o t
formulas
are
y € L, we s h a l l d e s c r i b e an a c c e p t i n g computa-
TmM t h a t r e c o g n i z e s y, i. e. o u r p r e d i c a t e w i l l be
L = I y 1 3 X : X represents an accepting computation of M on
input y taking exponential time at most}.
272
JANOS
X w i l l be t h e s e t o f I D ' S o f
- including
M ' s tape
SIMON
M ' s computation ( i . e. a d e s c r i p t i o n o f
t h e square scanned and t h e s t a t e o f t h e f i n i t e
t r o l ) . The o n l y problem i s t h a t t h e t a p e used may be o f e x p o n e n t i a l and o u r q u a n t i f i e r s ( e x c e p t f o r 3 X ) range o v e r s t r i n g s
conlength
o f polynomially
bounded l e n g t h . We overcome t h i s d i f f i c u l t y by r e p r e s e n t i n g each I D as 2' where each element stands f o r a s i n g l e square o f t h e Tm t a p e .
elements, Each
ele-
ment w i l l be a t r i p l e (i.e. a s t r i n g w i t h t h e c o n p o n e n t s s e p a r a t e d markers,
by
$ ) o f t h e form t i m e $ p o s i t i o n $ square. = ( t , p , n ) .
( t , p , 6 ) w i l l represent t h e contents o f t h e
p - t h square o f t h e T m t a p e a t
time
t. We assume t h e T u r i n g machine t o have a s i n g l e i n f i n i t e t a p e and a s i n g l e r e a d / w r i t e head (so t h a t pasLtiun i s w e l l d e f i n e d ) and i f a t t i m e f
t h e T m i s scanning t h e p - t h square i n s t a t e q . we s h a l l r e p r e s e n t t h e c o n t e n t s o f t h a t square as t h e p a i r (u,q) where u i s t h e symbol t h a t t h e s q u a r e c o n t a i n s . I t i s e a s i l y seen t h a t t h e l e n g t h o f t and p w i l l be p o l y n o m i a l l y bounded. I n o r d e r t o ensure t h a t X i s t h e s e t o f I D ' S ,
as claimed,
we
shall
w r i t e a p r e d i c a t e e x p r e s s i n g t h e f a c t t h a t f o r a l l s t r i n g s w E X, a ) w i s o f t h e f o r m (t,p,6) as d e s c r i b e d above (ensured by t i e f o r m u l a A d e f i n e d below); b ) f o r a l l t i m e s we have a complete d e s c r i p t i o n o f M ' s I D ( a n d
only
one), denoted by.8; c ) t h e s e t o f such I D ' S i s an a c c e p t i n g computation o f M on i n p u t y , guaranteed by C. D e t a i l e d d e s c r i p t i o n s o f each o f t h e s e a r e : A = Y Pw E
I'
x 3 't 3'p
r
where
= { M ' s tape alphabet} U {,ti's tape alphabet} x { M ' s state set}.
A c t u a l l y 3'n t i e s f o r b since We
3'6 : w = .t $ p $ n A n E
i s n o t necessary - o n e
r
could simply w r i t e out a l l p o s s i b i l i -
i s f i n i t e - we used i t o n l y f o r conciseness o f e x p o s i t i o n .
s h a l l use t h i s as an a b b r e ' v i a t i o n f o r t h e complete e x p r e s s i o n
which, when expanded, o n l y m u l t i p l i e s t h e l e n g t h o f t h e f o r m u l a constant.
u
n ~ 'r by a
273
POLYNOMIALLY BOUNDED QUANTIFICATION
B = V p t Wpp(t < 2" P
P
P
A p 5 2")
->
36 ~ ' U J[LO = . t $ p $ h A UJ € X ]
A
P
W t W ~ W U J W W ' W6 W O ' [ ~ = t $ p $A u~ J ' = ~ $ ~ $ ~ ' ] - > u J = w ' . C ) An a c c e p t i n g computation i s c h a r a c t e r i z e d by:
c l ) a t t i m e .t = 0 we have t h e i n i t i a l ID o f M w i t h i n p u t y; c 2 ) a t t i m e .t =
P'Mi s
i n an a c c e p t i n g s t a t e ;
c 3 ) i f a c o n f i g u r a t i o n i s assumed a t a t i m e d > O , i t i s a t t a i n e d as t h e r e s u l t o f a sequence o f l e g a l moves f r o m t h e i n i t i a l comfiguration.
Such d e s c r i p t i o n s o f a c c e p t i n g computations
used i n Meyer and Stockmeyer 1972, and 1973 f o r t h e
were
study
of
1ower bounds. c l ) i s ensured by
CI = O$I$y,
x
... A
qi E X A O $ 2 $ y 2 E X A 0 $ 3 $ g 3 6 X A
0 $ n $ y Y I e X A Wpp(p > M A p 5 2'")->
. .ybl, q,
where t h e i n p u t i s y = y,y,.
O$p$b € X ,
i s t h e i n i t i a l s t a t e o f M, and b denotes
t h e symbol f o r b l a n k i n M ' s tape a l p h a b e t . c2): c 2 = ~ ' U J 3't
3'p
36 [ W € X A UJ = t $ p $ 6 A 6 = u x q 6
where a stands f o r some symbol i n
M ' s t a p e a l p h a b e t and
unique f i n a l s t a t e o f M, such t h a t M upon e n t e r i n g q
q
d
]
9
denotes
the
accepts a n d
stops
c 3 ) : To make t h e e x p r e s s i o n s e a s i e r , l e t us assume t h a t M may
eitner
( i t can be assumed wlog t h a t M has such b e h a v i o r ) .
6
r e w r i t e a square o r move i t s head b u t n o t b o t h . T h i s causes t h e r u n n i n g t i m e t o d o u b l e a t most. Now
(.t,p,h)
i s p a r t o f M's configuration a t time t i f
( 1 ) 6 does c o n t a i n a s t a t e component (6 = u x q ) , a t t h e p r e v i o u s moment M was e i t h e r scanning t h e p - t h square i n s t a t e q ' , r e a d b
on
i t , and one o f t h e v a l i d moves o f M upon r e a d i n g a b i n s t a t e q ' i s
t o r e w r i t e i t as an u and go t o s t a t e q ; o r bl was scanning one adjacent
square, and an a l l o w a b l e t r a n s i t i o n was t o move t h e h e a d t o
t o t h e p - t h square.
274
JAFIOS
SIMON
( 2 ) 6 does n o t c o n t a i n a s t a t e component ( 6 = a ) and a t t h e
previous
move e i t h e r M was scaning t h e p - t h square, r e a d i n g a , a n d
i t s
s t a t e caused t h e head t o move; o r M was scanning a n o t h e r squareand d i d n o t m o v e o n t o t h e p - t h square, which a t t i m e t
-
1 c o n t a i n e d a.
We must, i n a d d i t i o n , r e q u i r e t h a t o n l y one move was made 6(y,a) = {(c,y',O),(a,q",L),(a,q"',R))
( c. 9. i f
t h e n i t i s n o t t h e case t h a t
( t , p , y x a ) E X and more t h a n one o f ( . t + l , p , y ' x c ) , ( t + l ,p - 1 , b x y"), ( t + l p, + l , y " ' x d ) E X . I t can be seen e a s i l y t h a t t h i s r e q u i r e m e n t i s s a t i s f i e d ( g i v e n o u r p r e v i o u s r e s t r i c t i o n s ) by ( 3 ) a t a l l t i m e s e x a c t l y one square has a s t a t e s p e c i f i e d i n i t . The e x p r e s s i o n s t h a t s t a n d f o r t h e f a c t s above a r e : C3(1) = WptWppWA E C X Q ,
0 < t 5 ZCn,
t $ p $ A E X ->
([(t-l$p$bxq')E
[ A = ~ X Q ->
X A(ux~,A)
(b,q')]
E
( s t a t e change o n some square) V (3c,q"
c E C,
q" E
Q, ( q , R )
A (t-l$p-l$cxq")EX V [3d,q"'
E 6(c,q")
A(t-l$p$a)EX]
( o r come f r o m t h e l e f t )
d e C, q"' E Q , ( q , ~ )E 6 ( d , q " ' )
A (t-1$ p + 1 $ d x 4 ' " ) E X A ( t - 1 $ p $ a ) EX])]
( o r come f r o m t h e r i g h t ) ;
C3(2) = WptWppW6 € C, 0 < t 5 ZCM, t $ p $ 6 E X ->
[ b = a ->
[
[ t - l $ p $ a x q ) E X A 9 € Q A [ ( a , g ' , R ) € 6(a,y)
A ( t - l $ p + l $ c ) E X A ( t $ p + l $ c x q ' ) E X] V [ ( u , Q " , L ) 6 6(a,q) A ( t - l $ p - l $ d ) A (t$ p -1 $ d x q " ) E X
V [(t-l$p$a)
]]
E X
(head was h e r e b u t moved)
( a c t u a l l y t h i s s u f f i c e s , we show t h e
E X
possibilities)
A[((t-l$p-l$b)
€ X
V ((t-l$p-l$bxq')
A (t-l$p+l$C) EX, E X
A ( ( t $ p - l $ c x y") E X
V ((t-l$p+l$bxq')EX
b,c E Z)
(head was t o t h e l e f t ) V (t$p-Z$c'xq''')EX))
(head was t o t h e r i g h t )
(didnotmovetop)
275
POLYNOMIALLY BOUNDED Q U A N T I F I C A T I O N
A((Z$p+l$c"
(R$p$a
x
x
4 " ) E X V ( Z $ p + Z $ c " ' x 4"')
q ) E X A ((Z$p' $ A ) E X A p # p'*
]]
E X))]
n E E).
Now t h e language L may be expressed as
L = {y I3'X
A A B A C1 A C2 A C3(1) A C3(2) A C3(3)).
I t i s c l e a r t h a t each o f
A, 8,
...,
C3(3) c o n t a i n s o n l y
predicates
t e s t a b l e i n p o l y n o m i a l t i m e (remember t h a t w € X i s counted as a u n i t operation). Also, o n l y 4 q u a n t i f i e r s were used - i f we c o u n t a l t e r n a t i o n o f
quan-
t i f i e r s , o n l y 3 have been used. ( A s l i g h t l y more c l e v e r encoding would u s e o n l y ~3 p r e f i x e s . Remember t h a t we do n o t c o u n t q u a n t i f i e r s o v e r
constant
s i z e d s e t s . ) T h i s proves t h a t e v e r y language r e c o g n i z a b l e i n NTIME[2P(n)],
p ( ) a p o l y n o m i a l , has t h e c l a i m e d r e p r e s e n t a t i o n . T h i s proves h a l f o f o u r lemma. The o t h e r h a l f f o l l o w s f r o m a
simple
P c o u n t i n g argument: i n a l C 1 formula, we have a p o l y n o m i a l l y bounded s e t
I f t h e bounding p o l y n o m i a l i s
P( ) , t h e s e t has a t most Zp(')
.
elements.
Given t h e i n p u t x, t h e n o n d e t e r m i n i s t i c T u r i n g machine t h a t w i l l accept t h e language denoted by t h e f o r m u l a w r i t e s , on one o f i t s t a p e s a v e c t o r length
of
2 P ( n ) , w i t h a 1 i n p o s i t i o n i i f f iis i n t h e s e t . T h i s takes e x -
p o n e n t i a l time, and t h e r e s t o f t h e s i m u l a t i o n i s s t r a i g h t f o r w a r d : e x i s t e n t i a l q u a n t i f i c a t i o n o v e r s t r i n g s i s t r e a t e d by w r i t i n g
t h e s t r i n g down,
w h i l e u n i v e r s a l q u a n t i f i c a t i o n i s t a k e n c a r e o f , by t r y i n g a l l
strings
bounded l e n g t h . The e v a l u a t i o n o f t h e p r e d i c a t e t a k e s polynomial time, c e p t f o r subexpressions o f t h e t y p e
of ex-
x € X, which a r e d e c i d e d by l o o k i n g up
t h e x-th p o s i t i o n o f t h e vector. A l l o f t h i s takes a t m o s t
exponential
time, so t h a t o u r lemma i s proved.
3, THE
MAIN
THEOREM,
F i r s t , we want t o show t h a t a l l t h e languages a c c e p t a b l e by a Tm nondeterministic time
in
276
JANOS
SIMON
( f o r i n p u t o f l e n g t h n ) can be expressed by a p r e d i c a t e .
i
zy
, i. e .
t h a t i s , t h e p r e d i c a t e contains a s i n g l e p o l y n o m i a l l y bounded
existential
q u a n t i f i e r over o b j e c t s o f type i, and a f i x e d f i n i t e number o f polynomiall y bounded q u a n t i f i e r s over o b j e c t s o f type a t most i - 1 , and t h e p r e d i c a t e
preceded by these q u a n t i f i e r s i s computable i n d e t e r m i n i s t i c polynomial time (assuming a l l membership predicates t o take u n i t t i m e ) . We s h a l l proceed by i n d u c t i o n . The case i = l has j u s t been proved. Our approach t o t h e general case w i l l be analogous t o t h e case o f e x ponential time: we s h a l l have an o b j e c t z o f type i, representinganaccepti n g computation o f t h e T m M ( t h a t recognizes t h e language w i t h i n t h e given bound) on i n p u t x . Again, we s h a l l want t o describe an I D as a t r i p l e ( t i m e , p o s i t i o n , square) and w r i t e down the predicates
A-C3(3) that w i l l
ensure
t h a t X indeed represents such a sequence. The problem i s t h a t t h e f i r s t two components now have 1enyth
a n d a l l s t r i n g s must have p o l y n o m i a l l y bounded length. The t r i c k we w i l l use i s t o use encodings f o r t h e numbers, j u s t as we used encodings f o r t h e I D ' S . I n t u i t i v e l y , t h e numbers " s t a y one l e v e l b e low" so t h a t t h e procedure may be c a r r i e d o u t i n d u c t i v e l y . More p r e c i s e l y , we s h a l l prove t h a t w i t h q u a n t i f i e r s o f type may o b t a i n o b j e c t s o f type
i - 1 we
i - 1 t h a t w i l l represent numbers. Also, t h e op-
e r a t i o n s o f forming p a i r s and t r i p l e s o f such o b j e c t s can
be d e s c r i b e d
w i t h i n t h e same formalism, and we may express t h e r e l a t i o n s
a=b a=O a = b+l
a i s t h e f i r s t (second) components o f a p a i r
w = (a,b), a
b, w representing numbers up t o - t i + l ( n ) , using a f i n i t e number o f
quan-
277
POLYNOMIALLY BOUNDED QUANTIFICATION
t i f i e r s of type a t most i - 1 , over polynomial time bounded predicates. Triples can be handled as the r e s u l t of two pairing operations. I t i s only a matter of going over our previous proof ( f o r t h e c a s e i = l )t o check t h a t only the predicates above were used t o write down formulae A - C3(3), so t h a t i f we prove the claim above we shall have proven the theorem. 16 A and €3 ate p o L y n o m i a y bounded objects (pbos) 0 6 t y p e i, x E r, thehe a t e pbos a6 t y p e i ( p e ~ b p w s i t h a kigheh bound) C and U such t h a t LEMMA 2 .
and
1 ) C 4ep4e~enL5 ( A , B ) , 2 ) I) 4ep4ezesenL5 ( A , x ) .
Mufieuuefi, t h e pfiedicate C = ( A , B ) , [D = ( A , x ) ) x = 6A.t I6econd)component 06 C ( U ) ate exp4ennibLe u i n g poLynamidLy bounded quantibiehd 0 6 t y p e s at most i - 1 (except doh i = O When an exintenLLd quanti6iehD needed 604 t h e Lat o n e ) . PROOF: By i n d u c t i o n o n i.
For i = 0, C = A $ B , D = A $ x ; w = A $ B i s deterministic computable, as i s the predicate (w,y) [ L O= x $ y ]
polynomia
time
For i > 0, A = {a,), 8 = { b 1 where a bk a r e objects of type i - 1. J k j' By induction c i k = ( n i , b i ) i s defined, as well as d . = (a.,x).Def ne C =
I, D Jk
{c.
- -
= {d.).Now C = ( A , B ) i f f W W E C , J
A = { aJ. 13bk [ b k E B
1
LO
=
(a. b.) J'J
1
a . E A , bk E 8 and
J
( a j , b j l E C]}.
This lemma j u s t i f i e s the use of t r i p l e s a t a l l l e v e l s . Now l e t us d e f i n e representation f o r numbers. For i = 0 we represent a number bywriting i t down. Since our numbers were bounded by 2' (remember the s i m p l i f y i n g assumption t h a t our polynomials were l i n e a r ) , t h i s could be done in p o l y nomial space. For i = 1, a number will be represented a s a s e t {(position,digit)), with the obvious meaning: p o s i t i o n will be a s t r i n g of polynomial l e n g t h . This enable us t o write - 2 ' positions ( i . e . our number will have length
- 2 ')
and represent values up t o 2 z n
.
I n g e n e r a l , we will assume as inductive hypothesis t h a t the r e p r e -
278
JANOS SIMON
s e n t a t i o n o f numbers up t o using an o b j e c t Ti (pij,digit)
t i + l ( n ) ( i . e. o f l e n g t h t i ( n ) ) c a n b e d o n e
of type i (i2 1) and t h e elements o f Ti a r e o f t h e form
where p . . r e p r e s e n t s *-J
a number ( t h e j - t h p o s i t i o n o f
the
number represented by T i ) and aXgLt i s t h e corresponding d i g i t . M o r e o v e r , we assume t h a t t h e s e t Ti may be defined using a s i n g l e e x i s t e n t i a l q u a n t i f i e r o f type i (and q u a n t i f i e r s o f smaller types). (S'ince we s h a l l use i n a formula which has a q u a n t i f i e r o f type i + l ,t h i s w i l l cause no
Ti prob-
Then we d e f i n e numbers o f l e n g t h ti+l( n ) as s e t s { ( T i , d i g i t )
lems.)
1
where Ti w i l l denote a p o s i t i o n . I n order t o g e t numbers we must a)
ensure t h a t a l l d i g i t s a r e defined;
b)
be a b l e t o deal w i t h these o b j e c t s as numbers 5 ti+,(n) i) ii) iii) iv) v)
t e s t f o r = 0, t e s t f o r = .ti+* (n) - 1 , t e s t f o r e q u a l i t y o f two "numbers", t e s t f o r successor-relationship, test for
<.
Note t h a t i f we can prove type
t h a t a) and b) can be done using
quantifiers o f
i + l a t most, we n o t o n l y s h a l l have proven t h a t we can d e f i n e
bers b u t w i l l have proven our theorem, since t h e predicates b \ i l a r e e x a c t l y those needed f o r t h e p r o o f o f t h e main theorem.
BASIS:
, i. e.
(i= 1)
-
num-
b\iv)
P O L Y N O M I A L L Y BOUNDED QUANTIFICATION
((Z',O)
(i.e.
e Ti <->
(z',O)
279
e Ti))
t h e r e i s a p o s i t i o n z which becomes 1
-
3J a l l d i g i t s higher
t h a n z remain t h e same, w h i l e t h e d i g i t s below change from 1 t o 0 . I n T . a l l d i g i t s up t o t h e ( z - 1 ) s t were 1 ' s ) .
f
T h i s concludes t h e p r o o f f o r i=l. L e t us suppose t h e theorem h o l d s f o r i. INDUCTION (i+ 1) a ) ( w P z ) [ ( ( z , o ) E T v~ ( z , i )
e
T ~ A) ( ( G O )
E T~<->
(Z,U I T ~ ) ) .
b ) I t i s easy t o see, i n a s i m i l a r way, t h a t i ) - v ) s t i l l h o l d o f c o u r s e w h e n e v e r we u s e < f o r e x a m p l e , w e
-
s h a l l have t o
s u b s t i t u t e t h e a p p r o p r i a t e formula,but by t h e i n d u c t i o n hypothes i s such a f o r m u l a e x i s t s and has q u a n t i f i e r s o f t y p e i a t most. T h i s completes t h e p r o o f o f t h e e x i s t e n c e o f n u m b e r r e p r e s e n t a t i o n s . I t a l s o shows t h a t < , = , +1, e t c may be d e f i n e d f o r o b j e c t s r e p r e s e n t i n g
numbers s m a l l e r t h e n
. t i + 2 ( n ) , and such a f o r m u l a may be w r i t t e n by
q u a n t i f i e r s o f t y p e i a t most.
-
C3(3) o f
i=l, and o b t a i n h a l f
o f the
Now we may s u b s t i t u t e these r e p r e s e n t a t i o n s i n t o f o r m u l a e A o u r proof of t h e r e p r e s e n t a t i o n theorem f o r
using
e q u i v a l e n c e f o r general i, i. e. i f a language L i s accepted by a nondeterm i n i s t i c Tm w i t h i n t i m e P
L={YlliZ
P
.ti(n), i t may be w r i t t e n as P
Q 1 x i * * . Q ~ X C R ~ ( X ~ ~ * . ,Z,Y)I . > X ~
where R
Qp's a r e i s d e t e r m i n i s t i c polynomial t i m e computable, a n d t h e L J p o l y n o m i a l l y bounded q u a n t i f i e r s o v e r o b j e c t s o f t y p e a t most i - 1 .
We s t i l l have t o show t h e r e v e r s e i m p l i c a t i o n , i. e. t h a t i f a l a n g u a g e L can be c h a r a c t e r i z e d by
then
L E N T I M E [ . ~ ~ ( ~ ( M]) )f o r some polynomial
p(
).
The p r o o f i s q u i t e s t r a i g h t f o r w a r d , and we j u s t s k e t c h i t . F i r s t p r o v e a s i m p l e 1 emma.
we
280
JANOS SIMON
LEMMA 3 . The nutnbm a 6 a b j e c A o 6 type i polynomi&y
bounded by p b :
ZP(0) a.
.c
=z2
:]i
Ib k n a h y a l p h a b e t 1 .
PROOF: F o r s e t s n1 = Z p ( I y I ) . The number o f o b j e c t s o f t y p e i + l i s ni c a r d i n a l i t y o f t h e power s e t o f objects o f t y p e i , i.e. 2 .
the
Thus t h e c a r d i n a l i t y ni o f an o b j e c t o f t y p e i i s 5 . t i ( p ( n ) ) . We
now
s k e t c h a T u r i n g machine M, t h a t r e c o g n i z e s L . F i r s t i t w r i t e s down
(nonde-
t e r m i n i s t i c a l l y ) t h e s e t Z , i n t a p e and t i m e Mi. F o r t h e q u a n t i f i e r s o f l o w e r types, i t a c t s as f o l l o w s : i f t h e q u a n t i f i e r i s e x i s t e n t i a l , i t w r i t e s down a c h o i c e f o r t h e o b j e c t s ; i f i t i s u n i v e r s a l i t g o e s i n t o a l o o p s t a r t i n g w i t h t h e empty s e t , i t t e s t s whether t h e f o r m u l a i s t r u e ,
if
-
so
adds a new element t o t h e o b j e c t ( w h i l e t h i s i s p o s s i b l e ) and r e p e a t s
the
procedure. I f t h e f o r m u l a i s f a l s e , M r e j e c t s . I f a l l p o s s i b l e v a l u e s
for
the s e t s a t i s f y t h e formula, t h e universal q u a n t i f i e r
was checked.
Since
t h e p r e d i c a t e i t s e l f i s computable i n p o l y n o m i a l time, t h e v a l i d i t y o f aEX can be checked by scanning t h e tape, and t h e enumeration o f an o b j e c t
of
t y p e i - 1 t a k e s a t most .ti(n) steps, t h e r u n n i n g t i m e o f t h e n o n d e t e r m i n i s t i c procedure i s bounded by
ti(p' ( n ) ) f o r some p o l y n o m i a l p'
( ),as claimed.
T h i s concludes t h e p r o o f .
4 CONCLUS IONS I
We o b t a i n e d a new c h a r a c t e r i z a t i o n o f t h e languages accepted w i t h i n i l e v e l s o f e x p o n e n t i a t i o n by n o n d e t e r m i n i s t i c T m s . Although t h e p r o o f s a r e q u i t e messy, t h e u n d e r l y i n g i d e a s a r e c l e a n and t h e r e s u l t i s i n t u i t i v e l y s a t i s f y i n g . Our theorem may be t h o u g h t o f , as c h a r a c t e r i z i n g t h e p o w e r o f t h e symbol ' b e l o n g t o ' i n an environment where a l l o t h e r o p e r a t i o n s
are
p o l y n o m i a l l y bounded. I t i s a l s o s a t i s f y i n g , t h a t w h i l e one cannot c a r r y t h e analogy between t h e a r i t h m e t i c a l h i e r a r c h y and t h e polynomial h i e r a r c h y f a r enough t o p r o v e p r o p e r i n c l u s i o n s , i t i s a t l e a s t p o s s i b l e t o p r o v e
that
g o i n g t o t h e a n a l o g o f t h e a n a l y t i c a l s e t s , one does o b t a i n a p r o p e r h i e r archy:
s i n c e NEXPTIME
NP.
F i n a l l y , a l t h o u g h many o t h e r h i e r a r c h i e s o f t h e e l e m e n t a r y s e t s a r e
POLYNOMIALLY BOUNDED QUANTIFICATION
28 1
known, they a l l correspond t o d e t e r m i n i s t i c resource bounds, so t h a t i t i s a very d i f f i c u l t open problem t o compare t h e l e v e l s i n our hierarchy t o , s a y t h e l e v e l s of t h e Meyer-Ritchie hierarchy, s i n c e t h i s would answerthequestion NTIME ti [ p ( n ) ] ] = DTIME ti [ p ( ~ ) ]] I
REFERENCES, Meyer, A . R . and L. 3 . Stockmeyer 1972,
The eqLLivdence p m b h i d o t h e g d a h e x p t e n h i o n h w L t h h q u u h i n g
he-
quhen e x p o n e n t i d tape,Proc. 13th IEEE SWAT Conf., 125-129. 1973,
Wohd phobYun kequhLMg txpontb~CLdt u p e , Proc. 5th STOC., 1-9
S c o t t , D. 1968, Some d e d i n i t i o n d nuggenfionh d o t au*omccta t h e m y , JCSS, 1 , 2 , 187-1 92. Simon, J . 1974,
On some central problems in computational complexity, TR
75-224, Dept. of Comp. S c i . Cornell U. Stockmeyer, L. J . 1975,
The polynomial-time hierarchy, I B M Res. Rep. RC 5379.
Departemento de C i e n c i a s d a Cornputasso U n i v e r s i d a d e E s t a d u a l de Campinas Campinas, Sao Paulo, B r a z i l .
Non-Classical Logics, Model Theory and Computability. A.I. Arruda, N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
ON RANDOM R, E. SETS by
RCBERT M. S O L O V A Y
ABSTRACT: C h a i t i n has r e c e n t l y proposed two e x p l i c a t i o n s o f t h e e n t r o py o r i n f o r m a t i o n c o n t e n t o f a r e c u r s i v e l y enumerable s e t . The f i r s t , I ( A ) , i s ( r o u g h l y ) t h e minimal l e n g t h o f a program f o r a u n i v e r s a l computer
that
enumerates A . The second, H ( A ) , i s t h e n e g a t i v e log o f t h e p r o b a b i l i t y t h a t an a p p r o p r i a t e l y u n i v e r s a l computer connected t o a random sequence o f
0's
a n d 1 ' s ( w i t h t h e u s u a l d i s t r i b u t i o n ) w i l l enumerate A . I t i s easy t o see t h a t H ( A ) 5 I ( A )
that
I(A)
+ O(1). Our main t h e o r e m a s s e r t s
s 3 H ( A ) + O Q o g H ( A ) ) . The p r o o f may be viewed as a c o n s t r u c t i v i -
z a t i o n o f a theorem o f de Leeuw, et. al. t h a t s t a t e s i f a s e t i s r e c u r s i v e l y enumerated w i t h p o s i t i v e p r o b a b i l i t y by a computer a t t a c h e d t o a
source
o f a random s t r i n g o f 0 ' s and l ' s , t h e n i t i s , i n f a c t , r e c u r s i v e l y enumerable. T h i s paper a r o s e f r o m a q u e s t i o n o f C h a i t i n concerning t h e r e l a t i o n s h i p between two measures o f t h e
"em5topy" o r "in~omu,Lioncontent" o f
a
r e c u r s i v e l y enumerable (r.e.) s e t . Our p r o o f can b e s t be viewed as a "conoZtuOtiueft sharpening o f an o l d theorem o f de Leeuw, Moore, Shannon,
Shapiro
and
1956.
(The word covm%uOtiue should be t a k e n w i t h a g r a i n o f s a l t . I have no i d e a hcw t o f o r m u l a t e , much l e s s prove, t h e main theorem i n t h e f r a m e w o r k o f c o n s t r u c t i v e mathematics,
5 la
Bishop ( c f . Bishop 1 9 6 7 ) ) .
The theorem o f de Leeuw, Moore, Shannon, and Shapiro 1956 c o n c e r n s a T u r i n g machine, M, equiped w i t h a
handam numbe~gcnemloa and a w r i t e - o n l y
o u t p u t tape. The random number g e n e r a t o r e m i t s an i n f i n i t e sequence o f
283
0's
284
ROBERT M . SOLOVAY
and 1 ' s w i t h no c o r r e l a t i o n s among t h e successive o u t p u t s , and with 0 a n d 1 e q u a l l y probable a s o u t p u t a t a n y s t a g e . From time t o time, M w r i t e s a non-negative i n t e g e r on i t s output t a p e , with the successive i n t e g e r s separated by c o m a s . In t h i s w a y , i n t h e course of time M enumerates a s e t A . Let S S u . Suppose t h a t the p r o b a b i l i t y t h a t S i s t h e set enumerated by M i s g r e a t e r than zero. Then, Theorem 2 of de Leeuw e t . a l . 1956 s t a t e s t h a t S i s r. e.. Our improvement i s a s follows. We g i v e a r e c u r s i v e procedure t h a t cons t r u c t s from a Godel number, e , f o r M and a lower bound, l/n, f o r the proba b i l i t y t h a t M enumerate S , a f i n i t e s e t D. ( I n f a c t , D, w i l l have a t most 1 / 2 ( ~ 1 + 1 elements.) )~ 0 will contain some Godel number f o r t h e r.e. set S. The proofof deLeeuw et. a l . 1956 w i l l be r e c a l l e d i n Section 1. One can show t h a t our more p r e c i s e r e s u l t can not be obtained by t h e m e t h o d s o f
.
d e Leeuw e t . a l . One of the key t o o l s i n our proof i s a r e c e n t Lemma of 0. A. Martin which we present i n Section 2. Indeed t h i s r e s u l t was "camm~i66ioMed"foruse
i n t h i s proof. W e c l o s e t h i s introduction with a rough d e s c r i p t i o n of how our p r i n c i pal r e s u l t bears on t h e problem r a i s e d by C h a i t i n . ( P r e c i s e d e s c r i p t i o n o f the r e l e v a n t concepts w i l l be r e c a l l e d in Section 5 . ) Let M, be a uvLiuehosLe Turing machine equiped with a random o r a c l e . Let P(A) be t h e p r o b a b i l i t y t h a t M o enumerates A . Let H(A)= -log2P(A)
L e t 1 ( A ) be t h e minimal number of b i t s of information needed t o s p e c i f y a program f o r enumerating A . Then Chaitin has proved t h e following: PROPOSITIONm
l)H(A) 5 l(A)
+
Co,
2) F u t i n d i n i t d q many A, we have H(A) 5 I ( A ) ~ - l ~ g ~ l ( A ) + C ~ l o g ~ l ~ g ~ l ( A ) .
Here Co a n d C 1 a r e c e r t a i n a b s o l u t e c o n s t a n t s . Of the two results , 1 ) i s t r i v i a l , while 2 ) i s q u i t e n o n t r i v i a l , and i s one of t h e main r e s u l t s o f C h a i t i n 1976. C h a i t i n ' s question was a s follows: I s t h e r e a r e c u r s i v e f u n c t i o n h
ON RANDOM R.
285
E. SETS
such t h a t
I ( A ) 5 h(H(A)) ? A p o s i t i v e answer f o l l o w s e a s i l y f r o m o u r main theorem ( c f . S e c t i o n 5 ) . I(A) 5 3H(A) t O(logH(A)).
The c o e f f i c i e n t 3 comes f r o m t h e exponent 3 i n o u r main theorem. I d o
not
know i f i t i s optimal. T h i s paper would n o t e x i s t w i t h o u t t h e q u e s t i o n o f Greg C h a i t i n a n d t h e mathematical a s s i s t a n c e o f Tony M a r t i n .
Their help i s g r a t e f u l l y
acknowl-
edged.
1, PRELIMINARIES
,
1 . l . w i s t h e s e t o f non-negative i n t e g e r s .
Except i n
Section 5
we f o l l o w t h e usual c o n v e n t i o n s i n which each i n t e g e r n i s t h e s e t smaller integers. tions
A
6
tion of
into
2 .
of all
I f A i s any s e t , A 2 i s t h e c o l l e c t i o n o f a l l funcA I f 6 E 2 and B C A , t h e n 6 l B E '2 i s t h e r e s t r i c -
t o B.
L e t u be t h e measure on 2 such t h a t
v(0) = v(1) = 1 / 2 . L e t p be
the
p r o d u c t measure on w2 which i s t h e p r o d u c t o f c o u n t a b l y many c o p i e s o f v. A subset
subset
27
of
A S w 2 i s &&i.teA!q b a e d '2
i f f o r some i n t e g e r n
a n d some
,
(An a l t e r n a t e d e s c r i p t i o n , which we s h a l l n o t use, i s t h a t A i s f i n i t e l y based i f f i t i s b o t h open and c l o s e d i n t h e usual p r o d u c t t o p o l o g y on w2 . ) L e t us c a l l t h e p a i r < n , V > a code f o r A . Then we can
effectively
t e l l whether two codes a r e codes f o r t h e same s e t , compute t h e measure o f a s e t f r o m i t s code, and compute a code f o r a Boolean c o m b i n a t i o n o f A and B f r o m codes f o r A and 8 . V i a a s u i t a b l e Godel numbering, we may
as w e l l
assume t h a t o u r codes f o r f i n i t e l y based s e t s a r e i n t e g e r s . 1.2.
We( 6 )
Here i s a p r e c i s e statement o f o u r main theorem. I f
be t h e s e t r e c u r s i v e l y enumerable i n
THEOREM,
Thehe 0 a
6
hecuhniue 6unc.tion,
6
w i t h Gijdel number
h ( e , n , k , I?)
€ w2
e
.
, let
huch t h a t ;
286
ROBERT M. SOLOVAY
id A g w, and
~ ( ( 6 :W e ( d ) .then
rn =
= A}) 2
borne k < n h( e,n,k,k).)
60h
Un, 2
( n + n ) / 2 , we have A = Wrn
II
1 and borne
+
(
~ O J L
The f i n i t e s e t mentioned i n t h e i n t r o d u c t i o n i s t h e n
Ue,n = {h(e,n,k,e): k < n
+
, II
1
2
< (n
+ n)/Zl.
The f u n c t i o n h w i l l be o b t a i n e d as f o l l o w s .
w i l l c o n s t r u c t an a r r a y o f
we
.
2
where
e,n,k,
Uniformly i n
sets
r.e.
N = ( n + n ) / 2 ( T h i s w i l l be c a r r i e d o u t i n S e c t i o n 3.) The number k encodes a c e r t a i n f i n i t e amount o f i n f o r m a t i o n needed t o m a k e t h e c o n -
s t r u c t i o n proceed c o r r e c t l y .
Since our construction i s uniform
t h e r e w i l l be a r e c u r s i v e f u n c t i o n
All ue, k,,
for
theorem,
qua
h(e,n,k,R)
k,n,e,
in
which g i v e s a Godel
number
r.e. s e t . Ue show i n S e c t i o n 4 t h a t i f k has i t s c o r r e c t v a l t h e n i f A i s as i n t h e statement
( t o be d e f i n e d below)
A = AL f o r some II w i t h II <
of
The e s t i m a t e we a c t u a l l y g e t o n t h e s i z e o f
U
e,
is
n(n
+
1)’/2,
b i t b e t t e r t h a n what we c l a i m e d i n t h e i n t r o d u c t i o n . O b v i o u s l y we c a n Can we g e t by w i t h a s e t o f s i z e
do b e t t e r t h a n n , 1.3.
6).
We(
a not
0(n2) ?
From now on, we suppress e f r o m o u r n o t a t i o n , and
rather than
the
N.
write
W(
( E v e r y t h i n g we do w i l l be done u n i f o r m l y i n t h e
6)
sup-
W ( 6 ) t o be t h e c h a r a c t e r i s t i c eth r.e. s e t , r e l a t i v e t o 6. We l e t W ( 6 , . t ) be t h e charac-
pressed parameter e . ) Also, we s h a l l t a k e function o f the
t e r i s t i c function o f the portion o f
W(6)
ennumerated b e f o r e s t a g e .t. F o r
o u r purposes, t h e c r u c i a l p r o p e r t i e s a r e : 1)
tl 5 .t* -> W(6,.t1)(n)
2)
W(6)(n) =
3)
W ( 6 , t ) ( n ) = 1 ->n <
4)
W ( 6 , t ) can be e f f e c t i v e l y
lini
5 W(6,t2)(n).
W(d,.t)(n).
t-+-
t. computed f r o m t and
6l.t.
uni-
ON RANDOM R.
287
E . SETS
formly i n e. 1.4. Thus
xA
A cw2, l e t
If
6
(6j = 1 i f
We s a y t h a t a
xA
denote t h e c h a r a c t e r i s t i c f u n c t i o n o f A .
E A and x A ( 6
)
6e
= 0 if
A.
sequence o f s e t s
t h e c o r r e s p o n d i n g sequence o f c h a r a c t e r i s t i c f u n c t i o n s converges p o i n t w i s e .
example, t h i s f o l l o w s from t h e bounded convergence theorem ( c f . Halmos1964, Theorem 260) a p p l i e d t o t h e sequence o f c h a r a c t e r i s t i c f u n c t i o n s . ) 1.5.
For
g E
p(g) =
For
b €
'2,
b E
let
u({6: W ( 6 )
= 91).
let p(b) =
For
%,
u(C6: W(6)In
= A}).
'2 ,.t E w, l e t P(b,t) =
u(C6: W ( f j , . t )
IM = b l ) .
The f o l l o w i n g lemma f o l l o w s e a s i l y f r o m t h e f a c t s r e c a l l e d i n 1.4.
1.6.
The f o l l o w i n g theorem i s
6
s p e c i a l case of Theorem 4 9 B o f Halmos
1964. I t i s used i n t h e o r i g i n a l p r o o f o f de Leeuw,et.al o f t h e i r theorem, which we s h a l l r e c a l l i n a moment. We w i l l n o t need i t f o r t h e p r o o f o f o u r own theorem
(which, o f course, i m p l i e s t h e de Leeuw theorem).
F i r s t some n o t a t i o n . I f
b and 6 i < m , and
f o r the concatenation o f
h ( i ) = ~ ( if o )r
(p(5;n)
b €
m2 and
. I.
6
E
?, we
use t h e n o t a t i o n
e., A-6 i s t h a t h E w2 s u c h h ( m t i ) = b(i). We w r i t e
6-6
that
i s t h e p r o b a b i l i t y t h a t o u r machine w i l l o u t p u t g g i v e n t h a t t h e
288
ROBERT M.
i n i t i a l portion o f
SOLOVAY
o u r random i n p u t i s n . )
1 . 7 . We can now p r e s e n t t h e p r o o f o f de Leeuw e t . a l . THEOREM, j d e Leeuw e t . a l . ) L e X g E ' 2 . actehisfic ,5unction 06 home r.e. A d . First,
PROOF:
consider the
16
case when
p ( g ) > 0, then g 0 t h e chm-
p ( g ) > 1 / 2 . Then i f
A = { n E w: g ( n ) = 11, we can enumerate A by t h e f o l l o w i n g r u l e : P u t n E A i f t h e p r o b a b i l i t y t h a t n i s w r i t t e n on t h e o u t p u t t a p e i s > 1 / 2 . More f o r m a l l y
A = {n: ( 3 X ) p ( I 6 : n E W(6,Z))) > 1/21. ( c f . 1.3.4.).
T h i s e x p r e s s i o n shows t h a t A i s r.e.
I n t h e general case, we a p p l y Theorem 1.6, and t h e f a c t t h a t p ( g ) > 0, t o g e t an
6
so t h a t
p(g;n)
> 1/2
.
I f we now p u t
t h e arguments o f t h e p r e c e d i n g paragraph a p p l y t o
1.8.
g,
W ' ( 6 ) = W(6"6),
w'
The argument o f de Leeuw e t . a1 i s t o o n o n - c o n s t r u c t i v e t o y i e l d
o u r theorem. F o r example, t h e f o l l o w i n g c o n s t r u c t i v i z a t i o n
&7&e.
o f the proof i s
( I t i s e a s i l y r e f u t e d u s i n g t h e r e c u r s i o n theorem.)
There i s a r e c u r s i v e f u n c t i o n f i n i t e s e t o f f i n i t e sequences, 0 E
then
so a g a i n A i s r . e . .
ue,n
I
p,(g;n)
h ( e , n ) such t h a t
h(e,n) encodes
U e t n , and i f p e ( g ) > 1 / n ,
a
t h e n f o r some
> 1/2.
1.9. The f o l l o w i n g t r i v i a l lemma w i l l be needed i n a moment.
LEMMA,Ue width
06 n + 1 open i n t e h v a t n whone u n i o n c o n t a i n n [ O , l ] .
can e d 6 e c t i v d y 6Snd a nequence
tebb than
1/n
PROOF: L e t a4. = i / n - (2i + l ) ~bi , =
i+ 1 -
( 2 i + 2 ) .~
06
ON RANDOM R.
Let
I . = ( a i ,b i ) . F i n a l l y l e t
=
E
properties.
... , g,
12. 5n
,..., In
Then lo
We can now d e f i n e t h e i n t e g e r k,
1.10.
289
E. SETS
be d i s t i n c t f u n c t i o n s i n w 2 w i t h
have t h e d e s i r e d
mentioned e a r l i e r . L e t
p(gi)
g,,,
1/n. I!e p u t
h
Roughly speaking,
A t e l l s us how much o f t h e measure space we must t a k e k,, A 6 I . (So r o u g h l y
c a r e o f i n o u r c o n s t r u c t i o n . Then f o r some
fi,
speaking,
encodes t h e
k0
l o g Z n h i g h o r d e r d i g i t s o f t h e b i n a r y expansion
o f A .) D u r i n g t h e course o f o u r c o n s t r u c t i o n
k a l l o w s us t o w a i t t i l l about
A o f t h e measure space i s a d w . Too l a r g e a k , and we w i l l w a i t f o r e v e r . Too small a k , and we w i l l n o t t a k e c a r e o f a l l t h e measure space t h a t
2, 2.1.
k,,
With the correct
we should.
HOW
t h e c o n s t r u c t i o n w i l l work c o r r e c t l y .
T O A V O I D PUTTING A L L YOUR EGGS IN ONE BASKET.
T h i s s e c t i o n g i v e s an e x p o s i t i o n o f a lemma o f M a r t i n u s e d
in
t h e p r o o f o f t h e main theorem. We b e g i n by i n c ' i c a t i n g t h e problem t h e lemma i s designed t o s o l v e . Suppose we a r e a t some s t a g e so f a r ,
Ao(n),
... , ANV1(h).
The s i g n i f i c a n c e o f by a d d i n g elements t o
b
i n t h e construction.We have c o n s t r u c t e d
Put
Ci(h) i s t h a t we can o n l y hope t o make Aiequalto Ai(b)
if
6
E Ci(h).
I f , f o r example, t h e r e i s a s e t
0, a l l i < N , we a r e W(6) = A f o r a l l 6 € Y,
Y
o f measure
1/n
such
Y rl Ci(5) =
i n a potentially
where i f
we may n o t be a b l e t o make o n e
Ails
W(6) that
disastrous s i t u a t i o n o f the
equal t o A. On t h e o t h e r hand, i t may happen t h a t f o r some s e t Y o f measure >_l/n,
290
ROBERT M. SOLOVAY
we have
A.
W(6,h) =
for all
h
2
6
ho,
then
Ci
E Y.
Y
e v e n t u a l l y , make one o f t h e Ai(h)'s w i t h
n
I f t h i s happens we Ci(h)
+
0, equal
w i l l c o n t r a c t , p o s s i b l y t o Y, as we add elements t o
2.2.
To s t u d y t h i s s i t u a t i o n , we f o r m u l a t e an i n f i n i t e
game.
must,
t o Ao.
But
Ai. two
person
A c o n f i g u r a t i o n i n t h i s game i s N - t u p l e o f f i n i t e l y based s e t s : < c 0,...,c ~ > -. The ~ i n i t i a l configuration i s < w 2 ?>.
,...,
P l a y e r I on h i s t u r n p l a y s a c o d e f o r a f i n i t e l y based s e t Y o f measure
.
>- I
If
n+1 Y, f o r some Ci
prevent
I
Y n Ci = with Y
0
for all
i < N, I w i n s .
n Ci # 0, and
from winning
I f not, I 1 r e p l a c e s Ci by
I 1 wins i f he can
t h e game c o n t i n u e s .
as d e s c r i b e d above
f o r the entire,
infinitely
l o n g game. The game d i f f e r s s l i g h t l y f r o m t h e i n t u i t i v e m o t i v a t i o n o f 2.1 i n t h a t
2 1 / n by
we have r e p l a c e d
1/n
at
h =
00,
>,&.
The reason i s t h a t a s e t which has measure
m i g h t have measure 1 > -). )I + 1
< 1/n f o r f i n i t e
(but w i l l
b
eventu-
a l l y have measure
THEOREM, ( M a r t i n ) L t l N = -2h . ( n + 1 ) . Then pPage4 11 h a a h e c u h h i v e w i n n i n g bLJmtegy ,504 t h e game ju&t d u d b e d .
2.3.
The p r o o f i s based on a p r o p e r t y o f c o n f i g u r a t i o n s , which we dub
p r o p e r t y M. I t i s e f f e c t i v e l y checkable whether a c o n f i g u r a t i o n erty
M.
has prop-
Moreover, t h e i n i t i a l c o n f i g u r a t i o n has p r o p e r t y M, a n d
c o n f i g u r a t i o n has p r o p e r t y M , t h e n f o r any move o f I ,
if
a
I 1 has a t l e a s t one
r e p l y t h a t r e s u l t s i n a c o n f i g u r a t i o n w i t h p r o p e r t y M. A r e c u r s i v e s t r a t e g y f o r I1 i s t o p i c k
Ci
minimal so t h a t
Ci n Y # 0 and t h e r e s u l t i n g con-
f i g u r a t i o n has p r o p e r t y M. (Another r e c u r s i v e w i n n i n g s t r a t e g y
i s easily
e x t r a c t e d f r o m o u r p r o o f t h a t p r o p e r t y M can be p r e s e r v e d . )
2.4.
We l e t T = { < i , j > :1
configuration.
A bijection
as a t r i a n g u l a r a r r a y :
Z.
J-,j
sj
< i < d.Let
,...,
YN >l-
be
a
h : T -> N determines an arrangement o f t h e Y's
. = Yh ( i ,j )
.
( C f . Figure 1.)
291
ON RANDOM R . E. SETS
Zl,l
z2,i
z2,2
... ... ... ... ...
Figure 1 .
...,Y N > h a p h o p e h t y M i6 d o h some LhhnglLean_ m y h a the6oi%wingp~op-
DEFINITION, The con6igutdtion < Y 1 , b i j e d o n h: T ->N, ehtq: L e i 1 s i, 5 i, 5
2.5. LEMMA,
.the huu&ing
... 5 ik 5
j ; then
The following l e m m a i s now evident. 1 ) The i n i z i a l c o n 6 i g u h a t i o n <w2,...,
..,YN >
2 ) 1eL < Y1,.
,...,
Yb>
have p h O p e h t y M .
@2> h a phopehty M .
Suppose YiS
Y l C- w2.
Then
ha4 phopehty M.
3 ) W e can t e l l ed~ectiucLyiuhetheh n n c q u e n c e 0 6 i n t e g e h s < x,, ,x N > -io a code a can6igiguhation with p m p e h t y M .
...
2.6. The following l e m m a will complete the proof o f Theorem 2.2. LEMMA, L e t
..., Y N >
Y be a g i n i t e l y b a e d subsei
i 5 N such t h a t Yi Yi by Y O
fl Y
e a c o n 6 i g u h a t i o n w i t h phopehty M . L e t 06
+ 8,
m h a phopehty M .
w2, w i t h
p(Y) >
n+l . Then thehe
0
an
and t h e condigiguhation obtained by heplacing
292
ROBERT M. SOLOVAY
Yi's i n a t r i a n g u l a r a r r a y , Z . . , with the $9 J p r o p e r t y guaranteed by t h e Y ' s h a v i n g p r o p e r t y M. Then t h e u n i o n o f t h e Z ' s
PROOF: We a r r a n g e t h e
>
i n t h e b o t t o n row has measure tersection with Y. Let
n.
n+ 1
be minimal such t h a t some Z .
j
,
JIk
Permuting t h e
jth
Yh(j,j)
has nonempty i n -
n
Y
Z. ;
f 1J
# 0. We show
by Y t h e r e s u l t i n g c o n f i g u r a t i o n h a s p r o p e r t y M.
j > 1 , we p u t Zk,P. = Z k , &
I f j = 1, t h i s i s e v i d e n t . I f Z'.
n,j
has non-empty i n t e r s e c t i o n w i t h Y .
row, i f necessary, we may assume
t h a t i f we r e p l a c e
o r j - 1 . Put
Z
Whence, some
, unless k = j
= Z. , f o r P. 5 j - 1 , and Z ' . = Z . for!L<j-1. J,P. JIP. j-199" = Y. We show t h a t t h e t r i a n g u l a r a r r a y Z ' e s t a b l i s h e s
j-1,P-
Finally, put
Z'. J.J t h e new c o n f i g u r a t i o n has p r o p e r t y M.
that
I
What rows c o u l d cause t r o u b l e ? The o n l y rows t h a t do n o t a p p e a r i n t h e array are the
j - l s t and j t h . The
i s 0 . k . The o n l y way t h e
<... <
j - l S t row i s o . k . ,
becayse t h e o l d j t h
row c o u l d cause t r o u b l e i s i f f o r some
jth
1 s i,
i k 5 j - 1.
But by our choice of j ,
Y
n
=
Zj-,,i
0, a l l i 5
j - 1 . Thus ( 1 ) i s e q u a l
to
u
~ ( ~ j - i , i Since t h e
+-
n+ 1
, as
2.7.
than
z.
J-Lih
)
+
p(Y).
j - l S t row o f t h e Z ' s i s o . k . , and p ( Y ) >
- , this
n+1
is
We s h a l l need t h e f o l l o w i n g e x t e n s i o n o f
k -
n+ 1
...,Yh >
lemma2.6.
be a can&LgwmLion with t h e p t u p e ~ ~ XM. y Le,t Z1, each 0 6 rnea6u.w g h e a t e t
be pai,wde d i s j o i n t & L n i t d q b a e d n&,
1. Then we
n +1
>
desired.
LEMMA I L e t < Y,
..., Z k
... u
can Bind did.tinCt intege~1.4 il,...,ikAuch
Mat
293
ON RANDOM R. E. SETS
16 we put ,Y;
2)
=
J
z.
Y l = Yh d o t
h
not an i . , then J
j '
(again, given c a d u don euehgtking i n
il'..., i k ) .
big&,
we can e 6 ~ e o t i v d y dind
PROOF: I t w i l l h e l p i f we t h i n k o f N cubbyholes, which a t t h e s t a r t c o n t a i n t h e c o n f i g u r a t i o n Y1, YN . We a p p l y Lemma 2.6 s u c c e s s i v e l y t o each o f Z l , . . , Z k i n t u r n . A t t h e jth t u r n , Lemma 2.6 s i n g l e s o u t a cubby-
...,
.
h o l e whose c o n t e n t s have non-empty i n t e r s e c t i o n w i t h 2 . a n d s o t h a t a f t e r r e J p l a c i n g t h e s e c o n t e n t s by Z . we s t i l l have p r o p e r t y M. We t h e n p e r f o r m t h i s J
rep1 acement.
F o r o u r c u r r e n t l e m m a w e must show no cubbyhole has c o n t e n t s r e p l a c e d more t h a n once
. But
t h i s i s e v i d e n t s i n c e t h e Zi's a r e p a i r w i s e
A f t e r a cubbyhole i s f i r s t used, a t t i m e i , i t w i l l c o n t a i n Zi. used a g a i n a t t i m e
i , we would have t o have Zi n Z .
j
J
t o our hypothesis.
3 THE I
3.1. < Ao,
+ (3,
disjoint. If i t
CONSTRUCTION,
We s h a l l be s i m u l t a n e o u s l y enumerating an a r r a y o f
...,AN-l
were
contrary
>
. We
let
Ai(b)
r.e. s e t s ,
be t h o s e elements p l a c e d i n t o Ai b e f o r e stage
6 . (We w i l l a r r a n g e m a t t e r s so t h a t A i ( b )
ZA.)
L e t Ci(n) =
I d : Wn(n E
(1)
A i ( ~ ) -> W i ( 6 , ~ ) ( n )= 1).
We w i l l a r r a n g e o u r c o n s t r u c t i o n s o t h a t ,
. ..
for all A ,
the
array
Lemma
( T h i s i s t r u e a t 6 = 0, by
n + l by Lemma 2.5 2 ) . )
A t each s t e p b , we
f i r s t check
whether o r n o t A i s opportune f o r a d d i n g new elements t o the opportune, we w i l l add elements t o t h e Ails such t h a t
Ails.
If b i s
i n an a t t e m p t t o pursue those
p ( 6 ) 2 l / n . We use Lemma 2.7 i n d o i n g t h i s so t h a t
property
6
M
i s preserved.
3.2. We now g i v e t h e p r e c i s e c o n s t r u c t i o n . R e c a l l t h a t o u r c o n s t r u c t i o n
294
ROBERT M . SOLOVAY
e,
n , and k. Here k < n + 1 . L e t I k = ( a k , b k ) be 1 = - n ( n + l ) . To s t a r t t h i n g s o f f , l e t ' s agree t h a t 2 A i ( 0 ) = 0, i < n.
depends on t h e parameters
N
as i n S e c t i o n 1.9. L e t -1 i s opportune and t h a t
We now d e f i n e what i t means f o r a stage, 6 , t o be oppohtune. L e t
t be
t h e l a r g e s t number < 6 t h a t i s opportune. L e t m = (o),,II = ( A ) ~(where,
as
usual, ( x ) , i s t h e exponent o f p . i n t h e p r i m e power f a c t o r i z a t i o n . o f x,and 1
(0)i = 0 by c o n v e n t i o n ) . Then
Let hl,
2)
p(hi,6)
...,hh
be t h o s e b i n a r y sequences o f l e n g t h R s u c h t h a t
1 1 - -. n m
2
6 i s opportune i f
<
Then
I;=, Jl
P(hi,6)
< bk.
Note t h a t we can e f f e c t i v e l y determine i f 6 i s opportune.
3.3. stage 6
+
I f n i s n o t opportune, we p u t
to
A i ( 6 + l ) = A i ( 6 ) , and proceed
1.
Now suppose t h a t 6 i s opportune.Since we o n l y add e l e m e n t s a t opportune stages, we have L e t Yi =
2 -2 n
>
m
A i ( 6 ) E t + l < R.
i d : W(d,n)lR
2 . By
=
hi}.
By 1 ) o f t h e d e f i n i t i o n o f
. 2 ) ' o f t h e same d e f i n i t i o n , p ( Y i ) >
n+1
opportune,
- . Also n+l
the Yi's
a r e c l e a r l y p a i r w i s e a i s j o i n t . We now a p p l y Lemma 2.7. T h i s g i v e s d i s t i n c t
...,ih< N
i n t e g e r s il,
, such
that: 1)
Yj
n
C . ( 6 1 2 0 ; 2) i f we r e p l a c e
"j
each C . ( 6 ) by Y . and l e a v e t h e o t h e r Ci's unchanged, t h e r e s u l t i n g "j J f i g u r a t i o n , say < C & . ,Ci(r-l > , has p r o p e r t y M.
con-
..
Let
Bi
hi(q)= 1 ) . From 1 ) and t h e f a c t t h a t sup A i ( 6 ) < R,we Bi. We now s e t A i , ( 6 + 1 ) = Bi. F o r i n t e g e r s i < N n o t o f t h e
= (yew:
A i , ( o ) C_ J f o r m i . , we l e t
get
J
Ai(6
+
J 1 ) = A . ( o ) . Note t h a t we have C i 5 Ci(6 + 1 ) . S o prop-
"
e r t y M i s preserved. Note a l s o t h a t o u r p r o o f o f Lemma 2.7 g i v e s f e c t i v e procedure f o r d e t e r m i n i n g il,..., ihand we s h o u l d
an
follow
efthis
procedure so t h a t o u r c o n s t r u c t i o n w i l l be e f f e c t i v e . We have now completed s t e p 6 ; t h e c o n s t r u c t i o n now proceeds t o s t e p 6 + 1 . The d e s c r i p t i o n o f o u r c o n s t r u c t i o n i s now complete. Our n e x t t a s k
i s
295
ON RANDOM R. E. SETS
t o show t h a t if k i s c o r r e c t l y chosen ( c f . S e c t i o n l . l O ) ,
the
construction
succeeds.
4, THE 4.1.
CONSTRUCTION
We l e t e,n,N
WORKS,
be as i n t h e p r e c e d i n g
now assume t h a t , w i t h n o t a t i o n s as i n S e c t i o n 1.10, t i o n , f o r t h i s 12, y i e l d s a sequence o f s e t s t h a t p ( g ) 2 1/n. L e t
B
= {PI: g ( n ) =
paragraph, but
we
A E l k . Our c o n s t r u c -
Ao,...,AN-l.
L e t gEWZ
11. We must prove t h a t f o r
such
some
i,
B = A . . 1
Our p r o o f w i l l proceed as f o l l o w s :
( a ) We show t h a t i n f i n i t e l y
stages a r e opportune. T h i s w i l l f o l l o w f r o m t h e c o n t i n u i t y AElk.
S e c t i o n 1 . 5 and t h e f a c t t h a t
many
properties
of
( b ) The n e x t s t e p i s t o a n a l y z e what
happens a t a l a r g e opportune stage. Our method f o r d o i n g t h i s i s t o t a k e a n i n c r e a s i n g sequence o f opportune stages and c o n t i n u a l l y s i m p l i f y t h e s i t u a t i o n by p a s s i n g t o a subsequence. The f i n a l u p s h o t i s t h a t f o r a sequence
o f elements o f
gl,. ..,g, 1
P(9i) 2
2,
Ii=,P(gi)
3)
certain
we have:
y
1)
h
?!,
2 a& *
F o r any i n t e g e r m, t h e r e a r e i n f i n i t e l y many stages o a t which i s t h e c h a r a c t e r i s t i c f u n c t i o n o f some A , ( & )
gm li
J
I t i s c r u c i a l , o f course, t h a t 5 b e among t h e
A 2 P(g)+
I
h
i =1
P(gi) 2
+
1
' bk'
n m. But, i f n o t ,
5i's.
This contradicts A E T k .
argument now shows t h a t f o r some i < N,
( c ) An easy
A . = B.
4.1. We now commence p a r t ( a ) o f t h e p r o o f . Towards
a
contradiction,
assume t h a t t h e r e i s a l a r g e s t opportune stage, say t. ( R e c a l l
that
i s a t l e a s t one opportune stage, a t l e a s t f o r m a l l y , namely t = -1.)
Our
...,n4.
By t h e
2 ZM0 3n3 Sn4
i s op-
argument w i l l c o n s t r u c t a s e r i e s o f v e r y l a r g e i n t e g e r s , no, t i m e we a r e done, i t w i l l be c l e a r t h a t some i n t e g e r
there
p o r t u n e . T h i s w i l l complete p a r t ( a ) o f t h e p r o o f .
4.3. To s t a r t t h i n g s o f f , l e t g,l
...,g,
be t h o s e elements a t w2
such
296
ROBERT M. SOLOVAY
p(gi) 2
that
1 7 . Since
a t most n t l
f u n c t i o n s g have p ( g ) 2
1
1
, we n + l
can
1 ) no 2 t + 2 ;
.e p i c k no so l a r g e t h a t p(g) < 7 , p(g) < 7-EW 2) no > n 2 + n ; 3 ) 1 < E ; 4 ) t h e f u n c t i o n s gllnO, ...,ghl no
are d i s t i n c t ;
5)
f i n d an E > O so t h a t i f
0
LEMMA,
as
1 7 - E.
n l > n O hU that wheneveh h E m2, 1 h f gilm, t h e n P ( ~ ) < T- E
and
m 2 n 1’
p ( g i ) = l i m p ( g i l m). An i n s p e c t i o n o f t h e d e f i n i n+p ( c f . S e c t i o n 1.5) shows t h a t p ( g i l m ) i s monotone n o n i n c r e a s i n g
By S e c t i o n 1.5,
i i o n of i n m.
>
T h e h e 0 an
hln, = gilnl, PROOF:
p(gilnO)
Pick
n1
2
no so l a r g e t h a t p ( g i l n l )
i n t h e statement o f t h e
Id:
W(6)1n1 = g i l n l
lemma, and W ( d )
1
-
p(gi) < M _ - E . Then i f h i s
p ( h ) i s t h e measure o f a subset o f
.
# gil
,By o u r c h o i c e o f n1 t h i s l a t t e r s e t has measure <
1
F-E
.
4.4. We s h a l l need t o c i t e t h e n e x t s t e p o f t h e argument again,
so we
i s o l a t e i t o u t a s a lemma.
PROOF:
B y t h e u s u a l C a n t o r d i a g o n a l a r g u m e n t , we may a r r a n g e f o r
(1) t o h o l d . I t remains t o show
p(g) 2 y
.
For t h a t i t suffices,
of Lemma 1.5, t o show p ( g l m ) 2 y , f o r any m E w . Suppose, towards a c o n t r a d i c t i o n , t h a t p(g1m) < y c i e n t l y large, B u t then,
hi has l e n g t h 2 m , p ( h i ) >
p ( g l m ) , and
.
by 1 )
Then f o r i s u f f i -
hilm = glm.
ON RANDOM R .
29 7
E. SETS
Contradiction! 4.5. LEMMA, Thehe 0 an
then h = gilm
60h
n2 2 n1
i5
name
that i d
6uch
m 2 n2, and p ( h l m ) 2
1
M - E ~
h.
PROOF: D e n y t h e l e m m a . L e t
liin hi(") = g ( m ) .
i +m
large, tradicts
1
Lemma 4.3,
(since p(g)>,-E).
( C f . S e c t i o n 4.3.)
hi #
by ( 2 ) . B u t by assumption
h . ] n = gilnl, J
I
some i 5 h
g =gi,
1
p(h.) 2 J
since
9.1". I J
If
. This
j
i s con-
;-E
4.6. By o u r c h o i c e o f k ,
Let
n3 > n2
11.1 Next s e l e c t (3)
h
p(h) <
2)
t o prove
1-E , that
f i n i t e l y many
h
p ( g i l n 3 ) < bk
.
(Lemma 1 . 5
l a r g e enough so t h a t t h e f o l l o w i n g a r e t r u e :
n4 2 n3
" k < Ci=1 P ( g i l n 3 ' m ) < bk
(Use Lemma 1.5 and
ak <
be chosen so l a r g e t h a t
2 4 '
if
t
n4 e x i s t s . ) . Next, we want f o r 1
n - E if , m
p(h,m) <
2 m4.
h E n32
Since there a r e o n l y
h ' s o f l e n g t h n3 we may a r r a n g e t h i s by 2) o f Lemma
1 .5.
F i n a l l y , we r e q u i r e
Since
5 p ( g i ) 5 p(giln3), L e t now 6 = 2
9, = (n)l,then
n
since
3
n
we may c l e a r l y a r r a n g e t h i s as w e l l .
n 5 '+.Then
6
2 n4
n 2 n4, we have f o r
.
I f we p u t m = ( A ) ~ ( = no), a n d
h €
R
2, p ( h , 6 ) >
1
1
-m
iff
298
ROBERT M. SOLOVAY
.
E q u a t i o n (3) now shows t h a t i f no 6 ' w i t h t < A ' < 6 i s opportune, t h e n 6 i s opportune. T h i s c o m p l e t e s o u r p r o o f t h a t i n f i n i t e l y many
h = gill.
stages a r e opportune.
4.7.
We now e n t e r t h e second phase o f o u r p r o o f . We have an i n c r e a s i n g <si:iEw>
sequence
o f opportune stages. By p a s s i n g t o a subsequence, r e -
p e a t e d l y , we s h a l l be a b l e t o a r r i v e a t a f a i r l y cogent p i c t u r e o f happens a t s t a g e bi".
"what
So as n o t t o drown i n a sea o f s u b s c r i p t s , we r e f e r ,
by a r e l a b e l i n g , t o each new subsequence as hi.
, ki
To s t a r t m a t t e r s o f f , l e t m . = (6.)
L O
L
l a r g e s t opportune s t a g e < A ~ . L e t quences h o f l e n g t h IIi w i t h
l e t ti b e
= (6.) ,and h l
the
be t h e number o f d i s t i n c t b i n a r y se-
fii
- mi' -
p(hlni) 2
Then
ti
5 n+1.
passing t o a subsequence and r e l a b e l i n g , we may assume hihas
Thus
some
by
constant
v a l u e h.
hi,l,...,h,
Let
L,fi
be t h e b i n a r y sequences o f l e n g t h ki w i t h
. By p a s s i n g t o
-> -n- - mI. assume, by
a subsequence,
l i m h . .(m) = g i ( m ) , a l l m 6 w L*j
gi's
N o t i c e t h a t we d.0 n o t c l a i m t h e
mi+,
4.8.
By a s u i t a b l e r e l a b e l i n g o f t h e
hi
t o prove next t h a t
I f n o t , by
---I 1 1 n m
i
g. is J
i n w2, we have
goes t o i n f i n i t y w i t h i. Thus
1
a r e d i s t i n c t and each
may
are distinct.
=
2 lim[
g,, ...,g,
L,j
we
.
, mi
> ti+,2 6h. > mi
l i m p ( h . .) X.3 J
...,g6
times, and r e l a b e l i n g ,
Lemma 4.4 t h a t f o r c e r t a i n f u n c t i o n s
j-r,
Since
JC
p(h. .)
=
n
. I s , we may assume t h a t
9 J
t o some
Lemma 1.5 l),we can f i n d II such t h a t
gi w i t h i 5
6.
g,,
We w i s h
and t h e f u n c t i o n s
299
SETS
ON RANDOM R . E.
L a r e d i s t i n c t . Now by Lemma 1.5 2), we can
gl(L, ...,g,(
f i n d t oso t h a t
I* i=1
(5)
p ( g i l E , t ) < fik
9
t 2 20
'
h . 1 ~= g . I R , j J 3 ) A . 2 t o . The sum in (5) i s , i f t i s r e p l a c e d by A . , t h e measure o f J J
Now s e l e c t j so l a r g e t h a t and
if
1 ) L . 2 L,
2)
1
for
i5
k ,
The s e t (6) i s a s u p e r s e t o f
(7) has measure > fik s i n c e
But t h i s s e t (5),
4.9.
gi
A , i s opportune. T h i s
J
and proves (4) as d e s i r e d .
L e t now
i5
with
g € w2
with
p(g) 2
A . Suppose n o t . Then s i n c e
.I
1
p(gi)
( C f . S e c t i o n 1.10.) B u t by o u r c h o i c e o f k , shows t h a t g i s one o f t h e 4.10.
Let
gi's,
CM: g(M)
8 =
= 11.
contradicts
say t h a t g i s o n e o f t h e 1 f o r 1 5 i 5 b,we have:
2
X < bk. T h i s c o n t r a d i c t i o n
say g j . We now show t h a t
B
= A f o r some p <
P
N.
A t each opportune stage o f o u r sequence, bi, t h e r e i s an i n t e g e r pi so
that A
Pi
( n . + 1 ) = {rn < L.: h . .(m) = l}. 4
4
may assume, s i n c e pi < N , that
A
P
4,J
By p a s s i n g t o a subsequence,
we
t h a t pi has some c o n s t a n t v a l u e p. Me s h a l l show
= 8.
Suppose n o t . P i c k no so l a r g e t h a t that
A ( m ) n M~ = A
that
2 nl,
P
P
n
M",
A n
P
ttO
# B n no. P i c k n1 2 no
f o r rn 2 M ~ .L e t A . be an opportune s t a g e
and such t h a t h . .InO = gjIno. I*J
Then
SO
such
300
ROBERT
B (since
n
n ={m < 0
ki 2 A
~
hi, j ( m ) =1}= A
MG
2-
M. SOLOVAY ( 6 . + 1 ) il no = A
P * -
P
n
no
n ~l ) .
T h i s c o n t r a d i c t s o u r c h o i c e o f M ~ .The upshot i s t h a t 8 = A theorem i s proved.
P'
and o u r
5, H vs, I , 5.1.
The purpose o f t h i s s e c t i o n i s l a r g e l y e x p o s i t o r y .
We
review
enough o f t h e work o f C h a i t i n 1976 and 1975 t o m o t i v a t e t h e n o t i o n s H ( A ) , I ( A ) and make t h e d e r i v a t i o n o f t h e i n e q u a l i t y f r o m o u r main theorem comprehensible.
I ( A ) 5 3 H ( A ) + O(log H ( A ) )
I n t h i s s e c t i o n , we do n o t i d e n t i f y t h e i n t e g e r n w i t h {m E w: m < Rather we i d e n t i f y each i n t e g e r
M
n a r y s t r i n g s t h a t corresponds t o t h e i n t e g e r s 0, 1, 2, 3,
10, 11, ... ( I . e. i f
M
MI.
w i t h a b i n a r y s t r i n g . The sequence o f b i -
..., i s
then
0, 1,
> 1, n i s i d e n t i f i e d w i t h t h e d y a d i c expansion
n . ) O c c a s i o n a l l y , we w i l l have t o encode a l l t h e i n t e g e r s <
of
Zm i n t o s t r i n g s o f
t h e f i x e d l e n g t h m. When we have t o do t h i s we f i l l o u t t h e b i n a r y
string
w i t h l e a d i n g zeros. (E. g., 3 w i l l be encoded by '0011' i f m = 4 . )
5.2.
Our f i r s t g o a l i s t o d e f i n e t h e ' i n 6 0 m u n a t i o ~c o n t e n t ' o f a f i n i t e
b i n a r y s t r i n g , A . We s h a l l g i v e a p r o v i s i o n a l d e f i n i t i o n ( o f t h e number K ( b ) ) and t h e n p r e s e n t t h e r e f i n e m e n t ( o f C h a i t i n 1975) which we s h a l l , by
anal-
ogy w i t h C h a i t i n 1976, r e f e r . t o as I ( n ) . The i n t u i t i o n behind t h e d e f i n i t i o n i s as f o l l o w s . We have a machine M
which a c t s as a decoder. We g i v e t h e T u r i n g machine a
t i o n o f t h e s t r i n g A . M decodes t h e d e s c r i p t i o n and o u t p u t s A . M
Turing descripw i 11
be
chosen u n i v e r s a l ( o r ' a l m o s t o p t i m a l ' ) i n a c e r t a i n p r e c i s e sense. The number o f b i t s i n t h e shortest d e s c r i p t i o n o f
b
i s the "in~u~nintioncontent"of
A.
Note t h a t a l o n g s t r i n g may have few b i t s o f i n f o r m a t i o n . the s t r i n g o f length than
F o r example
c o n s i s t i n g o f zeros w i l l c e r t a i n l y h a v e
less
lo3 b i t s on any r e a s o n a b l e encoding o f T u r i n g machines. 5.3.
F o r o u r p r o v i s i o n a l d e f i n i t i o n we use t h e f o l l o w i n g v a r i a n t
T u r i n g machines. A machine w i l l have t h r e e tapes, an i n p u t tape,
of
an o u t p u t
301
ON RANDOM R . E. SETS
tape, and a s c r a t c h tape. M determines a p a r t i a l f u n c t i o n , UM f r o m t h e s e t o f b i n a t y s t r i n g s , C*,
i n t o i t s e l f . UM(n) = t i f M s t a r t e d i n i t s
initial
s t a t e , w i t h s c r a t c h and o u t p u t tapes blank, and n on i t s i n p u t tape, w r i t e s
t on i t s o u t p u t t a p e and t h e n h a l t s . L e t In1 be t h e l e n g t h o f t h e b i n a r y s t r i n g A . (Our c o n v e n t i o n i d e n t i f y i n g i n t e g e r s w i t h b i n a r y s t r i n g s makes [ y l l = l o g a y l + O ( l ) ' f o r M 2 1.) We Put KM(n) =min { I Z l : A m a c h i n e , Mo fix, vM
(Here
, so
,i
U M ( t= )A ]
s u n i u e n n d i f f o r each machine, M, t h e r e i s a p r e -
t h a t f o r each b i n a r y s t r i n g x,
o"t i s t h e c o n c a t e n a t i o n o f n and t. As u s u a l , t h e symbol
f o r par-
t i a l l y defined expressions indicates t h a t t h e l e f t side i s defined i f f
the
r i g h t s i d e i s , and i f d e f i n e d , b o t h a r e e q u a l . ) I t f o l l o w s f r o m (1) t h a t
I n particular,
if
Mo a n d M1
a r e u n i v e r s a l machines,
I t i s e a s y t o m o d i f y t h e u s u a l construction o f universal
Turing
machines so as t o c o n s t r u c t a machine Mo u n i v e r s a l i n t h e p r e c i s e
sense
j u s t described.
F o r example, suppose we have Godel numbered a l l T u r i n g ma-
c h i n e s i n some s t a n d a r d way. L e t M
9
be t h e machine w i t h G o d e l n u m b e r g .
( T h i s i s a v e r y temporary p i e c e o f n o t a t i o n . ) chine
Then t h e r e i s a
Turing
ma-
M such t h a t UM( 041AX)
= UM
(x)
.
9
( H e r e Og 1 i s g zeros f o l l o w e d by a 1.) C l e a r l y t h i s M i s u n i v e r s a l . We f i x a u n i v e r s a l machine, Mo, and p u t
K(n) = K
MO
(A).
(This defi-
n i t i o n i s due t o v a r i o u s a u t h o r s i n c l u d i n g C h a i t i n and Kolmogoroff. C f . t h e
302
ROBERT M. SOLOVAY
paper C h a i t i n 1975, and i t s b i b l i o g r a p h y . )
5.4.
The K n o t i o n has t h e u n f o r t u n a t e p r o p e r t y t h a t v a r i o u s i n t u i t i v e -
l y t r u e f o r m u l a e a r e t r u e o n l y up t o a " l o g " e r r o r term. We g i v e one
exam-
ple. Let
h:
w x w E w
be a r e c u r s i v e isomorphism. ( F o r example, h( < a , b > ) = 2 ' ( 2 b + l ) K ( < a , b > ) = K ( h ( < a , b > ) . I n t u i t i v e l y , one would expect
+ O(1).
K(
Put
-1.)
5
K(a) + K ( b )
B u t t h i s can be shown t o be f a l s e . One can p r o v e t h e weaker r e s u l t :
K(
-<
K(a)
+
K(b)
+ min
[ l o g 2 K ( a ) , l o g 2 K ( b ) ] + O(1)
(The d i f f i c u l t y i s t h i s . L e t A , t be t h e minimal l e n g t h programs f o r a, b. Then f r o m t h e s t r i n g
b^t,
l e n g t h o f one o f t h e s t r i n g s
we can n o t r e c o v e r n and t u n l e s s we know t h e b
and t.
Thus knowing n o t o n l y t h e s t r i n g
i n addition t o
Ibl,
n b u t when i t ends means t h a t
b
has
up t o log21n1 b i t s conveyed i n knowing when i t ends.)
C h a i t i n (and i n d e p e n d e n t l y t h e Russian mathematician L e v i n ) have u r e d o u t how t o t a k e t h i s e x t r a i n f o r m a t i o n i n t o account. Roughly t h e y r e q u i r e t h a t as we a r e r e a d i n g t h e code word n , we a r e a b l e
fig-
speaking tell
to
when we have r e a d t h e e n t i r e word. The p r e c i s e concept i s captur,?d i n t h e n o t i o n o f an e n d - d e t e c t i n g T u r i n g machine, which we now d e s c r i b e .
5.5. We now p r e s e n t r e v i s e d d e f i n i t i o n s o f t h e n o t i o n o f T u r i n g rnac h i n e and o f t h e f u n c t i o n UM. We r e q u i r e t h a t t h e i n p u t t a p e r e a d i n g head can n o t move t o t h e l e f t . A t t h e s t a r t o f t h e c a l c u l a t i o n , t h e
input
tape i s positioned a t t h e l e f t m o s t binary d i g i t o f n . A t t h e e n d o f computation, f o r
UM(b)
t o be defined,
head be p o s i t i o n e d o n t h e l a s t d i g i t o f
b.
the
we now r e q u i r e t h a t t h e i n p u t Thus, w n i l e r e a d i n g n , M w a s a b l e
t o d e t e c t a t which p o i n t t h e l a s t d i g i t o f 6 o c c u r r e d . The n o t i o n o f a u n i v e r s a l machine i s e s s e n t i a l l y t h a t o f S e c t i o n (modulo t h e new d e f i n i t i o n o f UM).
5.3.
Once a g a i n i t i s e a s y t o s e e t h a t
u n i v e r s a l machines e x i s t ; we f i x a u n i v e r s a l rnachineMo
, and
put
303
ON RANDOM R . E . SETS
Again, t o w i t h i n O(l), t h i s does n o t depend on t h e c h o i c e o f u n i v e r s a l machine
.
F o l l o w i n g C h a i t i n we can g i v e t h e f o l l o w i n g p r o b a b i l i s t i c t i o n t o I(6).
(The p r o o f i s n o n - t r i v i a l . C f . C h a i t i n 1975.)
6E
s t a r t o u r machine M o w i t h
interpretaS u p p o s e we
w2 w r i t t e n on t h e i n p u t t a p e (and t h e i n p u t
r e a d i n g head o f Mo p o s i t i o n e d o n t h e l e f t m o s t d i g i t o f 6). Say t h a t UMo(6)= t,if M~ h a l t s a f t e r w r i t i n g t h e s t r i n g t (and n o t h i n g e l s e ) on i t s o u t p u t tape. Put
P ( t ) = p(C6: U MO
(6)
= t}). Put
H(t)= -log2P(t). Then
r(t)= ~ ( +t ~)( i ) . (The c o r r e s p o n d i n g r e s u l t when t i s r e p l a c e d by an r.e. s e t A t r u e , though o u r r e s u l t s w i l l p r o v i d e t h e weaker v e r s i o n :
H(A)
i s not
I(A) = O ( H ( A ) ) ,
= O(I(A)).)
5.6. We m e n t i o n b r i e f l y what i s known about t h e r e l a t i o n s h i p b e t w e e n I ( 6 ) and K ( 6 ) . I t i s t r i v i a l t o show K ( 6 ) 5 l ( 6 ) + O(1). The f o l l o w i n g unp u b l i s h e d r e s u l t o f t h e a u t h o r measures t h e c o s t r e q u i r e d t o make t h e c o d e for
6
"bel6-deli.tnLCing".
(This formula allows
Z(n) t o be "computed" f r o m K ( n ) . O n e
can
show
t h e r e i s no way o f "computing" l ( 6 ) f r o m K ( 6 ) w i t h an e r r o r term which i s (We p u t "computing" i n quotes s i n c e t h e f u n c t i o n K i s O(log2 K [ K ( n ) ] ) . ) not recursive.)) (4) has t h e f o l l o w i n g i n t u i t i v e c o n t e n t . I n o r d e r t o make t h e
K - s t y l e program f o r n
s e l f - d e l i m i t i n g , we must p r e f i x i t w i t h an
o f i t s l e n g t h . T h i s can be done i n
K(K(6))
+ O(log2K
[K(n)])
minimal encoding
bits in a
s e l f - d e l i m i t i n g f a s h i o n . The c o n t e n t o f (4) i s t h a t , t o w i t h i n t h e
error
t e r m c i t e d , t h i s s i m p l e procedure i s o p t i m a l . We remark t h a t u n l i k e K does s a t i s f y
r(
r(M)
+ I(m) + O(1).
T h i s f o l l o w s r e a d i l y f r o m t h e arguments used t o p r o v e 3) o f Lemma 5.7.
,I
304
ROBERT
M. SOLOVAY
5 . 7 . LEMMA, ( C h a i t i n 1975) 1)
I (11) 5 log2n + O(log2 log2n) ;
3)
I(M i m)
REMARKS:
=
I(n) + O(1og2m ) ;
2 ) through
1 ) i s t h e p r i n c i p a l r e s u l t f o r us.
4) a r e way-
s t a t i o n s on t h e way t o t h e p r o o f .
I n o r d e r t o make 1 ) t h r o u g h 4 ) l i t e r a l l y t r u e i t i s u s e f u l t o i n t e r p r e t l o g 2 0 and log21 as 1. ( A l t e r n a t i v e l y , we c o u l d j u s t i n t r o d u c e log'n = 10g2(max(n,2)) and s t a t e t h e lemma i n terms o f PROOF: F i r s t n o t e t h a t
log+.)
1) f o l l o w s e a s i l y from 2) t o 4 ) . L e t
n a r y s t r i n g i d e n t i f i e d w i t h n. Then In1 = log2n In1 + Z ( l . l ) + O(1) = log2n + 0(log2 l o g 2 n ) .
+
b
Z(n) = I ( n ) 5
O(1). Thus
The general approach t o p r o v i n g upper bounds on I i s
be t h e b i -
to
s p e c i a l purpose machines M and u s i n g I ( n ) 5 min{ltl: U M ( t ) = A }
construct
+ O(l),
which f o l l o w s f r o m t h e u n i v e r s a l i t y o f M o . F o r e x a m p l e , t o p r o v e 2), use a machine P1 t h a t works on t h e
follow-
i n g p l a n . I t reads t h e d i g i t s o f t two a t a t i m e . I t t h e n i n t e r p r e t s t h e m a s follows:
'Ox' means " p r i n t x on o u t p u t t a p e and s h i f t o n e s q u a r e t o t h e
r i g h t ' ' ( x = U,l);
'11' means " h a l t " .
sponding t o n, U M ( t ) = A
Then i f n i s t h e b i n a r y s t r i n g
f o r some t o f l e n g t h 2 1 n l
+
2
=
corre210g2n+ O(1).
To p r o v e 3 ) c o n s i d e r a machine which proceeds a s f o l l o w s . I t reads t h e f i r s t d i g i t o f t h e number t t o f i n d o u t whether t o add o r s u b t r a c t . I t t h e n s i m u l a t e s t h e u n i v e r s a l machine Mo r e a d i n g a segment Rl o f R t h a t encodes a number m . I t t h e n s i m u l a t e s Mo again, r e a d i n g a segment t, o f R . I f to= 0, i t o u t p u t s M + m . I f t o= 1 + O(1) 5 I(n) + O(log2m).
i t outputs
M
- m . Thus I ( Mi m ) 5 I(n)
+ I(m)
The p r o o f o f 4 ) i s s i m i l a r . The a u x i l l i a r y machine M s i m u l a t e s M o and reads an i n i t i a l segment o f t t h a t encodes a number m . I t t h e n r e a d s n e x t m d i g i t s o f t and w r i t e s them on t h e o u t p u t tape. I f
6
i s our given
s t r i n g and n ' i s a s t r i n g o f l e n g t h I ( I n l ) such t h a t UM ( n ' ) = 16
UM ( ~ ' " 6 )= n . T h i s proves 4 )
0
the
I,
then
305
ON RANDOM R. E. SETS We remark t h a t C h a i t i n shows ( i n C h a i t i n 1975) t h a t
Whence i t f o l l o w s e a s i l y t h a t
1(n) 2 l o g 2 n + log2 l o g 2 n , f o r i n f i n i t e l y many n.
5.8. We now r e c a l l t h e d e f i n i t i o n s of s e t , g i v e n i n C h a i t i n 1976.
I ( A ) and
The d e f i n i t i o n o f H i s s i m i l a r t o t h a t o f
H ( A ) , f o r A a n r.e.
H ( M ) g i v e n above.
L e t us
e n v i s i o n t h e Monte-Carlo machines o f t h e i n t r o d u c t i o n as b e i n g p r o v i d e d
6
t h e i r random s t r i n g
o f 0 ' s and 1 ' s o n a r e a d - o n l y i n p u t tape.We d e f i n e
a u n i v e r s a l machine as b e f o r e : Mo i s u n i v m d i f f f o r e v e r y M o n t e - C a r l o machine M, t h e r e i s a f i n i t e s t r i n g nM so t h a t Mo o n i n p u t nM*6 s i m u l a t e s M's a c t i o n on i n p u t 1)
MO[rMA6]
2)
I f M[d]
6: w i l l enumerate t h e same s e t A as M[d].
o n l y reads t h e i n i t i a l segment
b
of
6,
t h e n Mo[vM"d
]
reads t h e i n i t i a l segment sM*b. We p u t
P(A) =
~ ( { d :Mo [6]
enumerates A ) ) . We p u t H ( A ) = - l o g 2 P ( A ) .
Next, we d e f i n e 1 ( A ) . The d e f i n i t i o n presented h e r e i s t h a t o f C h a i t i n 1976. I t i s n u t t h e same ( t o w i t h i n O(1)) as t h e d e f i n i t i o n p r e s e n t e d i n t h e i n t r o d u c t i o n . We say t h a t a f i n i t e b i n a r y s t r i n g 6 i s an Mo-program f o r A , if Mo w i t h t h e s t r i n g n"6 on i t s i n p u t t a p e w i l l enumerate A o n i t s o u t p u t
t a p e and not head beyond n on i t s i n p u t t a p e . ( O f c o u r s e i t f o l l o w s t h a t Mo [ n A g ]
w i l l a l s o enumerate A f o r any g E
%,
since the contentsof
6
can
n o t a f f e c t t h e h i s t o r y o f M o [6"6] . ) Put 7 ( A ) = min((o
1:
b
i s a n Mo-program f o r A ) .
We can f o r m u l a t e t h e concept r e f e r r e d t o as I ( A ) i n t h e i n t r o d u c t i o n , c a l l i t now
I * ( A ) by
306
ROBERT M. SOLOVAY
We remark t h a t a l l t h e r e s u l t s r e f e r r e d t o i n t h i s paper a r e e q u a l l y
valid
I * ( A ) i n place o f I ( A ) .
for
LEMMA,
+ O(1).
I(A) 5 I*(A)
PROOF: L e t us r e f e r t o t h e u n i v e r s a l machine employed i n t h e d e f i n i t i o n o f I ( 0 ) as M1. We c o n s t r u c t an a u x i l l i a r y machine M as f o l l o w s . M s i m u l a t e s t h e a c t i o n o f M1.
Mlwill
r e a d an i n i t i a l segment n o f i t s i n p u t and o u t p u t a
number
j . M t h e n proceeds t o enumerate W . o n i t s o u t p u t tape. J
I t i s c l e a r t h a t i f n i s a m i n i m a l M L - s t y l e program f o r
j ,
t h e n nMAn
i s an Mo-program f o r enumerating W . . T h i s proves t h e lemma. J
5.9.
LEMMA.
I(A)<3H(A)
+
I ( H ( A ) ) + O(1).
O f course i t f o l l o w s f r o m t h i s lemma and Lemma 5.7 t h a t I ( A ) 5 3 H ( A ) t O(logH(A))
.
Also, by Lemma 5.8,
I*(A)
<-
3H(A)
Suppose t h a t
+
I(H(A))
i t s u f f i c e s t o prove
+ O(1).
H ( A ) 5 n. Then P ( A ) 2 2-'.
We n o t e t h a t i f N = 2',
Thus o u r main r e s u l t i m p l i e s t h e f o l l o w i n g . There i s a r e c u r s i v e f u n c t i o n 6 ( n , m ) so t h a t whenever A i s an r.e. s e t w i t h
H(A) 5 n
, then
m < Z3fl+1,
andA= W . . J To complete t h e p r o o f we now d e s c r i b e a machine M so t h a t i f H ( A ) 5 n, ( 3 j ) ( 3 m ) d(n,ni) = j ,
t h e n U ( A ) = j , and A = W . f o r some 6 w i t h In1 = I ( M ) + 3n + 1. M J M proceeds as f o l l o w s , on i n p u t A : I t f i r s t s i m u l a t e s MI and reads an i n i t i a l segment no o f 3n
+
(no). I t n e x t r e a d s t h e n e x t M1 1 d i g i t s o f n , i n t e r p r e t i n g i t as a number m < Z3'+'.Finally, M com0.
I t computes n = U
putes b(n,m) and w r i t e s i t on t h e o u t p u t tape. I n v i e w o f o u r m a i n r e s u l t , a s r e c a l l e d two paragraphs ago, M i s e a s i l y seen t o have t h e d e s i r e d p r o p e r ties: If M. L e t
H(A)
:n,
l e t no be a s t r i n g o f l e n g t h I ( n ) such t h a t UtI
m, j be such t h a t
d(R,m)
= j , A =
W., and m < Z 3 n + 1 . J
Let
(") 1
nl
= be a
307
ON RANDOM R. E. SETS
binary s t r i n g o f length
311
+ 1 t h a t g i v e s t h e b i n a r y expansion o f m ,
pre-
ceded i f necessary by zeros. Then UM(noo^nl) = j and ln0*n11 = Z ( n ) + 3 n + 1
REFERENCES Bishop, 1967,
.
I
E. Foundations of Constructive Analysis, McGraw-Hill
.
C h a i t i n , G. J. 1975,
A .thcohy v 6 phogkani n i z e 6ohm&y
J . ACM, 1976,
V O ~ . 22,
329-340
Algorithmic entropy of sets, I B M R e s e a r c h R e p o r t RC5799, 36 pages, IBM Watson Lab.,
de Leeuw, K., 1956,
i d e n t i c & .to i n d o m a t i o n t h e o f i y ,
Yorktown H e i g h t s , N . Y .
E. F . Moore, C . E. Shannon, and N. Shapiro
C o m p u R a b U y by p . t o b a b i k U L i c mackin~n, i n Automata Studies C.
E.
Shannon and J . McCarthey,
(Eds.),
Princeton
s i t y Press, P r i n c e t o n , New Jersey, 183-212. Halmos, P . R. 1964,
Measure Theory, van Nostrand, P r i n c e t o n , New Jersey.
Department o f Mathematics University of California Berkeley, C a l i f o r n i a , U.S.A.
,
Univer-
