INTRODUCTION TO T H E THEORY OF FINITE AUTOMATA
N. E. KOBRINSKII AND
B. A. TRAKHTENBROT T R A N S L A T E D FROM THE R U S S I A N
Translation edited by . I C. . Shepherdson University of Bristol
1965
NORTH -H 0LLAND PUBLISHING COMPANY AMSTERDAM
1965 N O R T H - H O L L A N D P U B L I S H I N G COMPANY
Nopart of this book may be reproduced iii any form by print, photoprint, microfilm or any other means without written permission .from the publisher
PREFACE
K
on
book
V
PREFACE
VI
by
j
by
abstract synthesis;
structural synthesis
do
PREFACE
on
by by book, on no
book. book
book book 5), 111 V
by 5
by book
A.
by S.
0. B. book. S.
on
Yu.
A.
on by
E. KOBRINSKII B. A. TRAKHTENBROT
INTRODUCTION
1.
for book, by
on
A a
by
At,
by
by
by
of
on
t,
on -
-
t
for This term must be distinguished from the term “determinate”, which will be used in
a different sense. 1
2
INTRODUCTION
by
by
internal state on memoryless automata. by
0.1. second
on dt=O),
on by
l
i
l
l
0.1 TABLE 0.1
0 0
0 1
1 1
0
0 0 0 0
0 0 0 0 1 1 1 1
1
0
0
1
0
0
0 1 2
0 0 0
3
0
4 5
6
7 8 9
0 - non-excited output, 1 -excited output.
0 0 1 1
1 0
1 0
1 0 1
3
INTRODUCTION
by
by
of
A
on 100 100 100 ....
by
2.
by by
0.2
4
INTRODUCTION
A *D
-
q
B
E
A,
C, D,
I
Z.
0.2b Z
{A,
{B, C } ,
(A, E, C}, (B,
no
loops
do is analogue automata1.
0.3
for
Fig. 0.3 See E. Kobrinskii, Matematicheskie mashiny nepreryvnogo deystviya (Mathematical analogue machines), Gostekhizdat, Moscow 1954.
5
INTRODUCTION
:
a l l x l - %2XZ
= bl,
+ %2X2
= b,,
- %1X1
0.3b to
3. logical nets. book.
by C . E.
by V. by
on by by V.
1
S. V. B.
V.
A. W. Burks and G. B. Wright, Theory of logical nets, Proc. IRE (1953) 41, no. 10.
6
INTRODUCTION
by L.
on 1953
by
W.
by A
1956
by C . 1956
on 3.
analysis of an automaton
4.
by
on p. 5 .
1
by
2
no.
1956.
3
1956; (1956) 261, no. 3, 4; (1956) 6, no. 2, (1959)3, no. 3 ; (1959)3, no. 6;
by
3,
1960;
1.
(1960)21, no. 2.
11,
(1960) 21, no. 3; by 21, no. 2; Yu.
(1960) 1958; 12,2.
M.
7
INTRODUCTION
by
by by
by E. F. A.
by
on by
by This approach is typical the “abstract” theory automata. This approach is typical of the “structural” theory automata. E. F. Moore, Gedanken experiments on sequential machines, Automata Studies, ed. by C. E. Shannon and J. MacCarthy (Annals of Mathematics Studies no. 34) Princeton Univ. Press 1956. 4 B. A. Trakhtenbrot, Operators which can be realised in logical nets, DAN SSSR (1957) 112, no. 6.
8
JNTRODUCTION
5.
This -
by 0.1
p.
by by no
no
by
A by
on
Known as inclusion tables. See M. A. Gavrilov, The theory of realy-contact circuits, Izd. AN SSSR, Moscow Leningrad 1950.
9
INTRODUCTION
1.
0 t
o
z
A
a-t, o-z by
is
1 on by
A
by by S.
V. M. A.
L. B. A.
A.
6.
7. As
10
INTRODUCTION
0
8.
by
is
on
As by
by C. by 0. by S. V. by
A.
is
A
28,
0. B. 1,
Coll.
2, 127,
1959, pp.
11
INTRODUCTION
-
-
by
von
9.
by ideaZising
no by
on
As
on
by
L.
on
L. by S. V.
book
See J. von Neumann, The logic of probability and the synthesis reliable organisms from unreliable components, Automata Studies, ed. by C. and J. MacCarthy (Annals of Mathematics Studies no. 34) Princeton University Press 1956. 2 M. L. Tsetlin and L. M. Shekhtman, Push-pull ferro-transistor circuits and an algebraic method of synthesisingthem, Coll. Problemy kibernetiki2, Fizmatgiz, Moscow 1959, pp. 139-179.
CHAPTER I
T H E ELEMENTS O F L O G I C
1.1. General remarks1 As
two by by
A on.
by 0
1.
0
1
by
0.1
p.
(xlxz...xm) (xj=O,l) code groups.
m
m=3
m 1.1, 1.2
1.3,
by
V. 51,
1958, pp. 12
1.11
13
GENERAL REMARKS
0
S f ( x l , x2, .... xm) 1 functions in the algebra of logic.
by
1.1.
TABLEI. 1 Xm-l
x3
x 1 ~.
.
0 0
0 0
...
0
...
0
...
...
...
... ... ...
1 1
1 1
1 1
...
... ...
0
0
... ...
...
... ...
...
...
... ...
//
.....................
1 1
2 , 3 , ....
0
1,
1.
m.
> 5),
1.1 by
1.3). 1
0
1.1 f ( x l , x2,.... x),
xi As
i = 1,2, .... m).
14
[I.1
THE ELEMENTS OF LOGIC
:
1. 2
complement
negation
x,
x”. A
x,
X
1);
x x x,&x2)’,
2
11. x 1 - x 2
conjunction
x2”. 111. x 1V x 2 ,
x1 x2 logical addition,
disjunction
x1 x2 IV. x 1 + x 2 , implication,
xl,
“x1
x2”
x2”.
xl,
x2”.
x1 x 2 is V . xlszx2, equivalence, x2”. x1 x2
“xl
x2”
“xl
TABLE1.2
0 .
1
1
0 0 1 1 . .
_
-
-
0 1 0 1
0 0 0 1
0
1 1
1
0
~
1 0 0
I
- _ _ _ ~
-.
We shall henceforth be indicating this operation by using either notation. The point will generally be omitted.
_
1.11
15
GENERAL REMARKS
by
...,
...,”
on,
by
EXAMPLE.
A
p. 4, B
C,
A,
C,
B,
: :
Z = ( A - D V B * CV A - E * CV B . E * D ) . on the algebra of logic,
logic of propositions mathematical logic.
logic of predicates.
on
16
II.2
THE ELEMENTS OF LOGIC
by 0. on
1.2.
of
of
:
q,
I. pl, q 2 , ..., x l , x2, ..., zl, z 2 , ._.
1
0
(%)
'u 91
2.
..., x,y , z,
for
23
(aV w , (%&B),
(a+%), (?I = 93)
3. on
1
2.
EXAMPLE.
((@&Y>
-+
(x v Y,)
v
x,y (x&y) ((x&y)-+(xV y ) )
by ((x&y)+(xVy))
1
(xVy)
z
(((x&y)-(xVy))z)
speaking "&" 0. B. Lupanov, A fizika (1958) 1, pp. 120-140.
"-"
1.1,
of circuit synthesis, Izv. vyssh. uchebn. zavedenii, Radio-
1.21
17
THE ELEMENTS OF THE ALGEBRA OF LOGIC
‘u
B
(2l.B) “&’
“*”
‘u
(b&Y>
+
.( &Y
z)
-+
9
z),
x&y+z,
(x &Z)
(P
-
“V”
“&’
(4
v
(x &Y>
f>>9
v
(P 4 71, (x &Y p-qV?,
v z,
(x & y )
+
-+
x
(x
v Y>)
3
v Y),
x&y+xVy.
v (x &z(2 v y)
X Z V Z , xyVxz(2V y).
‘u
B
equivdent
‘u
B
by 0
‘u by
1
23 “3
B”
‘21
B
18
[I.2
THE ELEMENTS OP LOGIC
TABLE1.4
I
__
x2
0 0 0
0 0
0
0
1 1 1 1
0 0
0
0 1 ___-
._
EXAMPLES. (xl &x2) &x,
x1 &(x2 8 ~ x 3 ) . 1.4; (x1&xz)&x3
x1&(xz&x3)). “0” “1” (x1&xz)&x3 x1&(xz&x3), ( X I &x2)
&x,
x1 &(xz
(x1&xz)&x3
XVy
&x3)
&(xZ&xj).
x&y. 1.5). TABLE1.5
in
1.4
1.21
19
THE ELEMENTS OF THE ALGEBRA OF LOGIC
“0” 1.5
xXy,
2Vy
1.
associativity, (xl (xl
= X I &(x, &x3),
&X,)&X,
v x,) v x3 = x1 v (xz v x3),
(x1 = x z) = x3
= x1
= (x, = x3).
: (xl &xz) &x3 = x1 &(x2 &x3) = x1 &x, &x3, (xl (XI
v x,) v x3 = x1 v (x, v x3) = x1 v x, v x3, = xz) = x3 = X I = (xz = x3) = x, = x,
= x3.
2. commutativity : x1 &x, = xz &xl, x1 XI
v x,
= x,
= x,
= x,
v
XI,
= x, .
3.
by
of idempotence
for 4.
(x1 ‘ j .2) (XI &xz)
&x3
= (x1 &x3)
v x3 = (xl v
with respect to disjunction.
v
&x3) x3) &(xz xg). (XZ
v
distributivity of conjunction dis~ribu~ivity
When referring to functions in the algebra of logic, we shall often use expressions of the typef(x,y) (“a function f in the algebra of logic, of the variables x and y”), g ( X I , x2, ..., xm) (“a function g in the algebra of logic, of the variables XI, XZ, ..., x,”) etc. When some function in the algebra of logic is referred to, one may imagine it to be specified by a tabulation defined by some formula in the logic of propositions.
20
11.2
THE ELEMENTS OF LOGIC
of disjunction with respect to conjunction. “&’,
V
“*”
V
(XI ~ 2 ) * ~=3 ~ 1 . ~ ~ 3 2 . ~ 3 ,
X1 ’ X 2
v
x3
= (XI
v X3)*(X2 v XJ.
5. :
x&O=O,
x&l=x, xvo=x, xVl=l. 1
0
x x x
x 1
x
x 6.
x&X=O,
xVX=l, -
x
x&X”. law of the excluded middle
X
x ‘‘x
X
7. xy,
xj
aj
0
1
x0 = X ,
x 1 =x.
x;’.xy ... X,bk
1.21
21
THE ELEMENTS OF THE ALGEBRA OF LOGIC
x 1 = a, , x 2 = a2, ..., xk = a,. xj aj) 5 x1x2x3x4 aj= xj= 1. 21xZx324 a1=04=0, a2=a3= 1 ) x1=x4=0, x2=x3=l. x , x l , x2,...
by 1 . Every function in the algebra of logic can be represented in
the f o r m : f ( X 1 , X 2 , X 3 7**.,Xk,Xk+lr '.-,x,)
( m 2 1)
where the symbol
V
x;'x;' ... xpf (alya2,..., a,, xk+ 1, ...,xm),
(1.1)
0 1 9 . . .,bk
denotes the logical sum over all alya2,03,..., a,. expansion of a,function with respect to k variables. ol,02,..., a,.
x 1= a l , x 2 = o 2 , ..., xk=ok. x k + l , ..., x,,,), 3, 5 7,
f(a,, a2,..., a,,
by az, ..., ok, xk+l,..., x,).
x1
f ( x l ,x2,x 3 )
x2.
By
f (XI,x 2 x3) ~
= 2122-f
OYx3)
v
21x2f
(O, x3) V xlfzf
v X3)
V xixzf
X3).
oj
a, =0
a2= 1.
x1
x2) 7 5,
on f(xl,x 2 , x 3 )
f(0, 1 , x 3 ) ,
1, x3).
2. Every function in the algebra of logic can be expressed by means of conjunction, disjunction and negation.
22
D.2
T H E ELEMENTS OF LOGIC
by
f (x1,x2,...)x,)
=
v...,
02 x,0 1x2
... xZm,”f(a1,a2, ...,a,).
a2,..., a,)
0
by
1,
v
- * * , x m= )
x;lx?
~ ( u I , u ~ , . . . , u=~1)
5
...x;m ,
(1.3)
a2,..., a,,)= 1 . principal disjunctive normal form
a,, a2,..., a, (1.3)
on
(1.2)
am
01,
x2, ...,
(1.3)
a (a,, ..., a ,) ...,x,) 1.
by
by EXAMPLE. x2,
1.6.
on
Solution.
x2,
V
V
V
-xy disjunctive terms
TABLE 1.6 Xl
X2
x3
1.21
23
THE ELEMENTS OF THE ALGEBRA OF LOGIC
(x;'
V xy V ...V xirn) (xzl V xy V ...V xzm)
1.
by
g
< g < 2").
(a1,az,..., a ,) g
g,
m
g = i= 1
g g(ol, az,..., a , ) [a,(g), az(g),
(a1,a2,..., a , ) g
..., om(g)].
m=4,
5.
13
rn Pj = X;'X~'
... x i m
C j = x;' V
x y V ... V xz'"
az,..., a,,,), 1 With the provisos noted above, relating to functions identically equal to 1 and those identically equal to 0.
24
II.2
THE ELEMENTS OF LOGIC
o1,Q,,
<j<2"),
..., D~
v2(s),..., a,,,(s)],
[a,(s),
s
O,<sc2".
= 3,
p0 -2 1 -x 2-~ 3 p, 1 _ - ~- 1-x
...,
2 ~ 3 , P7
=~
1 ~ 2 x 3 ;
c, = 2, v x2 v 2,,c1= Z1 v 3 2 v X3, ..., c, = x i v x 2 v x s . by
z by
i by (de Morgan's theorem 1).
by
A
v Z 2 ) . ( X 3 v x4 v X 5 ) ' X 6 .
= (xl Df
A = .f1-x2
v 2 3 ' 2 7 , * 2 5 v xg.
__ XI'X2
= XI
v 22,
vice versa:
-
Pk = cj;
Ck= P j ,
j k by
Pk = x1*22.x,= P5
cj = z1v x, v 2 3 = c,
(k = (j =
m m
A)
"="
m
by
by
1.21
25
THE ELEMENTS OF THE ALGEBRA OF LOGIC
by
6)
5
0.
rn = 3.
(x1'x2'23)'(21.22.x3) = 0. in
rn
A) by rn = 3,
1. (XI
v 22 v
x3)
5
6)
v (XI v v x3)= (2,v x,) v z,; 22
E2V x3V f2V x3
z, 22 V x3).
XIV x1= 1, (XI
:
v 22 v x3) v (2,v 22 v
x3) =
1
v z,
=
1-
C)
rn rn
by A):
rn A
= 212223
V 2122x3 V 21x223 V 2 1 ~ 2 x 3V
v
x12223
v
x122x3
v
Xlx223
v x1x2x3
X,,
x1;
Z1
x,
V. A = 21 (2223 V 22x3 V ~ 2 2 V 3 ~ 2 x 3 )V XI (2223 V 22x3 V ~ 2 2 V 3 ~2x3). by
Z2Z3 VZ2x3V x2R3V x2x3 A = (2223 V 22x3V ~ 2 2 V3 ~2x3) (2,V x,). 5,
2,V x,= 1,
by
A=X223V22~3V~223V~2~3.
x2 A=
Z2(23
V x3)V x2(.Z3 V ~ 3 )=Z2*1V x2.1 = Z2V x2 = 1 .
*
26
[I.2
THE ELEMENTS OF LOGIC
m=2, = (3, V
B
V
xZ)(X~
V 3 2 ) ( ~ 1V
~ 2 ) .
Df
(fl
v n2)(x1v x2) = xlxl v 21x2 v 2 1 x 2 v fzx2 = = x1 v x1(x2v x2) v 0 = x1 v i,.l= 21. (XI
v f2)(x, v x2) = x1-
B=x,x, =O. R f 1 Z 2 V RZ1x2 v Rx1R2 V R x 1 x 2 = R
( R V 21 V
f2)(R
V 21 V xz)(R v XIv Z 2 ) ( R V
V
(1.4)
~ 2= ) R
,
R
p
2'"- p A = Po V Pl V ... V P,
=p
- l),
Df
A = P,,,
v Pn,2 v _..v P z m p p .
on
A
= A = Pn+lV
Pn+Z V ... V P2m-p= CjCi... by
C
do
by
(1.5)
1.21
THE ELEMENTS OF
21
ALGEBRA OF LOGIC
K: K =21X223 V
Xl22X3
V X l X 2 2 3 = P 2 V Ps V P 6
a
Df
R = ~- 1- ~ -2 Vx 23 1 2 2 ~ 3V 2 1 x 2 ~ 3V ~ 1 2 2 2 V 3 ~
1 ~ 2 = x 3
= Po
v
PI
V P 3 V P4 V
P7,
K K=(21V2zV23)&(21
VX,VX~)&(X~V~~VX~)& & ( X I V x2 V 2 3 ) &(XI V x2 V x 3 ) = c O c 3 c , c 6 c 7
-
completeness of a system of functions
DEFINITION. A complete
fly
fly
f,,
...,A
f,,...,A.
(2,(xl Vx,), x1x2}
2,
by
by
1.3.
x1 + x 2 = 2122 V E1X2V x1x2= x1
= x2 = z12,V X l X 2 .
{2,(xl Vx,)}
V x,,
(2,x , x 2 } ~
X,X,=~~V~,, A Shefler function
X,VX,=~~~~.
by
xllxz=XlVP2,
by
1.7. X=xIx, x1
=(x11x1)I(x21x2)
28
[1.2
THE ELEMENTS OF LOGIC
TABLE1.7
I
x1
-
I1
X2
x1
1 x2
0
I
1 ...
/I
1
2
x2)=x1@x2, by
1 1 0
1.8. x1@x2 1.8
x,0x2
~~@x,=R,x,Vx,~~. x1Vx2) x@l=x,
1
on
on
x 2 , ..., x , ) :
1.21
29
THE ELEMENTS OF THE ALGEBRA OF LOGIC
1.
conservation Fzero: I
,..., 2.
conservation of unity:
f ( l , l , ...)1) = 1 . I f , ( x , , xz)=xlxz
2
x2)=x1Vx2, =0 9 fl(0,O) = 0 , fz f1(1,1)=1, "fz(L1)=1*
self-duality,
3.
f
*.
-7
oddness
xm) =
f (x1
f (4=fW 4.
5.
. .)xm). *3
f z (090)< f z (191). f(xl,xz,..., xm)
linearity:
f(x1,x2, ..., x,)
2
x,<x;, x,<xX;, ..., x,<xL, (xl,xz)=xlxz
monotonicity:
f(XI, x2, ..., x), < f(xi, x;, ..., xa). fz( x l , xz)=x1V x,
fl(090) < fl(L 1);
7 ~
= a,
0 alxl 0 ... 0 a,x,,l
1).
ai f ( x l , xz, ..., x ,)
on
not
even
by f(X1,X2,
x3)
=x 1 @x 2
0 x3
= (x1xz
v f122)x3
f'(Xl,Xz,X3)=X1O~,OR3
3, on
v
(XI22
v
21xZ)f3.
= f(X1XZ,X3).
In order that a
The operation@is commutative and associative, and we can therefore dispense with brackets here, which would indicate the order in which the mod 2 additions are performed.
30
iI.2
THE ELEMENTS OF LOGIC
system of functions rp,(x,, x2, ..., x J, rp2(xI, x2, ..., xm), ..., rpPk(xl, ..., x ,) in the algebra of logic shall be complete, it is necessary and suficient f o r it to contain: a function which does not conserve the constant 0, a function which does not conserve the constant 1, 3 ) a function which is not self-dual, 4) a function which is not linear, 5 ) a function which is not monotonic: for each of the proporties enumerated above there must be some function in the system which fails to possess it. CONSEQUENCE. l.
by A 1, 2
4 do
5 do
3
by
x1x2, xlOx,,l
by
1 1
no no by
(x1-+x2)
(x1x2).
is
0 by
x-+O=nVO=a,
S. V. Yablonskiy has shown that, from any complete system of functions, a complete sub-systemcan be selected, consistingof not more than four functions. See S. V. Yablonskiy, The superposition of functions in the algebra of logic. Matematicheskiy sbronik (1952) 30, 329-348.
1.31
31
THE MINIMISATION OF FORMULAE
1.3. The minimisation of formulae in the algebra of logic x2,...,
by
f ( ~ 1 9 ~ 2...,xrn) ,
=
v
f(a1,az,...,u r n ) = 1
x ; ' x y ...
2"
by 2"'
by
2,
As
equivalent
P elementary m (T= 0 (T = 1,
r
rank p(x,, x2, ..., xt,
..., xm)= 1
disjunctive normal form elementary disjunction
conjunctive normal form
EXAMPLES
V
1.
V
x t , x3,
2.
g
(dl
x2, x3,
=ZlV xgV X4
v x3 v x4)(21 v
v 24) x2,xj,
32
[I.3
THE ELEMENTS OF LOGIC
DEFINITION. A minimal
reduction of formulae expressing a given function in the algebra of logic by means of identical transforms.
by
N
N
= m.2’”.
(1.6) on
by
~~
-
occurrence XI,
1.31
33
THE MINIMISATION OF FORMULAE
by :
1. 2. 3.
f
fxV fxVf
f x , V f x , = ( x l V x,) by
(f V x ) & ( f VX)&f =f,
1'. 2'. 3'.
v x,>&(f v
x2)
=
f v (x1 &x,), do
on by by
by
pi
DEFINITION. pip =0.
pj
(o=1),
(o=
p'
= xlxz
p" = Z 1 x g ;
p'p" = 0.
DEFINITION. A absorbs
p 1V p , V ...V p , p
pj p+pl v p , v ... v p k
absorption p 1 V p 2 V...V p k p V p , V p , V ...vpk = p i V p , V ...v p k
p,
p + p 1 V p 2 V ... V p k = I , p
34
[I.3
THE ELEMENTS OF LOGIC
o n p (f f x V f =f )
f
p,
p 1 V p , V ...Vpk p 1 V p 2V.. . vpk).
fx,
by
p (fx
v...
THEOREM 4. For a disjunction p 1 v p , vpk to absorb the elementary conjunction p , which is not orthogonal to each elementary conjunction out of p 1Vp,V ...v p k , it is necessary and suficient that each conjunction pi can be represented in the ,form pi=p,!pr (the conjunctions p : and pi' can degenerate to satisfying thefollowing conditions: k
1.
VpY=l,
2.
p+p;p; . . . p ; = 1 .
i= 1
p;
pi
pi'
p,
ofp,. EXAMPLE.
D
V
= 2 1 x ' 2 ~ 3 X I X Z X ~V
2 1 ~ 2 x 3V
~2~3x4.
Df
4
V p2 V p 3
= R 1 R 2 ~ 3V ~ 1 ~ 2 V x f3 1 ~ 3 ~ 4 Df
p
= x2x3x4. Df
pl Vp, V p 3
3
1 See Yu. Zhuravlev, The separability of subsets of an n-dimensional unit cube, Trudy Yablonskiy, Functional constructions Matem. in-ta im. V. A. Steklova (1958) 51; S. in k-valued logic, &id.
1.31
35
THE MINIMISATION OF FORMULAE
p
+ p ; p ; p ; = x2x3xq -+ x2x3xq =
1.
4 p1 V p , V p 3
D
= 2122x3 V X ~ X Z XV~2 1 ~ 2 x 3 .
pi pr = p'-'x;
A
by
p,
is
p
adjacent
p>
x,
p? = p r - ' z z .
isolated no
p.
pi, p i ,
..., p;.
cohesion: pi
v p: = pr-Ixp v pr-12;
= pr-1.
- 1.
by -1
by s
v p; v ... v p;) v (p;-I v p;-l v ... v p ; - ' ) v ... V (p;-" v p;-" v ... v p;-) v ... v (p:-"" v p;-=+' v ... v p;-"")'
(p;
1..
"blindalley"
a
36
[I.3
THE ELEMENTS OF LOGIC
no
EXAMPLE.
by
f(xl, x2,x3)
V
~12223
V x ~ X Z XV~~
2122x3
1 ~ 2 x 3 .
no
2nd
X1X2,
(212223
9
2122x3)
9
(xlf2x3
3
xlxZx3)
9
(Xlf2x3
9
x122x3)
*
x ~ x ~ , X Z X ~ .
XiX2
V
V
~ 1 x 3 22x3.
By
is iiX2 V
{XlXZ.f3,
RlX2~3)
( ~ 1 2 2
~ 1 x 3 .
by ~ x1x2x3) 3 ,
by
by A
EXAMPLE.
A
=(al V x2 V x3)&(x1V Z 2 V Df
A = 2132 V
X1X3
V
V
~ 2 2 . 3 22x3
V
V
~ 1 x 2 ~ 1 x 3 ,
1.31
31
THE MINIMISATION OF FORMULAE
2nd
S
= 2182
V~
V
,
2 2 3 ~1x3
Df
by p3 = 1 &(Il &I2);
p1 = pip; = Il&I2; p2 = p;p; = I
3
&x,;
= ~2
p3 = p i p ; = 1 &(XI & x3);
p2 = XI
p1
= I2& I 1
,
1 &(x2&%),
&23;
P3 =
& x3;
p2 = x3 &XI.
V p i = 1. I
i= 1
p ; p ; p ; =I1I3,
x1x2
22x3.
S
A
this
A, :
-
-
permutation of different variants. A indeterminate coeficientsl.
2nd
1954
by
38
THE ELEMENTS OF LOGIC
n.3
is
(1.7)
by EXAMPLE. x 2 , x3),
by by
1.9.
1.31
39
MMIMJSATION OF FORMULAE
TABLE 1.9 x1
X2
0 0 0 0
0 1 0 1 0 1
1 1 1
0 0 0
0 1 1 -
AO-AO-AO-AoO-12 - A13 00 00 000 = A23 = A123 1 - 2 - 3 -A1 -A01 - 1 3 - AZ3 11 = A011 1 2 3 = A' - A:; = -
1
= A2 = A:: = AiQ = A'''1 2 3 -
1 -
= A'''-
123
l1
-
=
O1
=
- A101 1 2 3 = O* :
A11 12 All 12
v v
A001 123"I? A110 123 = l 7 A111 1 2 3 = 1.
do
we 110 A123
= A::: = 0 ,
40
12.3
THE ELEMENTS OF LOGIC
A:::
= 1,
A:
= 1.
f ( X I , x 2 , x 3 ) = XlXZ v 21xZ.,x3.
on n n
by up
TABLE1.10 X324j5
~
i i x324x5
x3x425
x3x4x5
I l l ~-
m' m"
f(xl, x2,..., xm)= 1
x;',
x y , ..., xLm,
1
all = 1.
by l. ~
This situation is similar to that which arises, e.g. in the binary coding of some alphabet, when the number of distinguishablecode groups is greater than that of the distinguishable symbols the alphabet. Thus, in coding the decimal numbers, 0, 1, 2, 3, 4, 5 , 6, 7, 8, 9, by uniform code groups of binary digits, the minimum number necessary in each group is four. Only ten of the sixteen possible code groups are used, and the remainder are unused.
1.31
41
THE MINIMISATION OF FORMULAE
formula
by
by
all :
no
by 01 1
1.11.
I11 1.12. TABLE 1.11 X2
x1
x3 ~
TABLE I. 12 x1 ~
0 0 0 0 1 1
1 1
____
x2
O 0 1
1 0 0 1 1
I
x3
i
0l
0 1
0
1 0 1
42
THE ELEMENTS OF
:
1.4.
2
k 2k
f(ul, u2, ..., ui,..., u,) essentially depends on
vi,
CONSEQUENCE. f(al,o ~..., , vi, ..., a,,)
aj ( j # i ) , ui
iji:
(1.10)
Z= u1 V u2V ... V uiV ...V v,, on ui. uiifvj=aj=O(j#i),andthefunctionZ’isequal EXAMPLES.
Z‘ = v 1 v 2 - ~ q - ~ u , 2 is
(j#i).
-
~~~
by
1
A
by on A
i (1957) 18, no. 2;V. 1955 ; I. Coll. by 1959; A. 1951 ; 1954.
1.41
EXPANSION OF FUNCTIONS IN
43
ALGEBRA OF LDGIC
TABLE 1.13
f(vl, v2, v3, v4)=Vl V O2 V O3 V V4
on
v2, v3, v4) v1 ci2 = ci3 = a, = 1,
f
(vl, u2, O3, u4)
= vl
.
u2,
cil =u3 =a4=
1,
v3
a l = a 4 = 1 , a2=0.
o l = a 2 = 0 , u4=1; ul=O, u 2 = u 4 = 1
u2, v3, a4)=v3.
v4, cil = u2= 1,
f('17
O29 O39 u4)
= '4
*
u3= 0 ,
44
[I.4
THE ELEMENTS OF LOGIC
TABLE 1.14
1
03 ~
04
~ _ _
I
~~
51
v v2 v 53 v 54 ~
61
i 1
1
03 03 03
04 u4 04
04 04
0 0 1
0 1 0
0 0 1 1 1
0 1 0 1 1
-
1
~
03
__
_
_
1 1.
03 03
v4
0 0 O 1
274
v4
~.
54
~.
1.14.
m =k + I f(x1, x2,
...)x k , yl, y2, ..., y,) > (xl, x, x g , ..., x k )
by y,, ..., y,) by ( x l , x2,..., 2k
1.10 xk)
by
fj(nl,n2?
3). nl, n,,
...)nk,
y1, YZ,
yl) 7
..., g k ,
j < 2k*
by 0.
k 5. Let aJunction P(w,,w l , ..., w n - J (where ~ = 2 ~ ) ~ - _ _ _.-
0 . B. Lupanov, A method synthesising circuits, Izv. vyssh. uchebn. zavedenii, Radiofizika (1958) 1, p. 120-140 (see Sec. 5).
f(X1,X2,
...,xk,y1,y2,
=
{$O
. * . , y l )= Cx1, x2,
.
.... x k , f (O, O, .... O, O, y l , y 2 ? .... Y l ) ]
$1 [ X l , X Z , . . - , X k , f ( O , o ,
.
.
. . . , O , l,yl,YZ? . . * ? Y l ) ]
....xk, .... 1, O, YZ?.... YL)] . . . . . . . . . . . . . . . . . . . . $2 [XI
$2k-1
(1.11)
x2,
[x1,X2,..-,xk,f(1,1,...,1,
l,Yl,yZ, . . . , y i ) ] } .
by on
1 Note that, as follows from the definition of the functionQ(correspondingto the sense of a strict "or" in natural language), if q j = 1, then w j @ q j = w j @ 1 = Wf;and if C i j = 0, then w j @ c j j = wj@O = w j .
46
[I.4
THE ELEMENTS OF LOGIC
by by EXAMPLES 1. (
on
isf(xlx2, y ) n = 2k= 4,
x2.
xI
F(w0, ~
1 ~ , 2~ , 3
F(wO,
=) W O
V W I V w2 V w 3 . < j <4):
c O 1 , c 0 2 , c03)
w1,c12,c13) F(c20,
c 2 1 , w23 c 2 3 )
F ( c 3 0 , c 3 1 , c32, w3)
= WO @ 00 7
@ cll = w2 @ c 2 2 = w3 @ c 3 3 . = w1
9
9
F
cji=O. by
$i
[ul (A90 2 ( j ) ,Y ] = 0
$i
[GI
( j ) ,02 ( j ) ,Y ] = Y
(j
( j # i),
o0
( j = i)
i
F(wo, wl, w2, w3),
our
$2,
__
-.
0 0 0 0 1 1 1 1
-
..
1:
0 1 1 0 0 1
1
1
0 1
'
0 0 0
I~
1 0 1
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0
1
1.41
47
THE EXPANSION OF FUNCTIONS IN THE ALGEBRA OF LOGIC
xlyx 2 : 00,01, 10
11
j). *o = %12Y
Y
$2 = XIR2Y
3
=flX2Y
3
*3
= XlX2Y
2
*1
by (1.1 1) ( X I , x’z,
x2,y)
Y) = G . . f 2 f (O,O, Y>
v 1 2 x z f (0,L Y ) v v Xlf2f(l,O,Y)V
xlx,f(1J9Y).
2 on (1.1).
2.
x2,y ) on F ( w O ~w l w~ 2 w3) ~
= w0w1wZw3*
c j i = 1,
(1.12)
by $i $i
( j ) ,0 2 ( j ) , Y] [nl ( j ) ,0 2 ( j ) ,Y]
=
# i),
1
=Y
o1
=
i) . 1.16):
TABLE1.16
$0
$1 $2
*3
v x2 v 7, = 21xzy = x1 v 1 2 v 1, = 11 v x2 V y , = 1, v v y.
= 2112y = x1
12
48
11.4
THE ELEMENTS OF LOGIC
TABLE1.17
i
~~~
0 0 0 0 1 1 1 1
"; 0 1
0 0 1
1 0
0 1 0 1 0 1
1
I
162
-
-
*3
I
1 1 0
1 1
0 0
1 1 1
1 1 1
I x,, y )
V ~ ~ V ~ , ~ , Y ) I ~
f(x1,x2,y)=~xl
&[X1
v xz V.f(1,O3y)] &[X, v 2, V"f(1,
on
f(xl, x,, y )
3.
LY)].
(x,,xz),
P(wo,w,,w,, w 3 ) = 0 0 v 0,v 0,v w3. pp.
cji ( j #
i)
1.14).
(ol,oz,..., a,),
j = j ( o l , o,,..., ...,O,O, Y,, yZ, ..., Y J ] , ...
F {$o [ul ( j ) ,o,(j), ...,o k ( j ) , f
..., = F { c j o , ...,c j j - I ,
..-,1,1, Y , , y,, .-, Y J ] ) = ..., Y J 0 ( j ) , ...)ck ( j ) ?Y l ? Y 2 , ...)
( j ) ,6,( j ) , ...,0, ( j h f (1,1,
f [o1( j ) ,cz( j ) , ..., o k W 3Y,,
@ C j j , c j j + 1, ..., c j z k -
I}
=
12,
( j ) ,o2
7
1.51
49
ELEMENTS OF THE LOGIC OF PREDICATES
0. B. k
F(wo,wl, ..., wN-l)
N>2k x2, ..., xk, yl,
..., y,)
by
$i
(1.13),
(zk< i
xi[ol(j), 0 2 ( j ) , . . . ,ok(j)] =cji
< ~ ) .
(1.15)
1.5. Elements of the logic of predicates 1.5.1. PREDICATES AND QUANTORS
..., xn)
by
x1&x,, x1 Vx,, 2,
propositional calculus
x2,
EXAMPLE 1. x1
1
x2 1).
1 1)
x2
x3
by (XI
v x2)
W. object domain 9JI. 1 Remember that “or” is taken here, inclusive-exclusivesense.
predicates
book, the strictly exclusive sense, but
an
50
P.5
THE ELEMENTS OF LOGIC
1,2, 3, ..., EXAMPLE 2.
P(x,y)
x
;
“x
r(x)
Q(x)
‘‘x
R ( x , y ) : “R(x, y )
y
x
y,
y
x
us
R(x,y ) .
EXAMPLE 3. by “S(x, y ) x
S(x,y )
y
x, y S‘(x,y )
S ’ ( x ,y )
y,
z
1)
x
y
y,
z
x,y
S(x, y ) S ‘ ( x ,y ) quantors -
logic of predicates, calculus of predicates -
is
on V. F(x),
A MF 1.
on
M, F(x) property F
M,
1.51
51
ELEMENTS OF THE LOGIC OF PREDICATES
MF. “x
F(x)
F”,
F(x) F(x)
..., x,) ..., x,)
M,
f ”. A
xi,x2, ..., x,
“
is
n
quantor ofgenerality
F(x)
M),
x F
(x)F(x).
quantor of existence
F(x) M)
x
F(x).
F
1,
F(x) F(x)
(a F(x) 2
if F(x) r(x) F(x, x)
“x
M a19a2, ...,a n7 F(x) F ( a l ) & F ( a 2 ) & ... &F(a,), F(x) F(aJ V F(a,) V
... V F ( a , ) .
fl
f,
+f, + ...+fk
f(x) dx. F(y)
F(z)
F(x).
52
[IS
THE ELEMENTS OF LOGIC
S(x,y)= F(X,Y)&[(J”(X)&J“(Y))V
(f(X)&J(Y))]&
’
Df
&(Ez) [ F ( x , z ) & F ( z , Y )
S’(x,Y ) = F ( x 9Y ) Df
(x)
[(J“
‘‘x
(1.16)
Lk ( Y ) ) v (?(XI & f ( Y ) ) ] Lk J”
[ F ( x , z) &F (z, Y )
C(x),
Q(z)],
-,Q
,
by
1.51
53
ELEMENTS OF THE LOGIC OF PREDICATES
x,
x”:
(Ev) ( E w ) { F ( u , x ) 8L F ( w , x) 8~n ( 0 , w , x>>.
F(x, y ) , C(x) (Eu)( Ew) ( ( E u ) S(u, u, x ) &(Eu) S ( w , u, x ) & n ( 0 , w,x)> .
by
“x
lI(x),
H,.(x), odd
(1.17) “x
by
y +y” :
y, x
( Y )S( Y , Y , x )
9
by
r ( x ) (“x (EY)S(Y,Y,X).
(1.18)
F, xl, x2,...,x,
..., % [F,
...,H , x , , ..., x m ] .
%[F, G , ..., H , xl, ..., xm] on xi, x2, ..., .x, variables, on T F, ...,
by
A ( x 1,..., x m ) = T [ F , G,..., H I .
by % [ S ,x] S(x, y , z )
EXAMPLE 4.
“x+y=z”,
“x - y=z”.
“x
% [ S ,x]
“x
54
THE ELEMENTS OF LOGIC
by by
(x) F ( x ) &( Ey) H ( x , y ) x) bound
bound
x
z, A
by by
by
by F( ), G ( , ) , H ( , , )
x, y , z bound
1.5.2. THE INDUCTIVE DETERMINATION OF FORMULAE IN THE CALCULUS OF PREDICATES
by
1.
by by
F @(Al, A,, ..., Ak)
2.
a,, ..., 21,
‘u,,
A,, A,,
..., A , by :
on
‘ui, bound
3.
@[‘ul, a,, ..., ‘?Ik]), bound ‘u[...x i . ..]
Formulae of this type are called atomic.
‘ul,?I ..., ,, ‘uk. xi
no
IS]
55
ELEMENTS OF THE LOGIC OF PREDICATES
(xi) (%[
...xi.. .])
(Exi) (%[
...xi...])
x
bound, %. 4.
NOTE.
3
% (Exi) rank of a quantor
3,
on 3.
(xi)
:
1. A 2. A
1
no k+l
k.
by
use by S(v, u, x), S(w,u, x), n(v,w,x) (Eu) S(u, u, x), (Eu) S(w,u, x)
(1.17) :
1 3
no 2 A , & A , &A,);
(Eu) S(v, u, x)&(Eu) S(w,u, x)&ll(u, w,x) (Ew)[(Eu)S(v, u, x)&(Eu) S(w,u, x ) & n ( v , w,x)]
3
;
(Eu)
3 (Eu)
(Ew)
2,
3.
(Eu)
3.
1,
56
D.5
THE ELEMENTS OF LOGIC
1.5.3. FUNDAMENTAL IDENTICAL TRANSFORMATIONS I N THE CALCULUS OF PREDICATES
by on by
'ul [(F7 ...)H7xl, ...,x],
'u2[F,
... 7 H , xl,
... x,,,] F7 G ,
..., H
al
xl,
'u,
...) x,. '21,
212.
EXAMPLE 5. ( 2 )H ( 2 )H
(z) &
G (x, ). & (EY) F ,.( Y> (x, ).
(z) &
Y
v (EY) F (x, Y ) )
7
A,&A,&A3, A,&(A,VA,), (A,&A,)V(A,&A3). dentical transformation.
do
of on by
1. ~
(Ex) A ( X I
(XI
A (XI
7
1.51
57
ELEMENTS OF THE LOGIC OF PREDICATES
by vice versa.
A,VA,
A,&A,,
A,&A,
A,VA,. (E y ) F (x,y )
2
A,
( z )H ( z ) &
(qav ( Y )F (x, Y ) ) .
11. : (XI
(4A1 (4
(4 ( X I & A2 (4) (XI
A2 ( X I
7
( Y ) ( X I A (X?Y ) .
( Y ) A (x,Y )
by 111. : :
(Ex)(A,(x)
v A&))
(Ex)A,(x)
v (Ex)A,(x),
(EY) (Ex) A (x, Y>.
(Ex)(Ey) A (x, Y )
111
i.e.
[ A , (4v
(41
(4v A2 ( X I (4& A2 (4) (Ex) A1 (4& (Ex) A2 (x) (Ex) (4 (4( E d A (x, Y ) , F Y I (4A (x,Y ) (XI
A2
9
(x) A1
3
7
?
*
A,(x)
“x
“x odd
A,(x)
58
D.5
THE ELEMENTS OF LOGIC
vice- versa.
23
(x)
(Ex),
x bound by
23
constant with respect to the given quantor.
[x] x,
0
[XI {a 0 B (x)}
0
2l
23 (x), x.
[XI
0,
:
'u & (423).(
(4{'u& 23 (x)} (x) {'uv I).(
7
a v (x) 23 (x)
2
(Ex) {a&23 (x)}
'u &(Ex)%
(Ex) {'u
'u V (Ex) 23 (x).
23 (x)}
(x),
EXAMPLE 6.
{W v X ( 4
(7)
-+
G ( 4&X(t)>. X(t)
by by
[(F) &
(7)
v (G (4& x
1.51
59
ELEMENTS OF THE LOGIC OF PREDICATES
x(t)VX(t)
1
do
V). 1.5.4. THE EXTENDED CALCULUS OF PREDICATES
extended calculus narrow calculus of predicates
of predicates,
P>W {F> (.
W>
(1.21)
(Y>H (x9 Y>>.
(1.22)
jV
F(x)VF(x) F.
F
bound
x
x;
on x, H
and x
60
[IS
THE ELEMENTS OF LOGIC
on x.
on
H.
on pp. bound on
by (EF) F > .(
c%
( Y ) ff
7. (
Y).
x). ( y )H
(x,y).
1.5.5. THE CALCULUS OF PREDICATES WITH BOUNDED QUANTORS
-
I.
. 11.
bounded quantors,
(Ez),,
t X (z) t,
X(z)
(T)~<~X(Z)
6,
1.51
61
ELEMENTS OF THE LOGIC OF PREDICATES
t,
z,
X
(t)
do ‘‘x< y” ; : (Ez) [z
[z]~<+~,
< t &X(z)], [zIzat
< t
( 7 ) [z
<
< t -+ X
.
(t)]
<.
bound by
t p, t
:
(EWw&(e>[Z(e)
v
(1.23)
z<@
C C t
bound X(t).
t,
~
(Et) ~
(e),<, .,[Z
z
X(t)]
t. (1.23) calculus of one-placepredicates with boundedquantors.
(ET) { ( z ) ~ T< (~t ) + X
(I)-(IV) X(t) (@)T<e
(t)}
62
[I.5
THE ELEMENTS OF LOGIC
(Ez)
( ET),,~(
v [XI& ( E z )
(Q)
&
(7)
r<e
T
Ea),
(Eo),<~(E~)~<~ ;
X(O)
bound,
z
z
x(7>].
T
~ = t ,t = k ,
t
t=z-k,
t=x+k,
k
t, z 5
=
(Eo) (Ee) T , t
Df o < t
=
= (7)
F.
Df r < t
rQeQo
x
z < 1,
by
(z)[z< t<
1- X ( z ) (Ez) X(x) (7) t
(Ez) X ( t ) t
f
X(x)
(2) t
X(z)
(8)
ELEMENTS OF THE LOGIC OF PREDICATES
Furthermore: 3.
F
4.
e q u (Ez)X(r) V X(t). 5-=t
(9 X(9 equ r
(4 X(4 &x(t). *
Similarly 3’.
(Ed X (4 ev ,$F! X (4 V (e < t &X (t>) e-Cr
4’.
(7) X(.t> ev e
CT> xW~(ei~+W)). e-Cr
63
C H A P T E R I1
OPERATORS. LOGICAL NETS
11.1. General remarks finite automaton by by circuit (logical net)
by
operator
1).
by
2
bounded-determinate operators2 3,
bound -
6
-
by
4 by by real-time processing
by discrete itformation 2.1 real-time device f l , f 2 , ...,f;,, on p. automaron operators
automaton correspondences. 3 0
autonomous device
64
2).
11.11
65
GENERAL REMARKS
input information processed information.
g 2 , ..., g n
outputs
alphabet, letters.
by
k
1'
f2
fm
k
-
-
91 * 92
----4
-
--
9n
A a word in A . t = 1,2, 3 , ..., p , ... If gi(t)
t
gi
T)
no by
0, 1, 2, ..., (k1
Otherwise known as the input and output channels.
66
[II.1
OPERATORS. LOGICAL NETS
EXAMPLE. 0
“0”
1)
“1”
“1”
m , , m,, ..., mk
A,A,...Ak,
m , *rn2..-rnk
..., Ak A , , A , , ..., A ,
3,
direct product
3= A , x A , x ... x A,;
A
direct power k m
A,
A,.
Ak
mk.
k
l.
alphabets of input channels output) alphabet channels
alphabets of output channels single input “amalgamation” of parallel no
Of course, a fully-determinateserial numerationmust be established for these channels for each system of k letters to be considered as a word.
11.11
67
GENERAL REMARKS
Fig. 2.2
binary alphabet, uniform binary code. EXAMPLE. 28
28
by
(0, 1)
a , b , v , ...
10 000,00110,01101, ...
A by k S
S
by
S
S :
S2
(2.4)
A
by infinite sequences of letters
2.2); B.
68
[II.2
OPERATORS. LOGICAL NETS
( t = 1, 2, . .., p, ...) by A'
A.
A).
(2.5) g(M2)...g(P)-. .
(2.6) 8
on
E A'
s(t)= e[S(t)];
(2.7)
9(t)EBf.
to classify these operators. 11.2.
of
real-time g(t)
non-anticipatory. on
t, on
T>
(T)
on
f(t),
on
on
DEFINITION. An g ( t )= 8 [ f ( t ) ]is non-anticipatory operator, t,
T<
t.
determinate operator g(t)
f(l)f(2)...f(t -l)f(t).
determinate operators by
1
1. Truth operators (memoryless operators). r g(t) f(t), f (T) T > t (condition of determinacy), on T < t (absence of memory).
I
by on (7)
by
11.21
69
THE DETERMINACY OF OPERATORS
A :
0
1
1
0 crl o2 ... om
rl r2
by
... r,,
n 1'
=
x2?
*..7
xm),
z2 =fZ(x1,~2,..*,xm),
. . . . . . . zn
= fn
( x 1 7 x2,
*.*7
xm).
by (m=n = 11.1:
by
: z =2.
23
3, q2(xl,x2, x3)
0 0 0 1 1 1
ql(xl,x2,x 3 )
by
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1
0 0 1
0 1 1
1 0 0
70
m.2
OPERATORS. LOGICAL NETS
TABLE11.2 x1
22
___-
0 0
0
0 1 1 (1 1 0 0 1 (0)
1 0 1 0
1 1 0 0 1 1
1
0 1
-_-
by
11.2
11.1
by
V 2 1 x 2 ~ 3V 2 1 x 2 ~ 3V XV~ 2 1 x 2 2 3 V 2 1 ~ 2 x 3V
ZI = 2 1 2 2 x 3 ~2
=X ~ X ,
ZI = 2 1 x 3 z 2
V~
= 2, ( x 2
2 2 3 ,
of
...f ( t ) ...
f(
... g ( t ) ...
~ 1 2 2 2 3 .
v x3)v X ] (x2 v x 3 ) .
2. Constant operators.
on
x I x ~ X ~ ,
t
on
t
z>t by
A g (1) g
generated by the constant operator. autonomous devices, by
on
no
.. g ( t ) ...,
11.21
71
THE DETERMINACY OF OPERATORS
by A
by fix
trees, m
infnite information tree a$rst-rank node root of the tree, call first-rank branches.
0 ni
Fig. 2.4
Fig. 2.3
-
-
0, 1, ..., m- 1
2.3 m = 3;
0, second-rank nodes; m
- second-rank
branches -
m on
m
(i = 1, 2, 3, ...), i-th level of the tree.
mi
2.4
f ( t ) E A‘. 0
on. by
on on A.
101 ... .
on on i >1
2.4
12
LI1.2
OPERATORS. LQGICAL NETS
i, f ( l ) f ( 2 )...f( t ) ...
g(1) g(2)
...
t=t,,
do
t,,
z >t ,
on g(z)
A
B.
T,
A,
loaded tree TA-B,
B
Fig. 2.5
T.,
on
T,
B, B.
A
2.5,
2010.. .
1011 ...
by
g ( t )= 0 [ f ( t ) ] , t 6 t,
by g(t)
B
g’(t)
z
A by t,-
T.
do
B r
, In particular, g ( t ) is uniquely determined by the values of f(1). f ( 2 ) , ...,f( t).
t,,
11.21
13
THE DETERMINACY OF OPERATORS
....f ( t , )
t, A
t, g(1),
.... g(t,)
:
r,
EXAMPLES. 1. The truth operator
by
by
........................ ........................ ........................
x
k . 4 1
0
i
\
r
1
0 4
0
0 4
0
(a)
................................ ................................ ................................ ........... ..................... ?\;pf 4 h
0
b
O
4 D
0' 0,
li
1
Y'
Fig. 2.6
on
all
is
101001000100001 ...,
2. The constant operator by
2.6b
All 1,
0,
14
III.3
OPERATORS. LOGICAL NETS
on 3.
by
0
2.7b,
11.3. Bounded-determinate operators (operators with finite weight)
>1
y
by
T,
>j .
y
T, f ( l I f ( 2 )...f ( j - 1 ) (j-
bough of rank j
T,
a
T. T
T
y.
T,
8,
0
a residual operator of rank j by 0, single residual operator of rank 1).
DEFINITION. distinguishable by indistinguislzable.
0
1
11.31
75
BOUNDED-DETERMINATEOPERATORS
trees
boughs
on EXAMPLES , 1.
truth operator
2.
constant operator
j 101001000100001 ... by 3.
(j-
bough bough
0 0
f
f
..f ( j -
Or
1)
8,
Or. afinite number of (functionally) distinguishable states.
DEFINITION. determinate operator) it
bounded ( a boundedoperator with Jinite memory
unbounded.
DEFINITION. weight k of a bounded-determinate operator
76
tII.3
OPERATORS. LOGICAL NETS
1,
CONSEQUENCE. truth operator. REMARK. a.
2
by basis of a deternzinate operator
K
K,
on
K
K K
Canonical equations of a bounded-determinate operator. 8 K. Q= q l , ..., g K internal alphubet alphabet of states), 6' Q,
Q, 1)
f(t) g(t),
by q(t)
by
on Q
11.31
I1
BOUNDED-DETERMINATE OPERATORS
on
( a sequence of states). q(
EXAMPLE.
2.8. 1 1 10... on
1001 ... qoqlqlql ...
by
... z ( l ) z ( 2 )...
...,
40
Fig. 2.5
..
x'(
.... If 8
2'(1)2'(2)...
z(t)=z'(z) 5.
x(t)=x'(z) t
is x(t) =
.'(.),I
4 ( t ) = 4'(2)
z(i) = ~ ' ( 7 )
q(t
\
+ 1) = q ' ( 2 + [ x ( t ) = x'(z)]
[ q ( t )= q'(t)], [z(t)=z'(z)] [ q ( t + I)=q'(z+ q(t+l) We shall declare that the state qo is ascribed to the
z(t)
the tree.
78
[II.3
OPERATORS. LOGICAL NETS
@(x,q )
Y (x,q )
A
Q
x, z , q
is 11.2
p.
by
EXAMPLE. 0 q1
2.8,
11.3. TABLE11.3 II
:
z ( t ) = x ( t ) * q ( t )v a ( t ) y ( t ) ,
q(t
+ 1) = x ( t ) * q ( t )v
z(f>=x(t)Oy(t), q(t+ I)=x(t)@q(t) q( 1)=0.
NOTE.
x ( t ) , q(t), z ( t )
p. 66);
zl( t ) , z2(I),z3( t ) , ..., z n ( t )
z(t)
z 2 , ..., z,,
n
qo
11.31
79
BOUNDED-DETERMINATE OPERATORS
. . . . . . . . . . . . . . . . . . . . . . . I
. . . . . . . . . . . . . . . . . . . . . . .
xl,
I
.... xm,zl, .... zn, q l , q2, .... qk
@
Y
is
bound
@, Yy
the canonical equations (functions1, tables) on the alphabets X , 2, Q 2 Q is ~~
The following terms are also used in the literature: “internal ( Y ) transfer function” and “external (@) transfer function”.
80
[II.3
OPERATORS. LOGICAL NETS
TABLE 11.4
0 1 0 1 0 1 0 1
0 1 1 0 1 0 0 1 -
z(t) x(t)
8
q(t+ q(t);
q(1)
q o = q l ; Fig. 2.9
q,, 00,01, 10, 11.
\ 0
Fig. 2.9
Fig. 2.10
q1
11.31
81
BOUNDED-DETERMINATE OPERATORS
11.4.
2.10
8,
by by
2.9. 8, q( = q,,,
by
by place-by-
place sequential summing operator,
If
q1 as
by
O(8,
x(t)
z(t) q ( t + 1) q(t)):
z ( t ) = z,x,q
v x,z,q v 2,R,q v x,x,q = XI 0 x, 0 q , q ( t + 1) = XlXZ v x,q v x,q . by
11.5 q
00,01, 10, 11; y1
y,
q.
TABLE11.5
8, 11.4
82
q(
tII.4
OPERATORS. LOGICAL NETS
10, (K=2),
00
10
01
qo);
11
qi,qj indistinguishable with respect to the Y,
canonical functions @
q( 1) =qi, q(
=q j ;
distinguishable. A
q,, q j Y
@
‘uGy
K
K
u ‘,
the common bounded-determinate operator of the pair of canonicalfunctions SP, Y, weight of the common bounded-determinate operator, partial bounded-determinate operator fixing the initial condition. no
do
(e,}
‘uQY 8
do
‘u @, Y
11.4. Elements (elementary automata) and circuits
A
K
11.41
83
ELEMENTS A N D CIRCUITS
by on
by
by
circuit.
complex automaton elements,
elementary automata,
-
by
by junctional algorithm of the circuit). element
algorithm
circuit
by Q , m input channels X , (m=O, 1,2, ...), 3) n output channels Zn(n= 1,2, 3, ...), 4) element.
cell
zl(t)=
[~l(t),xz(f), -*-,xrn(t),q(t)]
. . . . . . . . . . . . . . .
7
z n ( t ) = G n [ ~ l ( t ) , ~ Z ( t ) , . . . Y ~ , ( f ) , q ( ‘,) ]
q(t
+
= !P [ ~ l ( t ) , ~ z ,...,xm(t),q(t)], (t)
xi(t),z j ( t ) ,q ( t ) X i , Zi
Q.
Our definition of an elementary automaton coincides in the main with that of a finite automaton, used by the authors of studies in which the behaviour of automata is considered abstraction from their design.
84
p1.4
OPERATORS. LOGICAL NETS
by poles (input
output by
by on zj=Qj(xl, ..., x,, q ) Y(xl, x 2 , ..., x,, q )
q(t).
t = 1,2, 3, ..., internal memory, x l , xz,..., x, q,
on external alphabets.
k volume of the internal memory Q zero. z i = D i ( x l , ..., xi, ..., x,, q l , ..., qk), on xi.
Qji
subordination
xi causally determines xi
precede
zi
imaginary variable
xi;
zi
Qi.
by by
xj
x,
no xi Qi on
xi
(2.11).
{‘3JIi}.
. .-
~
A. W. Burks 41, no. 10.
G.
Wright, Theory of logical nets. Proceedings of IRE, 1953,
11.41
85
ELEMENTS AND CIRCLTTS
(fm,}
DEFINITION. A circuit
(aj}
set of poZes of the circuit set
of cells
set of channels
graph nodes
branches (channels) poles
cells
DEFINITION. A
no
input
p+
a>+p
p
1. If a
a
g
a)
p
p), 2. 3.
p
CI
a>
a>
-p.
-+p, b>
+y,
a>
+.y.
by
EXAMPLE.
'#Ix, 1.
fm,, '%Re, fmd
%Ix
2. fm,
q(
=0
:
z(t)=x(t)m(t), q(t+ 1)=x(t)@q(t). 1
A group may contain many specimens of the same element out of the set
{mm,}.
86
tII.4
OPERATORS. LOGICAL NETS
3. $'T,I
2 z(t) =xl(t)@xz(t).
4. % I,
z ( 0 = 4 (4
q(t
9
+ 1) = x ( t ) .
z(t)
md,
i
1 Fig. 2.1 1
3,
Fig. 2.12
Fig. 2.1 3
r-I
I I I I I
I
I I
I I 1
I
I I L- -------
Fig. 2.1 5 x(t)
2.15, 2.16, 2.17
{mE, m, ma,
2.18
md>.
Fig. 2.14
11.41
87
ELEMENTS AND CIRCUITS X
$4
Fig. 2.16
Fig. 2.17
88
D1.4
OPERATORS. LOGICAL NETS
no
no 2.16b, ‘%I?,
z4
!Illd. z1
2.15, by 2.17
z3. A
no
by 2.18.
2.15
2.17
+z4
Fig. 2.18
z4
z3.
proper improper a vicious circle arises in the transmission of infornzation
11.41
89
ELEMENTS AND CIRCUITS
function of the circuit.
{YJli}. on :
I.
of
no of
by :
by
2. by logical net.
11.
zp
z,
90
[II.5
OPERATORS. LOGICAL NETS
21
. .. ... “” .. .
YZ”
t
111, -
z,,
z,
logical net,
11.5. Logical nets
N
E(N)
Note that Burks and Wright here use the term “completely proper logical net”; their term “logical net” corresponds to our term “circuit”.
11.51
91
LOGICAL NElS
z1(t)= Z 4 ( t ) O X ( t ) O 4 ( t ) ,
4(t
+ 1) = Z 4 ( t ) X ( t ) v 4(t)z4(t) v 4 ( t ) x ( t ) , (4 = 24 ( t ) 0 P (4
22
P(t
+ 1) = Z 4 ( t )
3
0P(t), ~ ~ ( t ) = ~ ~ ( t ) ~ z ~ ( t r)( ,t )z, r~( t( +t l)) = z , ( t ) . ,
*
(2.12)
92
[IIS
OPERATORS. LOGICAL NETS
E(N) by E(N)
5 by
A A
or
E(N) A
C
B
0 the system E(N) is proper the proper organisation of the circuit. A,
A, B
C,
C by SZ, n, v
m,p
9,
E(SZ2)
i . . . . . . . . . . . . . . .
f2
m+p
2.20b.
That is, 0 is an operator with anticipation.
n+v
I
11.51
93
LOGICAL NETS
zn+l
zn+v
Fig. 2.20
E(Q,),
1
by
EXAMPLE. A
S
2, S
by
do no
94
nI.5
OPERATORS. LOGICAL NETS
Q
xi
xi
4 of
,I. J
.
.
.I
. .....
21
2"
Fig. 2.21
by
by
f2 by xi
xi)
xi.
EXAMPLE. by 2.
zi xj
feedback loop
~
111,
Q
11.51
95
LOGICAL NETS
zi(t) xj.
B
zi
zi
xi
zi(t)
xj(t).
non-subordination zi
zi
xj
d by
by [xl( t ) , . . . , ~ ~ - ~
Gi As
zi
....
xi. Q, Q,
no
Q,
Q,.
52,
Q, Q,, Q,
9, cascade.
96
[US
OPERATORS. LOGICAL NETS
A z
by
x l , x2.
I
z to x1
zn (0)
Fig. 2.22
(a)
Fig.
I"' Fig. 2.23
z by
Fig.
is
loop.
x2)
11.51
91
LOGICAL NETS
%TIi) DEFINITION 1.
{mil. 2. An 3.
xi, xj
4.
xi,
zi
5.
$
identification of the inputs
3 4 coupling to the out-
put. cascade loop,
2.24
2.25. 2.23).
For any logical net N , the corresponding systenz of equations E(N) is correct, and consequently speciJies a boundeddeterminate operator 0,.
Fig. 2.25
N
DEFINITION. A
%,
implement
0,
8,. by
1-4
1.
8, 2.
ON,
0,. %*, ON2,
SN, 8,. 1
In the following exposition, the words “embodying the set 9Jb” arc often omitted.
98
tII.6
OPERATOKS. LO(;ICAL NETS
XN
3. x(t)
xi(t)
xj(t),
illN, 4. x,(t) zi(t)= ai[xl (t),
@ i [ x , ( t ) ,. . . ] ,
illN,
...I,
XNp.
N k,, ..., k , k=k,-k,..-k,
v
k
of volume of the
illN. internal store
(1)
N.
11.6. The realisability of bounded-determinate operators. Problems in the
analysis and synthesis of automata
of a 3: :
any bounded-determinate operator is
physically realisable. “t-ealisability”.
.
. . . . . .
(2.15)
11.61
99
BOUNDEDDETERMINATE OPERATORS
qi,,qi2,..., q i ,
no
% WL) ,%TII; z,
%TI:
EXAMPLE. %TI,
2.26 on
z1
+ 22 Fig. 2.26
4's
z2
4 z3 x1, x2, x3, x4.
z2
4. z3
4, 6. on
100
[II.6
OPERATORS. LOGICAL NETS
m
n n
m ~1
=
(XI, ..*,xm),
. . . . . . .
z, = @, (x,,
N, f2(x1,.... xm), fi[x,, .... y,, ....
..., x,).
z , =f , ( y , , ....
N2
z2= z2=
.... x m ] ) , N,
N, is
complete
:
by
(2.16)
. . . . . . . . . . . . . . . *I q k ( t + I ) = Y k [ X 1 ( t ) ....tX,(f),41(t),...,qk(t)], J Y,, .... Uk
....
z,, .... z,, q1 in the
.... q k ( t ) of
@,,
.... @,
11.61
BOUNDED-DETERMINATEOPERATORS
~ ( t=) q ( t ) , q ( t
101
+ 1) = x ( t ) .
2.28,
'i
.....
; i : , i t
,I..
.*[,
.
I
Fig. 2.28
4z2
Fig. 2.27
9X= {%Il, ..., %In} 0,
8.
3
I
by V.
1 V. B. Kudryavtsev, Questions of completeness for automata systems, DAN SSSR (1960) 130, no. 6; id., A completeness theorem for a class of automata without feedback, DAN SSSR (1960) 132, no. 2.
102
“1.6
OPFRATORS. LOGICAL NETS
111.
on
Analysis of an automaton.
Synthesis of an automaton.
for
!Dl
9X
so,
IV-
VII, by
0’
0
1. The
of .y(I),
_..
DEFINITION 2.
z(t)=O
z-shift (z
[x(t)]
z’=O’[.x(t)],
t : z(t+z)=z’(t).
z(t)=U [ ~ ( t ) ]
xo
z’(t)=U’[x(t)],
...
by
0‘
11.61
103
BOUNDED-DETERMINATE OPERATORS
2’(2), z’(3), ...,
t(l)x0 ... X O ,
t(2)x,
... xo, ...
-.(a- 1) times
by
...)
(a- 1) letters
...)
z’
(a- 1 ) letters
...)
z’(3),
...
(a- 1 ) letters
DEFINITION 3. 0 with a shift
Y
T
synthesis with a shift (in a time-scale)
N realises (in a time-scale 8.
N
C H A P T E R 111
O P E R A T O R S OF P H Y S I C A L ELEMENTS
111.1. Introductory remarks
on
fast operation
reliability,
cost, on on
book
a
on
2
1. valve transistorised flip-flops,
104
3 magnetic
III.21
105
VALVE AND TRANSISTORISED ELEMENTS
111.2. Valve and transistorised elements 111.2.1. PHYSICAL PRINCIPLES
on triodes
3.1) R,
E,
3.2).
pentodes
Egi
on
f+
€+
Fig. 3.1
Fig. 3.2
on
by by rheostat coupling 3.3). by capacitive coupling
is
3.4).
by
See A. M. Bonch-Bruevich, The use of electronic valves in experimental physics, Gostekhizdat, Moscow 1956.
106
[111.2
OPERATORS OF PHYSICAL ELEMENTS
Ri
by C
ORa 4
-
ii T---
l
A
6.1
TC
OR2 I
i -
d EFig. 3.3
Fig. 3.4
C R, R,R,/(R,
+
is
R, R,))
C R , R 2 / ( R l+ R,).
R, E,
R,Ri/(R, + Ri) is
for
t
f+
+ E- - R , / ( R , + R2).
ia
ORa
r
0.
4---A
k Fig. 3.5
E.3
Fig. 3.6
111.21
107
VALVE AND I’RANSISTORISED ELEMENTS
on by by
negation
4
inverter,
i9
Fig. 3.1
3.5 V.
-
by i, on E,
E, i,=f(E,) i, on
3.7 ig = q (E,). 3.5.
A 1
E, EL
A
up on by
T, 3.8.
Ea=E+,
108
[III.2
OPERATORS OF PHYSICAL ELEMENTS
on
on c1
Eg=Eeq
3.9)
by 1
b
E,.
of
;
-- -.1
OR.
'9
/€+ Fig. 3.10
Fig. 3.9
T is
U,
on
by V
3.10).
A
U,,
111.2)
109
VALVE AND TRANSISTORISED ELEMENTS
EL
A As
up on
U, =E ,
T,
1 pentodes
Z -
9
4. 3.12
Fig. 3.11
no by
3.11.
A no Z
R, 1
E,,
U,.
cathode follower U,, E, U. If
3.13
double cathode follower
R,. E, 1
If
I10
DII.2
OPERATORS OF PHYSICAL ELEMENTS
R,,
on
power amplifiers.
diodes
Fig. 3.13
--..-
........ ..... ..... ..... . 1 €2
Fig. 3.14
conjunction
diodes disjunction m
on
3.14.
R R,,), 1, 2, 3, ..., m :
A
m
R & Rfd. Two E, E,
1
111.21
111
VALVE AND TRANSISTORISED ELEMENTS
up
R
(Uh=E+),
1 E , - E,,
R.
't
Fig. 3.15
R 9 R,,
U.
0
E,,
by is
on
i,
V
I) 3.15.
R,,# 0,
by
E, :
U
=
U,
+ A'U .
on
A'U
R,,
(3.5)
m,
A'u= f (Rfd,
mo)
mo on A'U by 3.16
112
OPERATORS OF PHYSICAL ELEMENTS
tIII.2
3), a=
1/R
E , - E,
A‘U
As
on
Fig. 3.16
no of
(Rout)
-g_,FE’ R
I
A
E,,
Fig. 3.17
Uzz E , .
A
U,, U = U,+A‘U+A”U,
(3.6)
111.21
113
VALVE AND TRANSISTORISED ELEMENTS
R, -gR by
A"U R,,,IR.
on
Rout Rout
by
R.
by
E r n * - E - T 2
. . .... .... m
&
Fig. 3.18
Fig. 3.19
C,
R
E,.
'c
3RC, < 2 ,
(3.8) R.
by
R,,
on
R :
(3.9)
114
IIII.2
OPERATORS OF PHYSICAL ELEMENTS
3.19
As
disjunction
0, 1,
El,
E,.
R $ R,,
E, on on E+ : U
= E+ - A’U.
(3.10)
Rout.
(3.11) AU
on by
AU
by by
point-contact
junction transistors.
111.21
115
A N D TRANSISTORISED ELEMENTS
by by
A book, :
f
E+
Fig. 3.21
collector, emitter, base
3.21,
C
E
116
DII.2
OPERATORS OF PHYSICAL ELEMENTS
a
_L Fig. 3.22
by :
of -
-
by by on 3.22
by
111.21
117
VALVE A N D TRANSISTORISED ELEMENTS
TI
T,. i, =f( U,) ib= 'p( Ub)
(ib=
U, =
'C
Fig. 3.23
3.23.
three
cut-oflregion,
I1 I11
linear amplifring condition,
saturation
region,
I
111.
vice versa. ib = cp
(Ub)
U, =
3.24.
We
TI E-
3.25) by
a R,
A UL = U,
TI.
1
118
[III.2
OPERATORS OF PHYSICAL ELEMENTS
T,.
i , = q (U,)
3.26),
do by
U, a= R,R,/(R,+
R,),
R,
ub
U, > 0 5 V u, =0.1 v
I ib Fig. 3.24
Fig. 3.25
3.25)
Fig. 3.26
T,.
UL
TI
T, 1.
11r.21
119
VALVE AND TRANSISTORISED ELEMENTS
111.2.2. THE REALISATION OF ELEMENTARY LOGICAL OPERATORS
on
0
I
Z = I
z,
Y
TX2
(a)
Fig. 3.27
Fig. 3.28
on on -
-
of
negation.
1,
high
1.
T ( x ) = x.
(3.12)
120
[III.2
OPERATORS OF PHYSICAL ELEMENTS
1, disjunction
K , (XI,
x2)
= x1
v x2
9
K(x)=x.
= F, "4
Fig. 3.29
b.
3.28
by
A
3.29.
by 111.1. of P ( x , , x2) = 21 v
by
22
3
0 1 0
1 1 1 0
111.21
VALVE AND TRANSISTORISED mwmrs
121
3.30 on As by 3.31, 3.32).
by by
-p(b)
Fig. 3.30
t Fig. 3.31
i-
jz=x1vx2'x3
(b)
(a)
Fig. 3.32
122
[III.2
OPERATORS OF PHYSICAL ELEMENTS
Dd(x1,x2,..., x,) = XIv ~2 v ... v x,, D, (x1,x2, ...,x,) = xlxZ ... x,. by
3.28
3.30,
on on on It by by Fc by by
by
Fd by F,
by Fd
by
3.33 by F, z=
xi=O. R,
z=o.
U,,
111.21
123
VALVE A N D TRANSISTORISED ELEMENTS
on
3 2
1
P E+
Fig. 3 . 3 3
z.
z = T (XI) T(
T
~ 2 ) ( ~ 3 )= 21Z223
.
(3.16)
by T, (xl, x2, ..., x,). 3.34
A
if is
by
124
[III.2
OPERATORS OF PHYSICAL ELEMENTS
( V xi), n
E-
by
- ibR,n
2 U,.
A, (U,)
3.35
n
by
9‘
ORC
Q. Fig. 3.34
?
Fig. 3.35
by
of of
z = T ( x , ) V T ( X , ) V ... V T(x,,)=Z;-,Vi!zV
... Vi!”.
(3.17)
A i (i= 1, 2, ..., n) x,=O.
P,,(xl, x2,..., x,,). As by by by
F:
F:
rn
111.21
125
VALVE A N D TRANSISTORISED ELEMENTS
z’ = x;x;
...x; ... xj, ... x,,,
z” = xnxn 1 2 ... Xk
by
by Fk
z’
... xi”....:x
x;
xj”
F:.
Fi u‘
u”.
3.36b. I
,I
z = z z .
Fi
Fd,
v x; v ... v x; v ... v x; v ... v xi, zn = x; v x; v ... v xk“v ... v xj”v ... v x; z’ = x;
z = Z‘V
F,
Zl’.
Fd z’ = x’ x’x’ 1 2 39 z” = x; x;
v v x;
FZ
126
pII.3
OPERATORS OF PHYSICAL ELEMENTS
F,
Fd
versa
vice u’
Fig. 3.37b.
u”,
(jj
(jji
OR1 OR2
111.3. Flip-flops 111.3.1. THE PHYSICAL PRINCIPLES OF FLIP-FLOP OPERATION’
AJEip-flop by
A See L. A. Meyerovich, L. G . Zelichenko, Pulse Technique, Sovetskoe Radio, Moscow A. M. Bonch-Bruyevich, The Use of Electronic Valves in Experimental Physics, Gostekhizdat, Moscow 1956.
111.31
127
FLIP-FLOPS
on
3.38,
Fig. 3.38
Fig. 3.39
E,
R,.
R U
i
3.39
R, i
by U = E - iR, i = f (U)
(3.18)
128
pII.3
OPERATORS OF PHYSICAL ELEMENTS
by
do 1/R
PI, P,
P,
P,. by U,, by i,.
by
R
i, by by Pik)
of by
on
E,,
b on on El.
on
by by
3.40;
U' =f (El),
111.31
129
FZIP-FLOPS
3.41.
by
T', on
on
T",
by
J
Fig. 3.40
Fig. 3.41
Fig. 3.42
no
on on
by
on
on T'
A
U'
T"
130
pII.3
OPERATORS OF PHYSICAL ELEMENTS
T'
T" (A)
3.40), e. :
U'
=f
U'
= E'
(3.19)
(E'),
-e.
(3.20)
(3.20)
(3.19)
3.41.
e
45"
on
Pik) e@)
(U,e ) a
e@)
U
3.43).
11,111). I I1 U
m\ II-
I-
__
Fig. 3.43
by for
III.31
131
FLIP-FLDPS
I11
by by 3.42, on staticflip-flop,
(a)
Fig. 3.44
by
3.44b
3.38.
R
R", by
3.45
I11
E,
a=
(l/R;)
132
[III.3
OPERATORS OF PHYSICAL ELEMENTS
PI,, PI,I).
PII
Pill on
E,
by
\m Fig. 3.45
A
B
(T'
A B (T'
A
T"
T" on
3.44
by
50 -
-
by
III.31
133
FLIP-FLOPS
3.44,
by
3.46). 3.44
3.45)
of
t
U'
U"
Fig. 3.46
A
A
T",
T'
B
A
3.45
PI T' T" 3.46
of on
A
a
134
rIII.3
OPERATORS OF PHYSICAL ELEMENTS
by
R,
R,
C, by by
of
3.48
I
PE,
1.-
i
ORC
p. Fig. 3.48
I ,
111.31
135
FLIP-FLOPS
by on
Re Re
111.3.2. LOGICAL OPERATORS OF FLIP-FLOP CIRCUITS]
by
Fig. 3.49
qA,
q B= FA.
v‘
v“ cpA=l,
qA=0.
d’
0
1.
v’
pA(t)= 1
qA(t)=O
qA(t+
no
qA(t+
=O
on
See M. L. Tsetlin, Non-primitive circuits, Collection “Problemy kibernetiki” no. 1, Fizmatgiz, Moscow 1958, pp. 2 3 4 5 .
136
[III.3
OPERATORS OF PHYSICAL ELEMENTS
d(t)= 1
d’(t)= 1
has : TABLE 111.2 UU ( t ) -
0 0 1 1 0 0 1 1
0 0 0 0 1 1 1 1
0 1 0 1 0
1 0 1
~
: VA(z
+ l) = ll’ ( t ) q A ( t ) v v ” ( t ) ( P A ( t ) ,
(3.21)
z ( t ) = v”(t)(PA(t).
(3.22) (3.22) 3.50.
(3.21)
(3.22)
qB(t+
zB(t). ‘PB(t
+
= v”(t)’PB(t)
v
v’(t)(PB (t),
(3.23) (3.24)
111.31
137
FLIP-FLOPS
v‘(t)=v“(t) = v ( t )
: qA(t+
I)=u(t)@-,(t)V
~(t>(~A(t)=v(t)0(~A(t),
(t> = ( t ) q A ( t ) .
FL-fl
Fl -
+%
+PA
Fig. 3.51
3.51
Fig. 3.52
3.52 by
If
T, T’
zA(t), qA(t)=O
x”(t)= 1.
qA(t+
+
(PA(t
ZA(t)
v (PA(t)-’f’(t),
= @A(t)x”(t)
= @A(t)x“(t).
cpA=O.
x’(t)
x”(t) = 5’(t),
x‘(t) = v”(t),
+ l) = @A(t)5’(t) v P A ( t ) 0 ” ( t ) 7
qA(t
ZA(t) = @A(t)V’(t).
v’
U”
by
a
dynamic flip-flop.
a z(t+
1, z(t+2)= 1, ...,
138
[111.4
OPERATORS OF PHYSICAL ELEMENTS
on u‘= 1
u”(t+s)=O.
of
t, t + s
3.54. z ( t ) = v’(t)V z ( t - 1)6”(t - l ) u a ( t -
ZA
ZE
3.53
v,(t)
u,(t)
= Z(t)
= u’(t)
v z(t -
- 1).
111.4. Ferromagnetic elements 1 111.4.1.
~~
on no. 2, 1959, pp. 1958.
by
111.41
139
FERROMAGNETIC ELEMENTS
on by by
Fig. 3.55
I
Fig. 3.56
B B=B(H),
H.
on on H, B =B ( H )
by
140
[II1.4
OPERATORS OF PHYSICAL ELEMENTS
+B,
-Br.
+B, 1.
by
by
transformer
A
I) by
1,
11) 0.
1, +B,.
+B,.
+B,, (3.30) -B,,
-Br. pulse
d,B
= B,
+ B,.
(3.31)
A 0
by
u = - wsW
(3.32) S
121.41
141
FERROMAGNETIC ELEMENTS
:
AB u = - ws-10-8,
(3.33)
z
on
t
(Br,Bm),
(3.35) on U,,/U, on
a = Br/B,,,,
coeficient of rectangularity 0.9-0.98.
a
no on
by by write-on read-out. 0
1,
0
142
[IIIA
OPERATORS OF PHYSICAL ELEMENTS
Fig. 3.57
m
by A, 1
0
a 1
0
by no A
, T,
(C).
1, A
0
A,
no 1
0 on
111.41
143
FERROMAGNETIC ELEMEN
(B). A,,
no A,,
on
on
For
by
by by
In is
bound
on
0
In by
by
Devices which ensure that pulses are of the specified duration. Strobing consists of “excising” narrow sections from the output pulses by means of synchronising pulses at strictly determined moments. This produces an accurate synchronisation of the pulses which are fed to the cores in subsequent stages.
144
G
[III.4
OPERATORS OF PHYSICAL ELEMENTS
H
single-crystal circuits
n t
Fig. 3.59
(D) pulses
td
A,
3.60
0 by
A,. A
A,
,
A is
by
zr,
by A,
A,
z,
by
by Two-cycle circuits,
odd
111.41
145
FERROMAGNETIC ELEMENTS
A
odd
on. Three-cycle transformer circuits
T,
146
rIII.4
OPERATORS OF PHYSICAL ELEMENTS
supby
good ferrodiode choke elements1 by 3.63
Win
C (0,
W,,,.
D,) by
E
B. (11)
do holding duty
See L. M. Shekhtman, An algebraicmethod of synthesisinga ferrodiode choke circuit, in collection “Voprosy radioelektroniki” series 12, no. 18, 1959.
111.41
147
FERROMAGNETIC ELEMENIS
A on
3.64),
S
3.63). AQPA
( Wout)
on
Z
Fig. 3.64
iOut = E/Z
no
ACDEF on
S
EGQA A
Z no t (t+
148
[III.4
OPERATORS OF PHYSICAL ELEMENTS
(t) ( t + 1).
3.65 by C,
A
3.64).
C2 by
A on C, C,
A on
no
C,
by
C,
A
C,
by
on
t, (t+l),
vice
C,
versa.
( W,.), a
( Wi). A
III.41
1 49
FERROMAGNETIC ELEMENTS
A
A
ih
ih Hrh
Hrh
- Hih < H O by
Hih Ho ih
A on
4 a 111.4.2. OPERATORS OF FERROMAGNETIC ELEMENTS
3.55,
I1
(b)
(C)
Fig. 3.66
b
by by 1
A
by
by 1 1
0 0,
0 0
1.
-
~-
-
~-
0 0 0 0
0 1 0 1 0
0 0 1 1 0 0 1 1
1 1 1 1
0 0 0 0 0
0 0 1
0 1 0
1
0 1
1
1
0
0
1
'
_ _ x z.
q,
y,
111.3. 111.3 :
by
(4 = 4 (t>Y (4 4(t+
9
i
(3.36)
[4(t)Vx(t)]Y(t).)
1
m
by on
be
on
3.67 t
111.41
151
FERROMAGNETIC ELEMENTS
1
q (t)
y
u ( t ) =0 z(t) = 0 ,
q (t
+ 1) = j j ( t ) .
(t+ u= 1 z(t
q(t
+ 1) = q ( t + 1 ) U ( t + 1) = j j ( t ) , +
=0.
*-z=g
Fig. 3.67
Fig. 3.68
H y)
H
y);
(3.38)
G(1,Y) =
y ( t )=0 z(t
+ 1) = x ( t ) ,
x ( t ) = 1, G(x,O) = x,
by 3.68). u ( t )=0),
q(t
+ 1) = XI ( t ) v x2(t) v v * a .
q (t)=O, Xk(t)
on z(t
+ 1) = 4 ( t + 1) =
(t)
v x2 ( t ) v ... v Xk(t)
m
152
DII.4
OPERATORS OF PHYSICAL ELEMENTS
I
m m =1= 1, z(t
+ 1) = x(t)Jqt). by
G (w) H (x, v). by by z = x l x 2 . . . x k = % l v d f 2 v
...v%k.
%F- 1I k y - y ~
xp-
U
1
-
... . .....
X
(‘
Fig. 3.69
x l , x 2 , ..., x k .
+ 1) = d f j ( t ) , xi”(f+ 2 ) = v X S ( t + 1) = v Z j ( t ) ,
XJ(t
z(t
+
k
k
j= 1
j= 1
k
=
v j= 1
%j((t) =
xl(t)xZ(t)
.a.
Xk(t).
III.41
153
FERROMAGNETIC ELEMENTS
LT=
3.
by
by 3.70).
Fig. 3.70
’k
Fig. 3.71
on
3.71):
3.72
.’
( k = 2 ) by
(3.40);
z(t
+ 2) = x i (t)xz ( t ) = G [G(xi, O), H(1, xz)]
(3.41)
Here, and in certain cases later on, we shall omit the time parameter ( t ) in the operator notation to make the formulae more easily comprehensible.
154
[II1.4
OPERATORS OF PHYSICAL ELEMENTS
1
by
on
Fig. 3.73 3.72
a=2.
by
~ = 2 ,
on
3.73; b
is A on
q 1,
0.
go by x
by z
y
i,
~-
by
p. 1 4 6 .
111.41
155
FERROMAGNETIC ELEMENTS
TABLE111.4 4 (t>
1 0
1 1 1 0
1
111.4 4(t
+
= 4(02(0
v Y (9.
(3.42)
z (t)=O,
z(t+l)=q(t+l),)
4(t
+ 2) = 4 ( t ) .
(3.43)
) on
q (t)= 1,
t y (t)=O.
by (3.42)
(3.43), z(t
+ 1) = x ( t ) .
R(x,O) = 2 . q ( t )= 1 z(t
+ 1) = y ( t )
(3.44)
x ( t )= O =y.
by by
by
156
[III. 4
OPERATORS OF PHYSICAL ELEMENTS
by
on
A
3.74,
Fig. 3.14
q ( t ) = 1, z(t
+
= x(t)
v y(t)= by
v R(1,y).
u=2:
CHAPTER IV
A N A L Y S I S O F AUTOMATA
IV.l. General remarks 11,
on
on
{%R,},
{‘$Xi} 2.16b
zl,
This 11,
us
~
_
_
Here, and very often later, the term ‘net’ will refer to a logical net. 157
158
[IV. 1
ANALYSIS OF AUTOMATA
K.
minimising
1.
2.
by
3. 5.
In 11
%Ii
by
by EXAMPLE.
Wd).
of
3; <8.
4).
IV.11
159
GENERAL REMARKS
by
(or (attainability,disting~is~~ability, periodicity)
on by
by
2 4 V.
body
2 by by
by
1. IV.l
IV.2
11.4
IV.3, z(t) q ( t + 1)
TABLEIV.1
TABLEIV.2
TABLEIV.3 __
\
1
1
\
XI
0 0 1 1
0 1 0 1
0 1 1 0
-
0 0 1 1
0 1 0 1
I
0 0 0 1
I/
0
1 1 1
0 0 1
0
o/o
1
1
1
1/0 1/0 0/1
0
~
l/O 0/1 0/1 1/1
160
[IV.2
ANALYSIS OF AUTOMATA
by
2.
transition 4.1
diagram.
If qi
on
qj,
Fig. 4.1
by
qi
qj,
IV.2. Attainability attainability
distinguishability,
8, z(t) = @[ . ( 4 7 4
4 ( t +, “= q
4
by
9
.
K
q( l),
by K
-
S
a
xl,x2 ..., xs)
us 11/Cx1,411=42, +[%42l
= 4 3 ~ . . . ~ ~ ~ x s ~ ~ s l == r4es + l
(4.1)
N.21
161
ATTAINABIL.lTY
xl, x2, ..., x, r, xl, xz,..., x,. This r
p
by
r
p
S
on by Yu. DEFINITION. q p irreversible.
reversible q
q,
p;
Fig. 4.2
by
EXAMPLE.
4.2.
q2 00
q2
qo,
q2 forgotten, C
8.
1 Yu. Ya. Bazilevskii, Questions in the theory of logical time functions. In collection “Voprosy teorii matematicheskikh mashin”, Fizmatgiz, Moscow 1958. 2 Anticipating matters, we may note that when such an operator is realised in a real physical device, the return to an irreversible state is sometimes produced by appropriate external intervention.
162
tIV.2
ANALYSIS OF AUTOMATA
A
closed no 8
irreducible
reducible).
A
0. qo
qo;
0 q,,
4.2) q1 4.1).
4.1 An
by
e
Q(0)) q
d
by
no 1. The degree of attainability of any bounded-determinate operator is strictly less than its weight, e(9) 8 by
q l , q2, ..., qs+l
Proof. (4. x,, x2, ..., xs
p
0,
r.
qi= q j , by x i + l ,x i + 2 ,..., xi, qi S 1
+
IV.21
163
ATTAINABILITY
by by
K
8 no
V,
! I ?
by
V
01,029
.... 0".
(4.2)
.......................................... ......................................
.............................. ............................ ........................... .........................
Fig. 4.3
23 (4.2).
23 EXAMPLE.
K
4.3 K
on
e(0)
8
by
00.. .0100...01... 1 00.. .O ( K
164
[IV.3
ANALYSIS OF AUTOMATA
V, from below. 8
z
y z.
p y
p - 1;
by p.
no up
no
p
... 100 ...0100 ...0100 ... 0 ... ... 100 ... 0100 ... 0100 ...0 ...
1 zeros
V"V v \ N
1 zeros
I zeros
1
I
p.
I,
21.
no
IV.3. Distinguishability 8,
by -
-
f12
;
is
are these operators distinguishable or not?
IV.31
165
DISlXiWXJISHADILTTY
by gy,
8,.
8,
on by do
1. A
by v 2. A ;
by
8, x = x ( l ) x ( 2 ) . .. y’(1) y’(2)... y‘(S). distinguishes 8, tinguishable by x . 8,
8,
8, y = y ( l ) y ( 2 ) ... y ( S ) x 8, 8, dis-
8,
2.
8,
DEFINITION 1. 8, “ k 8 2 ) k-indistinguishable a fortiori 8,w,8, y>k.
DEFINITION 2. 81+k8,)
k-distinguishable by k; 8, Wke2). 81 “ k 8 2 )
8, 81”k82
8,
8, strictly k-distinguishable 81-k-,8,. k,
E. F, Moore, Gedanken experiments with sequential machines. Automata Studies, ed. C. E. Shannon and J. McCarthy, Princeton 1956. 2 This statement is true in general for determinate operators.
166
[IV.3
ANALYSIS OF AUTOMATA
8,
8,a 02,
02,
strictly k-distinguishes qi
qj
0;
-
-
0. A, A.
k
A
3. degree of distinguishability
0
xi, x2,..., xk x2,x 3 , ..., xk qi qj
qj; qJ. xi.
qi q: by
x2,xg,..., x k
on qi
x,. c,, ... ti,
q:
qJ,
by
qj
qi y < k - 1, qi qj.
qJ
xl, c,,, ..., ti,
by y+ 1 < k
y
0 no
( k+
0, C. q,-kqj q1" k q 2 qn
q j WIrq,), q2- k q 3
qjWkqj)
p,
q, " k q 3 ) .
{By)}: B y , By', ...)23%)
(4.3)
IV.31
167
DISTINGUISHABILITY
p2
(4.3)
on
THEOREM 2.
of
of
is is
0”).
of
p
(4.3),
{By’},
p+ I,
-
-
‘23?+’).
By’
B?’’) ‘23?+’).
or b)
23:
y
<
By), y=p
(p+
+
{BY)},
...,( !By‘”} by
BY)
(0 p
by
By),
qj,
qi
1
+ qi
x,
qi
(p
{By’}.
That is, any two distinguishable states fall into different classes.
168
[IV.3
ANALYSIS OF AUTOMATA
A
3 l . In order that two partial bounded-determinate operators O1 and 0, of weight < K should coincide, it is necessary and suficient that they should not be distinguishable by any simple experiment of length 6 1. by
8,
O1
8,
0
8,
O,,
by 1.
0 8,
0,
0,
by
0,
:
1 ... 1011 ... 10 ... 11 101 _ _ 101 . _..
1
___
a
by
of
by
K.
1.
by
V,
EXAMPLE. 3 3 q’
by
q” 6 q’ q”.
- 1 = 15.
_.__ ~~
See E. F. Moore, Gedanken experiments with sequential machines. In collection “Automata Studies”, ed. C. E. Shannon and J. McCarthy, Princeton 1956; A. Trakhtenbrot, Operators which can be realised in logical nets, DAN SSSR (1957) 112, no. 6.
IV.41
169
PERIODICITY
IV.4. Periodicity periodic
by
A w2
{ U (n)} period),
periodic U (n)= U (n + w )
A
A p transient process
U ( p + 1) U (p+
n. periodic p>O .. U ( p + l ) length
p,
the
pre-period). w
p
LO
U(1) U ( 2 )... q p ) [ u ( P + 1) ..-U ( p + 4 1
p
+ LO
p <s < t,
reduced length
s= t
9
{ U (n)}.
w),
U ( s ) U ( s + 1) ... u(s + m)...) U ( t ) u ( t + 1) ... U (t + m)...
+
U (s+ m) = U ( t m)
m =0, 1, 2, ... . V (s)
{ V (n)},
V (t) p
<
t-s
s+ ( t -s), t.
170
[IV.4
ANALYSIS OF AUTOMATA
4. A bounded-determinate operator 0 of weight K converts any periodic sequence { f ( n ) } , having a period w and a length p of the pre-period, to a periodic sequence { g ( n ) } with a period Q < Kw and with a reduced length
{q(n)}
{f (n)}.
+
...<"f(97cl(W(f(i
+
(g(n)}.
do KO+ 1 < f ( p + 1),q(p+
I))...
K 0
K),
(w+ ( w + 1)
t=s by ( f ( s ) ,q (s))
( f (t), q(t)));
+ f (t +
f (t)
+ m), ... f ( t
+ m)
q(s)= y ( t ) , q(s) q(s
+ l ) . . . q ( s+ m ) ,
4(t)
t - 1 < /L+ Kw.
p < s < t < p + KO+ 1, Q=t-s
by
EXAMPLE. An
IV.4, 111 ...
q(l)=O,
01 [110]
by
An 101010 ...
110 IV.5
011 1
0)
IV.41
171
PERIODICITY
TABLEIV.4
0 1
112 117
010 011
111 114
013 012
117 116
011 014
012 017
114 115
-
TABLEIV.5 7
0
010
1
011
112 117
Oil
Oil
113 112
014
-
014 ~~~
Input : State: output:
1 0 1 0 1 0 1 0 1 o 0 1 2 2 3 4 1 7 4 4 0 1 1 1 0 0 1 0 0 o
-
. . .
. . . . . .
period
8
K. 0, 0, 0, ...,
p =0, o = 1,
0
0...P(P++)l,
P(l)...P(P)[S(P+ p
+C2 > K
(44,
p
8
+D <0+K - 1=K. by q l , qz, ..., qp, q p + l ,..., qp+*: CJ(a, qi) = P ( i )
9
y ( a , q i ) = 4i+l
(4-5)
172
[IV.4
ANALYSIS OF AUTOMATA
THEOREM 5 1. A constant operator generates a periodic sequence the reduced length of which exactly equals its weight. Any periodic sequence having a reduced length p is generated by some constant operator of weight K = p . 4
NOTE. on
p
on
0.
k
k
4 g (B)= 8 [ f ( n ) ]
k,
. . . , f ( n), g(n)=O.
g(n)= 1 ,
(00 ... 0 , l )
--
1) ones
(w-
... 10
w,
(0-
w zeros
---
--A
(0-
1)
by
k
1) zeros
00 ... 00 ... 00 ... 011
o
D=wk. (q,,, q l , ..., 4,-
by
k
k,
w w
4,
by
a I
by
to
IV.41
173
PERIODICITY
k
by
k?
by
101001000100001..., 0, g (n)
by
K = 2.
4
4
K,
K.
{ g‘(n)} (000 ...
(w -
(4.8)
{ g’(n)>
8
(g’(n)). w + 1,
4
K=2. 4.
THEOREM 4’. Given that a bounded-determinate operator 0 with a weight K acts on an input sequence with a period w, and that coo, ol, ..., w, are all divisors of w (including 1 and w itsew). In such a case the corresponding output sequence has a period whose length can be only one of’the numbers
wo,2wo, . . . , K w0 , w1,2w,,...,Kw,,
. . . . . . . . . . . .
w,, 2w,,
..., Kw, .
(4.9)
174
[IVS
ANALYSIS OF AUTOMATA
of
IV.5.
1
pp.
A
1V.5.1.
{%Jl,}
4-5
I1
(mi}.
61. In order that a circuit comprising a system should be a logical net, it is necessary and suficient f o r two conditions to be satisjied: no terminal should be f e d by two or more output channels, and the circuit should contain no improper loops.
by
Prooj:
2, 3
4
p. 97),
{mZ,}. by
3
do
4,
3
no
4 zj
bound
xi),
by ZJ
xi.
M
{mi} :
2
N, no
M,
N, 3
N,
A4 ; N2 ;
M,
N,
M, do no by
a,
IV.51
175
ANALYSIS OF A N AUTOMATON CIRCUIT
bound
a,
M N2 no
by
M.
M, 2.15 4 2.17
11,
m,
2.15
2.15
by
2.18 2.17. treelike
{mi>,
CONSEQUENCE. by
{%ti}.
on no
tl,TZ,..., ti,..., t, j 3 A
ti
i
The set of terminals in a logical net is partly ordered by the relation of subordination: an extended description of the numeration of a partly-ordered set is given below.
176
[IV.5
ANALYSIS OF AUTOMATA
0 0 bound zl, z2, ...
...,
z,>z,>z,>
n n+ 1 n,
n.
per contra, al
a1
no
p, y , ..., 6. az)
by
a,
p, y , ..., 6. al,az, a3, ...
no
a1
2.
1,
z2
x,z4
2.17,
EXAMPLE. z1
z3
2.
2.
IV.5.2. TO COMPILE THE CANONICAL EQUATIONS
by N E(N), N,
by on p. 91
2.17.
TV.51
ANALYSIS OF AN AUTOMATON CIRCUIT
1I1
E(N) 24 ( t ) 21
= ( t )9
(4 = z4 (4 0 x ( t ) 0 4 ( t )
9
(4.10)
r
by
As :
(4.11)
(4.12)
for
178
[IV.5
001
110
111 .
Values 1101
01 10
0111
Values I
0
'
'
"
I/I
000 001
010 111
011 010
j
the states 001 100
I
-
~~
output variables
001 100
I
1
111 110
I
1100 01 10
0001 101 1
010 111
100 101 _ -
zl, z 2 , z 3 , z4
IV.6.
6,
II), by
2.17,
z2
(4.1 Z2(t)
=r(t)@p(t).
1V.5 z4,
1V.4 IV.5.3. TO MINIMIZE THE ALPHABET OF STATES
A
(5 QiQz
..-Qs
(3
(4.13)
complete
IV.51
179
ANALYSIS OF AN AUTOMATON CIRCUIT
1,2, ...,
{BY)} 3
by qj,
qi
By)
+
(p
+
x
qj
q;
{By)}). by p = 1.
p
p
+ 1.
{By'} with
qi
andysis
to
{By)}
B$p)
qj
23:"
:
by
qi
qj
{By)}. {B:p)},
{BY)}
p
EXAMPLE. 2.17,
z2
IV.5): P = 1:
{0,3,4,7},{1,2,5,6}, P= {0,7},{3,4>,{1,6>,{2,5), P = 3: (71, (3941, {1,6), (21, (51 {3,4}
{1,6}
6, by
3.
4 6
*
6, by 3
4 1
IV.7).
180
[IVS
ANALYSIS OF AUTOMATA
TABLE IV.7
24
TV.6 8 z4:
1: {0,2,4,6}, {1,3,5,7}, {2,4}, 71, P = (21, (41, (71, (31,
P
=
P = 2:
7
-
by
CHAPTER V
METHODS O F SPECIFYING OPERATORS
V.1. General remarks
11.
by
on
by 181
182
[V. 1
METHODS OF SPECIFYING OPERATORS
on by
000,001, ..., 110, 1 1 1
1.
0, 1, ...,
x (0 6 x < 2. z(t)= 1
x
+1
7).
0, 1 ; x(t)= 1
x(l), x(2),
..., x ( t ) 3. 0, 1, ...,
000,001, ..., 110, 1 1 1 0, 1 . u,
z(t)=l t,
1, z
t,
c 1)
1;
z
A(o<1
1.
1
4. z(t)= 1 ..., x ( t ) 5.
0, 1 ; x(t)= 1
a, b, c ; abcbcba. abacaba
x(l), x ( 2 ) ,
0, 1.
v.11
183
GENERAL REMARKS
0010101
001 1011.
4
el, 8,, 03,8, e3.
V.l TABLEV.l
__. XlXZX3
01
00
I
10
I
.~
000 001 010 01 1 100 101 110 111
011 1 0/11 011 1 011 1 0/11 0/11 011 1 0/11
OlOO O/OO o/OO
Oleo 0/10 0/10 ojro 0/10
1/00 1/00 1/10 1/11 1/10 1/10 1/10 1/11
l1 1/11 1/11 1/11 1/11 1/11 1/11 1/11 1/11
any (the recognition of possible realisation),
on no
184
w.1
METHODS OF SPECIFYING OPERATORS
on by
by by by 3 000,001,010,011,100,101,110,111,011,111; 000,101,001,111,010,111
o,o, 0, 0,0,1,1,1,1,1; 0,0,1,0,1,1 In
5.2, I , 1, 1,0, 1 1, 1, 1
02),
5.3
1 , 1,0, 1, 1
Fig. 5.2
Fig. 5.3
0,).
v.11
185
GENERAL REMARKS
by
by O1
As
by inclusion tablesl,
2.
by
by
3
by
by by See M. A. Gavrilov, Teoriya releyno-kontaktnykh skhem (Theory of relay-contact circuits), ANN SSSR, Moscow-Leningrad 1950. M. L. Tsetlin, 0 neprimitivnykh skhemakh (Non-primitive circuits), in collection “Problemy kibernetiki” no. 2, 1959.
186
[V.2
METHODS OF SPECIFYING OPERATORS
A 4,
5. V.2. Fragments of trees 8 ,u
by
by
0
8. complete tree of height p. by :
by 1. 2.
on
no
of by
v.21
187
FRAGMENTS OF
5.4
5.5
Fig. 5.5
Fig. 5.4
by
11.
5.4, by
ct
10.
/3
5.5,
6,
y 1.
of
2
5.4
2
5.5, 1.
2
IV,
p
2 5.6 10 ...01
p. (p+ 1
a,fl
00 ...01 (p 10 ... 00 (p+
5.7
188
V’
METHODS OF SPECIFYING OPERATORS
V,
V”
V‘
Y”
complete trees basis
Fig. 5.6
O
\d/o V
v V”
no by the basis 23 of a complete tree, IV, 2 degree of attainability
by
V
by e ( V ) ) ,
v.21
189
FRAGhENTS OF TREES
(or,
2
3
IV
by 5.8),
Y
EXAMPLE.
3,
a, j?, y ; =
e(V)= 1 .
Y 0
o(
Fig. 5.8
by
p,
0
p :
A.
bound
B.
by h;
Y V , by h+ 1, h+2, h + 3 ,
on
... 1,0,1,0,0,1,0,0,0,1,0,0,0,0,1)
...
V,
h+ 1, ..., h + k - I ,
190
METHODS OF SPECIFYING OPERATORS
h + k , h+ k+ 1, ~..,
on
V,
k
k
V
bound
K.
V
:
V
K K-continuation)?
V
V weight of a$nite tree,
V for V on K
h,
h
23 K,
V
K.
K 8 by
K,
K,
: 1.
23.
V,
by 8.
23'
2.
As
23'
by
23, 23'
23,
do
y
p
23
y.
v.21
191
FRAGMENTS OF TREES
REMARK. For
V
23,
23
23, 2 0,
3.
qi
23’
qi
by
EXAMPLE.
5.9
K=5;
23
q l , q2, q3,q4, q5.
23’ on
2.
K,
V
by on p. 190, by
41
Fig. 5.9
0
0142
1
0193
0/42 1/q4
0145 1/94
0191 0142
0191
1/42
192
w.2
METHODS OF SPECIFYING OPERATORS
THEOREM 1. For a tree V of height h to have a unique continuation which preserves its weight, it is suflcient that it satisfies the condition h 3 e(V)
+
+1 B
6
Proof.
>
23'
y
V),
Fig. 5.10
%,
y
y.
Corollury K
3,
IV).
h
V
h 3 2K - 1 . 8 by
bound
h
+e(O) +
h3
V
8
V,
e ( V )G e ( 8 ) ; V
bound
0.
by
8. 1,
v.21
193
FRAGMENTS OF TREES
8, 1. An
<s, e(0) 6 r :
V
s+ r
2. on pp. 190, 191.
+
;V
< K,
V
K
h = 2 K - 1.
h.
by
bound
<s.
123,
s f 1;
qo. (m
+ 1)’
<2 ;
2,
q,, qo.
s
m, 3
m is the number of input letters.
by
194
Iv.2
METHODS OF SPECIFYING OPERATORS
3
A no
v
m+m2+
of
... +ms= {(mS+l-
of ms OGK,
1
v
m-1
qo
Fig. 5.11
s=K- 1 v < KmK-'
+mK,
(5.2)
of 2 K - 1,
(5.3) EXAMPLE.
5.1
by of 1.
no
0
4;
2 2 x 4- 1 = 7,
v.21
195
FRAGMENTS OF TREES
TABLE V.3
I
92 X
q3
0 1
A
b Fig. 5.12
28- 1
+
h=
26- 1= 63 18
@(0), @ (0) 1 = 1 + 3 1= 5, 5.12 of us
+
by
@( V )+ by K.
+
V )+@( V )+ 1 6( V ) = h + V
V. redundancy of the tree, 1, V of
THEOREM 2. If a tree V with a weight possesses a redundancy 6 , then in the class of all bounded-determinate operators of weight < + 6 it admits a single continuation (of which the weight is in fact K ) . V,,
Proof.
V
V bound
2
196
[V.2
METHODS OF SPECIFYING OPERATORS
V,
Bl
K,
B B1
23
V
by
V, 0,
v v < e ( V ) 2,
+
23 ;
B
by
Fig. 5.13
x>e( V ) + 2,
3h-
V)
23.
V) 0,
V 0,
x, e ( V )+ 1
j?,
fi
5.13).
V
+
V,
+
I>h-e(V) -1
V,
+6(V).
6(V).
V,
(23;)
p=
2, ..., K
(Bqd'"') V),
V, by IV, 3,
...
{Bqd(")+'),(B:d(Y)+2), K + 1, K+2,
less
{By("m)}, Vm
K + 6.
K+b.
K,
8, V
KGK).
K
k2R-1
K
v.21
197
FRAGMENTS OF TREES
K + 6 2 I?,
6, h,
h V
K). EXAMPLE. 0.
K< 3
1,
0 K
5.14.
V
,
- 0
ly
a
Fig. 5.14
/?.
a
1 V )= h -
@( V ) = 1. V ) -@( V ) - 1 =
TABLE V.4
1/92
1/90
198
P.3
METHODS OF SPECIFYING OPERATORS
V.3. Transition matrices
transition matrices. transition matrices of states matrices
transition
reactions
by synthesis of an automaton circuit, an automaton circuit.
analysis of
V.5. TABLE
v.5
X,
Xij X, Y ( X 2 , qi) = q j
9
qi 0
EXAMPLE.
qj.
11.4)
V.6, q (t) ~
by q
q ( t + 1) by q’.
~-
no. 2, 1959.
,
v.31
0 1
199
TRANSITION MATRICES
0
1
00,01,10
11 01,10,11
00
IV
:
i 6 k)
Xir, Xiz, ..., Xi,
no
X i l , ..., xi, k
xij= x. j=1
matrix
of reactions)
IlQs~ll QsL
(S G m,L G k), q,
@ (xs, I 4) = Z L . QSL
of
Q
X.
Xij QSL X i j ,Q S L )
EXAMPLE.
0 by
V.7. We shall write these pairs in the form of fractions.
200
w.3
METHODS OF SPECIFYING OPERATORS
QsL
1
00 no
q1 qo.
vice versa,
no
XqlqZ ,,.qk,,;,;.,.
,;.
(a,, az, ..., a,)
. . . . . . . . . . . . yk
( a l ,a Z ,
. * . Y ~ , ,41,7 2 , * * . )
4
qk) =4k
*
X,,,,,,,,,, ,;..,,; M,,,,,,, _ _ _ ,,,k,q;,q;, ... ,,;(xl, ... , xm), no
1
by
M,,,,,
EXAMPLE.
Moo = ZIT2 V XlxZV xITz = MI, = XlXZ
7
= ZIZz ,
Mll=x1Vx2.
V Xz,
v.31
201
TRANSITION MATRICH
X,,,, ...,qk,,,;, ...qk, (xl, ...,x,) X q 1 ,...,qk,qI’, ...,qk*(x17
...
m) =
7
= y?”(xl)
..
-7
x,, ql, ...)q k ) . yp’(x1, ...)x,, q l ) ..., q k ) * ... ....yp’(x1) ...,xmy ...)q k ) . (5* 5)
EXAMPLE.
x,,,,,= [zlv x2 v 01 v X,2,j.[Xl v v 01 v R,x,] 22
=21
v x2.
by
TABLE V.8
00
00 01
fl
v XZ
P1 xz 0 0
10
11
x1
01
10
11
0
XlPZ
0 0
v Pz 0
XlfZ
Q,,, ...,am,B1,_.., Bn(ql,... 4,).
Qal
=
,...,a,,p1,...,pn(417
qk)
0 x1
v Pz 0
21x2
21
v x2
Q,,, ...,a , , B 1 , ...,Bn,
=
(a1, ...)a,, 41,...)qk) & ... & @F(a,, ...,urn,4 1, ...)q k ) .
(5.7)
202
[V.3
METHODS OF SPECIFYING OPERATORS
(5.7),
EXAMPLE.
(5.6)
V.9)
TABLE V.9
I
00 01 10
41vq2
1 q1vq2
11
q1vqz
~
I by
1.
Q1,
..., CP,), Yl, ..., Yk).
Qi
Yj
cPi
Urj.
by Yi(i= 1,
by
q,=
!PI, by
q l , qz
V.8.
v.41
203
LANGUAGE OF PROPOSITIONAL LOGIC
%*,
qil q!f
Yl),
3)
by
(5.6),
ql(t+
@ji
zi=1,
@ji
V.4. Some further study of the language of propositional logic
no
on 2" (xl,
..., x), (zl,
( q l , ..., 4,).
..., zn)
2k
204
w.4
METHODS OF SPECIFYING OPERATORS
.... Qn, Y1, .... Yk
. . . . . . . . . . . . . . . . . . . * I on
z i ( t ) ,x j ( t ) , q Y ( t ) predicates (output, input
t
internal
{xj(t)}
{zi(t)}.
predicates propositional logic.
I,
truth operators. 1.
t
x1
z
(4= ( 4 21
2.
x2
( t ) 0 x3
.
x(t);
t> 1, z ( t ) = 1
x3
t
x
t = 1, z(
1.2
1
z
x , ( t ) , x 2 ( t ) ,x 3 ( t ) . 0
z
by
w. z (t)= (t
>
[x(t)0 x (t -
v (t =
w.
(5.11)
v.41
205
THE LANGUAGE OF PROPOSITIONAL LOGIC
internal 41(t
+ 1) = x ( t ) j
(5.12)
41 (1) = u
q1
=u
0, I).
q1
1)
> 1) [ X ( t ) 0 41 (t)]
z ( t ) = (t
v ( t = 1) w . t> 1 q2(t),
t = 1.
by
t> 1
42(t+1)=1, 4 2 (1) = 0.
l), : (t) = 42
(9 (0 0 41 v q 2 ( t ) w 41(t + 1) = x(t),
7
42(t+1)=1, 41 (1) = u
9
4 2 0 ) = 0.
~ ( t )ql(t+ , I), q 2 ( t + x ( t ) , q1( t ) , q 2 ( t ) z ( t ) = w , ql(t+l)=O, q , ( t + I ) = l j .
23 x=q1=q2=0 V. 10 3,
00
w.
01
do
206
w.4
METHODS OF SPECIFYING OPERATORS
3. t.
wi
by
x(t)
t
t = 1,2, 3
by
>3
qz, q3, t
V.11.
t<3 1) q2(t)
q2(t+
qz(t),q 3 ( t ) 0
V.12
by q2(t+l) q2( t )
q z ( t + 1)
q3( t )
+ 1) =
=q2(t)@
TABLEV.11 t
t>3.
v.41
207
THE LANGUAGE OF PROPOSITIONAL LOGIC
'42(1) = 0 , 43(1) = 1 .
Z
( t ) = q 2 ( t )9 3 (4 [x
v
0 41
q2
(t) 4 3
( t )w 1
v
v 42 (t> (0 v 42 (4 j.3
w2
43
(t>w 3 >
41(t+ l ) = x ( t ) , 42 (t
+ 1 ) = 4 2 ( t ) 0 43 ( 9
43(t
+ 1) = & ) q ' 3 ( 4 = w +,
q1
q2
7
q3( 1) = 1.
=0
by z ( t ) = @ [ x l ( t ) , x l ( t - 1 ) , x 1 ( t - 2 ))"., x , ( t > , x , ( l t>v,
by
,...]
t = 1, 2, ..., v
z(t)
t = 1, t = 2 , ..., t = v , t > v , xi t-s, 41 ( t
+ 1) = x ( t ) ,
42(t
+ 1) = 41(%
.
. . . . . .
4 s (t
+ 1) = qs-l(t).
NOTE. 1)
q(t+l)=x(t),
2)
t= 1
2, ..., v, V.9),
4.
by z (t)
= @ [ x (~t ) , x1( t
-
...,z ( t - l),z ( t -
w is an arbitrary constant (0 or 1). [a]denotes the least of the whole numbers m, for which a
< m.
...]
v
208
[V.4
METHODS OF SPECIFYING OPERATORS
q l ( t + l)=x(t), q2(t+ l)=ql(t), p l ( t + l)=z(t),p2(t+ 1) = P I ( $ ... . (5.14)
z ( t ) = @ 1x1 ( t ) ,41 (0,42 (0,...
( 0 9
...
P2
9
+ 1) = x ( t ) , q 2 ( t + 1) = 41 ( t ) , 41 ( t
. . . . . . .
Pi ( t
+ 1) = @1.t
(t>,41 (Q, q.2
( 0 7
.*-,PI (t>,P2(&
P2 ( t + 1) = P l ( 9
7
7
by z(t)=[z(t-l)@x(t)](t> p l ( t + 1) =z(t)
p l ( l ) @ x ( l ) =x(l).
p,(l)
0x(t), P1 ( t + 1) = P1 ( t )0 x ( t ) z ( t ) = P1(t)
p,(l)
no
in t - 1 , t -2 z(tz(t-2),
...
(5.14)
VI.
z(t) quantijiers)
by
v.41
209
LANGUAGE OF PROPOSITIONAL LOGIC
EXAMPLEI. as B,
A
AB, z1
A. 2
z1
AB,
z2,
B.
A A B B)
A
z.
A(t) =O AB
B (t)=O
K ( t ) . B( t ) ; 0, A)
A(t), B (t)
A=O
(B).
W ) ( 4 (Ex) [ A (4* B(4 *
*
*
T C t
z(t)
B= 1,
7<eCt
A(@). B
-
by
V A ( ~ ) * B ( ~ ) * ( E T ) [ A ( T ) - B ( TA ) *( Q ) * B ( Q ) ] . (5.15) r
<<@
EXAMPLE.
x(t),
z ( t )= ( E T ) x ( T ) .
z(t) x(l), x(2), ..., x ( t -
1
(5.16)
7
1 The example is borrowed from the book by M. A. Gavrilov, “The theory of relaycontact circuits”, which contains another method of solving this problem.
210
IV.4
METHODS OF SPECIFYING OPERATORS
is no Shall on elimination of guantijiers.
(zE< r4 by 9
rsr
z ( t + 1)
+
~ ( t 1) =
(5.16') rst
I,
V~ ( t )
( E z ) x ( t )= 1st
(5.16)
T i t
(Ez)x(z) rct
~(t); z(t
+ 1) = z ( t ) v x ( t ) .
t = 1, =0
~ ( 1= ) l<
no
1
z
by (5.16), z(t
+ 1) = z ( t ) v x ( t ) , = 0.
by
to by
z ( 4 = 41 ( t ) [92 (0 41(t+l)=L 42(t+ 1)=41(t).
v 93
7
v.51
211
THE LANGUAGE OF THE LOGIC OF PREDICATES
9(t
z
( 4 = 4 (9
+
= 4( t )
9
v x (t)
q(1) = O .
A z(4=
z(t
+
(4x (4 7
= Z(t).X(t),)
= 1,
(5.17')
i
q( 1) = 1.
V.5. The language of the logic of predicates V.5.1. THE ELIMINATION OF QUANTIFIERS
on
I, by
5), by
by
by
z(t),
'21:
'21. 1. '21 2. A
3
t.
'21,
by
1
2,
by % ( t ) t). %(x,, x2, ..., x, t ) ;
t, x1
x.
x2,
212
IV.5
SPECIFYING OPERATORS
METHODS
xl, xz,..., x,, z(t); xl, xz,..., x,
9l %(xi, xz,..., x , , t )
z(t).
e
z
on
x1
xz
t.
on
t
~ ( t )
by '2l(t). t-verijiability, z
DEFINITION.
'zc
23
t-verifiable
strictly t-verifiable)
9l
23
by z <
z
by
z < t).
NOTE.
LEMMA. In any sub-formula of the type [z]A<&?3 of the t-formula 9l(t), the object variable z is t-verifiable in the range of action of the quantifier [z], i.e. in formula 231. on
% (...m...)
!JJl
by
% (...
by F ) ,
.)
23 (...F . ..)). substitution equivalences,
m-%( ...m...) m B(... m...) m+ %( ...m...) -+
m-23(...T...),
G*B(...F...), m-t - B ( ... T . . . ) , +B(
... F . . . ) .
The notation [T] serves to express that we are concerned with a quantifier of either two types: a quantifier generality (T) or an existential quantifier (ET). For the meaning "<", see p. 61. 2 We shall be using some of these equivalences later, separately from'this lemma.
v.51
213
THE LANGUAGE OF T H E LOGIC OF PREDICATES
!Dl !Dl
23(...T . . . ) . [ z ] A < 7 d23 t
23
z
[z]A
(ET)~<,,~23,
(Ex) !Dl.
ztt,
[ z ] ~ ~23 < ~ ~ ~ ~
%(t),
t;
z1
T~
by
[z2], [z3],
...
[zk]Ak+l
leading
"<"
...
z
[ z ] A l < r < r23; l
NOTE.
[zk]Ar+l<7k
by on
by
( z ) ~ ~ ~ z (t
+ 1) = (4 [.l(Z) r
2,
(E&ie$r
v ( E d x2 (dl bl(9v 6!k
r
1.
(EL?) x2
t<eGt
.
(5.1 8')
R
2
For which we will retain the notation B ( t ) instead of the obvious notation 'u ( t By applying equivalence 4, stated on p. 63, to the right-hand side (5.18).
+ 1).
214
[V.5
METHODS OF SPECIFYING OPERATORS
do
[w],+,~~,
p. 63) (ECO),-,~~~K
COG
by (Em) K V g < t - K : ,
by
m
Q
(Em),
[o])
c
by by
K V K:, (CO)~+,<~K %(t) 'II(t).
As
W(t)
by
W(t),
[o],+,~,
%'(I)
2.
R. 3.
'II(t), 'II'(t)
%(t)
W(t) concomitant by
%(t)
'II 3
K
W. %
W, 'W (r) 'II(t),
a
%"(t),
[o]~-,~~~
4. '&(t)
no
R-2
[CC)]~<~~,.
on :
5. '%(I)
no
[ z ] ~ < ~ < ~[z],-,~-,,
‘rm LANGUAGE OF THE LOGIC OF PREDICASES
v.51
215
bound
(~)~<~+,23by T
( E z ) ~ < ~ < Jby ~ F. (Er) 23
(Er) 23-T
t
(Er) 23&o
t<7<w
(Er)(t
z(t
F.
23;.
[z],~,~~
is by T (Ee)t<estxz(e)
[r]t
F
+
=
(4(x1(4 v 7<e>).x,(t>.
r
%(t), %(t).
by
(E@),
+
(Ee) x ~ ( Q ) V~ z ( t ) ] * ~ l ( f ) .
~ ( t
(5.18)
r<e
7 i t
V.5.2. EXPANSION INTO REGULAR CONCOMITANTS
A
%(t) :
6.
by
%(t),
t. 4, 5
A regular.
on t
t). x1( t )
xz(t),
(z),,~.
by By the lemma proved on pp. 212-213.
6,
216
fV.5
METHODS OF SPECIFYING OPERATORS
+
1x1(7)
= I
v (EL?)
x2
T<e
v
x2
'XI
(4.
of
z(t) . z(t
+ 1) =
[Z(t)
v
v
= (7) I<
1
(Eel
7<0<1
x2
3
1
by
by
'u, @ [Xl
( t ) , ...,x,, (4, El
..-,Ev( t ) ] , xi
d, -,
1 ' 1.
Gi
2,
1
on.
I,
v This
will
Uz(t)]&...1,
v.51
LANGUAGE OF
ri
no
no bound
U (t)
by
217
LOGIC OF PREDICATES
z.
(7)
(7)[riV U i ( t ) ] , EXAMPLE.
x1( t ) (Ez),
xz(t)
(Et). :
A V B.C
( A V B ) - ( AV C ) .
x2(1)
v fl
t-x2
v 21 v 21
r<e
K.2 r<e
r<e
[XZ
[x2 r<e
v x2 v x2
r.2
&L
[
(E7) by on
A&(B V C)
r.2
v 21
@>I7
v x2
xz
i<e
x2(t)
r
7
*
r<e
A&B V A&C.
x2
v
L.2
r<e
v 31
'X2
(4.
(~),<, r
by
v 21 (@)I*
x2(e)
r<e
[x2(e)V XI
-x2
218
tV.5
METHODS OF SPECIFYING OPERATORS
A&(AVB)
A.
: % ( t ) = x1 (
t ) W{ T < f
% ( t ) = x1 (t)*(Ez) T
(el v [x2 x2 ( Q ) v x’2 (t)*(Ez)
(e)
T<@
T<@
x2
T<@
T
v 21 ( Q ) l . X 2 (t>> (x2 ( Q ) v 21 (e)) 3
*
T<@
(e) X 2 ( d
(Ez)
r
(Ez) 7
v 21
(e) (x2 r<@
on
a[..., x(t), ...] a[ ..., T, ...I
x(t)l.
a[..., F, ...]
% ‘ by
by T on
F p. Corollary.
‘u[ ..., x ( t ),...]
...I V 2 ( t ) - % [..., F ,...1.
x(t)-’U[ ...,
on p. 217. by ‘u[x1(t), x2(t)],
‘uCT,
(Ez) T i t
‘u(T>F) %(F,T) %(F,F)
(XZ
v (el),
Z<@
(Ez) r
xz(e),
F, F.
That is, occurrences in which both the predicate variable x and the object variable f are free.
v.51
219
THE LANGUAGE OF THE LOGIC OF PREDICATES
'u = x1 (t)'XZ(t)'(EZ)
(XZk?) 7
7<e
v 21 (e))v x1 (t>%
xz(e>-
( t W ) 7
7 < ~ < t
V.5.3. DESCRIPTION OF THE ALGORITHM
z ( t ) = 'u [ X i ,
... x, )
t] ,
'u
'u,
by
'8, 'u&'u, 'u&'u&'u,... . 'u
by standard form,
-
23
v
D
@[rl, ..., rs]
..., I',,
@
1.
2.
B. 1
This remark also refers to formulae with predicate quantifiers.
v.
220
1.
cv.5
METHODS OF SPECIFYING OPERATORS
'II
A [ w ] ~ < ~ ~ ~ )
B
x(t) z(t) =
...,Bs(t)].
~[~l(f),...,xm(f),Bl(t),
Bi(t),
2. o
[o]e+w
Bi(t+
Bi(t + 1)
= ~i[Xl(~),.,.,X~(t),
Gj(t>
3.
Bl, ..., 2
a,,Dz,... .
Bl, BZ,...,
r,,r,, ..., rk);
'u
'Ir(t) = @
[Xl(f)
,..., Xm(t),rl(f),...,rk(t)], rl(t)> ...)rk(t)],
r 1( t
+ 1) = y1 [XI ( t ) , -..,Xm(t),
rk(t
+ 1) =
. . . . . . . . . . . . . . . . . y k
[xl(t) ,..., xm(t),rl(t),...,rk(t)].
ri(t) qi(t),
i
by canonical
As
equations
I Because in a regular formula there are no irregular occurrences of x(t)which would have be replaced by x ( t
+
v.51
221
THE LANGUAGE OF THE LOGIC OF PREDICATES
q l , q2, ..., qk,
ri(t)
3. Any t7formula 'u(x,, x2, ..., x,, t ) withoutpredicate quant$ers specijies a bounded-determinate operator which transforms a system of input predicates xl, x 2 , ..., x, into an output predicate. V.5.4. A N EXAMPLE TO SHOW THE WORKING OF THE ALGORITHM
1
3
182,
by z(t)= ( E 4
{ x 1 ( 4 8z
o
(7)
v
[XZ
u
(El)
.
x3
o
A r , ( t + l),
(t
v a(<El <4 r x3 = C x z ( 4 v ( 4 x3(411 v o
+ 1) = ( ast E 4 {XI (0)8z (4 L.2 a
4 bl(4& a (7) irst
a
(7)
t
= (ED) {x1(.)
&
a
(7) a
= ( 4{x1(5)& o
U
B
rl ( t +
= x1 ( t )
v [ z (0x 2
v r 2 (t)
9
Tz(t)
( E 4 {x1(4 & a
(7) o
L2 .
(7)
v a(<El <4r
x3
(E4
x3
a
r2( b ) rl( t ) A,
B,
r2(t),
rz(t+ 1) = [ z ( t ) & ~ ~ ( t ) & ~ V~ r( 2t () t]) .
=
222
[VS
METHODS OF SPECIFYING OPERATORS
no :
z ( t ) = 41(t)9 q1(t+
qz(t
1>=.1(t)V
[41(t).x2(t)lVq2(&
+ 1) = [ q l ( t ) * X 2 ( o * X 3 ( t ) l q1
= 0, q2
v
= 0.
V.5.5. TO EXTEND THE LANGUAGE
by
2 by
1
=%(x, 1,
x z(t),
0
1
odd
by
'u
1
0,
t 1.
by by
by
(4 = (Eo) [.I a
(0)
L% (z)
(43
*
U
(5.18),
is by
0-
k
0
is
__ _____ I See also C. Elgot, J. Wright, Quantifier elimination in a problem of logical design, Michigan Mathematical Journal (1959) 6, 65-69.
v.51
THE LANGUAGE OF THE LOGIC
223
PREDICXTES
k
by ~(t)=(Ea)(x,(ouct
(7) ~ 2 ( 0 - 7 ) ) . u
k,
by
x 1(c- 1) do
3 by Whatever thebounded-determinateoperatorz(t)= T [ x , , ...,x, there exists a t-formula a [ x , , ..., x,,,, t ] of the extendedpropositional calculus with bounded object quantiJers, which specifies this operator.
by
T
Proof. As :
z (4= @ [ X l ( & x 2 ( t ) , rl ( t ) , r2 ( t > ] r,( t + 1) = ul, [ X l ( t ) , x 2 ( t ) , r,( t ) , r 2 r 2(t
+
= y 2 [Xl(t), x 2
rl(1) = 1
2
2
(9,rl ( t ) , r 2 TZ(
~(t)=(EI',)(E~~){~,(l)&~~(1)&(z)[~~(z)-1
= 0.
7
E
u2(x2(Z -
x2 (Z - I), r,(Z - I), r2(Z @ (XI
&
(4, x 2 (4, r1 ( t ) , r 2 ( t ) ) > ;
r(1) t-
1 3',
3,
Any t-formula % [ x , , ..., x,, t ] of the extended calculus of single-place predicates with bounded object quantijiers specijies a boundeddeterminate operator which transforms a system of input predicates x l , ..., x, into an output predicate.
pp.
224
rY.5
METHODS OF SPECIFYING OPERATORS
Proof.
xi(t)
bound
'u
t ( E r ) D [... T(t)...I (ET)[D( ... F...)], by (T)[D( ... T . . . ) & D ( . .F...)]. .
by
(T)D[ ...r(t) ...]
r
D, r(z)-z
r
(ET)D
(ET)[T(t)-D] on
D
3 t.
As r ( t ) - D (...T...)VT(t)D(... F...), (ET)[ I ' ( t )-D(. .. .)] V (ET) [ r ( t ) - D (..,F.. by ( E r ) D [...
.)I. ( E T ) D [...
D [...T..
[. ..F.. .] by
z, ...,
bound,
r(e)-e
r(Q), r ( z ) , ...
( E r ) D[... ( E T ) [ D (...
.] V (ET)D [. ..F..
F...)].
( E q D [... r ( t )...] by by
T(t)
(r)D(... r(t) ...). by :
(Ez) G+).(r)[Y(t).r(4 v rct
*
z<e
by That is, the elimination
y(t) quantifiers is assumed
'%[...y(t)...]. have taken place already.
v.51
225
THE LANGUAOE OF THE LOGIC OF PREDICATES
:
a[... a[...
(el
= T
Z < b < t
= (ET)(X(T)*(~) T i t
(r) by Bl [... r(t)
by B2[... r(t)
Bl[... T . . . ] = T ( T ) v (e) Bl[ ...F %z
=r ( T )
[.-.
v
F, r<e
= (Q) x(Q), ?
B,[ ... F
Y (2)
-
.{
(2)*
T
(0W < T > v
F.
= r
x T<e
- ( T ( dv
v
v u(t>*(E4{x(+(1')[ T i t
rCe
T<e
: {x(T)'(r) TCt
[W v r < e < t
v
r<e
T
REMARKS one
1.
zl, z2, ..., z,,
several
. . . . . . . . zn ( t ) =an t] .
2. T),
:
z
(9 =
z i t
e
v
x3
@
-
226
w.5
METHODS OF SPECIFYING OPERATORS
(5.18)
+ 1)= z ( t ) V [ x ( t ) & r ( t ) ] , r(t+ 1 ) = [r(t) v XJ(t)].X&). z(t
by
Q
by
+ ki,
ki
terms)
ki x(t+s), r ( t + s ) , ...,
s
%(x, t ) , z(t)=
@ [X(t),X(t+
9 j ( t + 1)= Y j [ X ( t ) , X ( t 4 j ( 1 ) = xj [x(l),
1), . . . , x ( t + S ) , q l ( t ) , . . . , q k ( t ) ] ,
1
+ ~),...,x(r+S),q~(t),...,q~(t)], s>O),
do bound by
e
t C f
+ 1 ) = ( E z ) ( E r ) [ x ( ~ + 3).(~)T(e+ 4)], e
r
[X(T
7Ct
e
e
V.51
227
THE LANGUAGE OF THE LOGIC OF PREDlCATES
1.
q(t) q ( t + 1)
= (ET)[(e)r(e
+ 4).T(t + 4)];
e
I'(t 4(t
+
+ 1) = 4 ( t ) .
(4 = P (4
P(t
+
:
7
1) = P ( t ) v x ( t y(1) = 0.
+ 3),
by :
t =k , t < k . bound e + k l < t + k , by T, k 2 >k , . t+kl=t+kz, t+kl
x(t)
go p. (Ez),
<
z(t). do ( A (z) .B (7)& <@ < [A(@) B(e)]1
(5.25)
ql(t+
by
q 2 ( t + 1)
on
228
Iv.6
METHODS OF SPECIFYING OPERATORS
41(t) AB, B
B
q2(t)
A
A
NOTE. B
A
0
1
z ( t ) = A(t).W)V [ A ( t ) O B ( t ) ] . ( E z ) { A ( z ) . B(4.
(el
r i e i t
z i r
z(t). z ( t ) = A(t)B ( t )
4(t
v [ A ( t ) 0 B ( t ) ]4 ( t )
Y
+ 1) = 4 ( t ) [ A ( t ) @ B ( t ) ]v A ( t ) B ( t ) , q(1)
=0.
(0) by physically permissible, do
00,01, 10, 11.
01 by 10
V.6. The expressive power of a logical language 3, 4
logical formula),
by book by
3‘
5
V.61
229
EXPRESSIVE POWER OF A LOGICAL LANGUAGE
by
by by
no
by S. C.
regular expressions,
by V.
event in the alphabet { x l , x 2 , ..., x,} by on
e, : disjunction, multiplication
iteration.
V S,
S,
S,, S,.
S,.S,
S,
by S,.
S1V S , V ... V S,,
S , S , ....* S,,,
“V S
‘ I - ” .
e , S , S - S , S . S * S ... , . {S},
by
A
elementary events e.
regular events
m+ 1 x , , ..., x ,
”
230
IV.6
METHODS OF SPECIFYING OPERATORS
by
by
regular expressions,
1. xl, x2,..., xnrre
2.
%.TI, %
(rm v
(rm-%), pm>.
3. { x ,y , z>
EXAMPLE.
({(.
v Y ) )* ( z . z > ) z’s.
by
by :
{xVy}zz.
T S,, to
T
recognize S,. representable in ajinite automaton
THEOREMI. For an event to be representable in ajinite automaton, it is necessary and suficient that it should be regular.
suficiency.
9JI a
a* ( t ) ,
by
by 21m See S. C. Kleene, The representation of events in nerve nets and finite automata, “Automata Studies”, ed. C. E. Shannon and J. McCarthy, Princeton U.P., 1956.
V.61
231
OF A LOGICAL LANGUAGE
W
by W.
91m
bound on no
B(t),
B7(t)
5,
B(t) by [ A ] l c r by
two-sided
one-sided
[A]rcl
EXAMPLE.
( E d rx (4)- (4y e
o<e
by B(t);
( E d cx
(0)
rCa<e
r<e
I-
B7(t). BI(t) t.
t
B(t) (z+
21m, 21n, ... xl,..., x,
W,%, ... . (t),...,
x1
:
x2,..., x,
F e.
xl, ..., x,,, e. ...)X m ( t ) . t = 1, F).
1.
21m
am,3"
% 8. !I,
:
2.
91mV2l"
WV %. 1
23 2
1V.6
METHODS OF SPECIFYING OPERATORS
3.
?lX-%. 4. (ET)
(WLk (W. 74t
P<7
r (e)-+(4 ? &I? Lk (7)(4( WT. ( 7 ) . 7
U
r
{mZ}.
r
Explanation. A z1< z2< ... < t , (z,
{m}
t
+
on,
9X. 4,
( E r ) { r(t ) .(0) (Z’(c) a4t
= am(c)V (ET) (r(7) Z
{?lX}.
(0)))
+
‘E?(4))
C H A PT E R VI
OF
VI.l. General remarks by
as
on on 3
optimize minimum minimum synthesis
on minimum forms negation, disjunction conjunction. 1
See Chap. I, Sec. 3. 233
234
p1.1
METHODS OF SYNTHESIZING AUTOMATA
two
minimum formulae, by -
-
by by by
on
by by by
2
VI.21
235
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
single-output
by
3 of multi-output
by
VI.2. Synthesis of a single-output logical net by by
by 3. m
f (~17x2,
xm)
=
v....
a1,a2,
f (cl,cz,...) fTm)[ X y x ‘ ; ’ ... xZm]
(6.1)
, a
by xp = Z i = T ( x ! ) , x?,
by
ui=O,
T(xi).
o,(Dk[Xl,XZ,...,T(Xk) ,...,T(Xj)],..., Dk[X1,XZ,..., T(Xj),..-,T(Xm)]}-
(6.2)
(6.2) Dk, T
1954.
xi
236
WI.2
METHODS OF SYNTHESIZING AUTOMATA
EXAMPLE (6.2),
x12.,Vx2X3VX1x3.
v X,X, v X,x3 =
:/IT$r:+T:!
2 = xlXz
= Dg{Dk[xl,
Dk,T
x,.
by
z(xl, x2, x3),
T(x3)I,Dk[T(xl),x3]).
T(x2)19Dk[XZ9
6.1.
~-
~~~
-
I~
1 5
.
3
Fig. 6.1
by
(6.1)
(6.3)
z=Vpi i
K , ( p j 7pk), on
= K , {...K~{K~[K~(P~,P,),P~I,P~) ,. . . , P i ) , by
by
pi
K,
(6.4)
(6.4). As
3,
k
by by
-
by
on
VI.21
231
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
T,(x,, x2, ..., xk),
by
ci=1,
by
T (x) xi = T (T(xi))9
(6.5)
x,x2%3%4
by
T,[T(xi), T(xz), X3, x4]-
xj by
Xj. 6.2
= K2 {K2 [T,
1, by
(k4,T2 (& x41, T2 (XI?2 3 ) ) -
Ti) - 10. by
23 8
METHODS OF SYNTHESIZING AUTOMATA
by
by
[VI.2
Z
by -
z = T(P). z,
P(xi, xi),
(6.6), oi = 1,
x;'
v x y v ... v x
2
=P
{ ... P { T { P { T [ P ( X ~ ' , ~ - a , 2 ) ] , X ~ )... X 22,). ~;4), (6.7)
2,
by
(6.7). EXAMPLE 2.
z
=XI
v x2 v x3 v X4
Df
(6.7).
z
: z =P{T[P(Ri,xz)],T[P(f3,x,)]},
v1.21
239
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
fb)
Fig. 6 . 3
j.2
Fig. 6.4
on
6
of
8 1, by
6.4 It -
z =
z = (21v xz)(2.zv
x3)(x1
on
v 23) = = T ( P ( X , , ~& Z~ > ( x z3 3, )
7
p(21,~
,
3 ) )
xi
240
M.2
METHODS OF SYNTHESIZING AUTOMATA
(XI' V x? v ... V x:~),
P ( x , , x2, ..., x,)
by do
by
m
+
N 6 2 m - 1 ~ n 2 m - 1= 2"-'(m
+ 1).
(6.8)
2"-'m 2m-1
2m-1
by
by vice versa
:
8
w
ri
by
by
EXAMPLE 3. z = x1xzx3x4
v
Df
by do 10
vI.21
241
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
: z = 2123 ( ~ 2 x 4V
z=2123(x2 v24)(22
v
2.224).
x4)=
fz
Fig. 6.5
6 6.5).
by on
is
pyramid circuit,
decoders,
rn matrix
4
circuits. 16
X
0
ol,c2,03, o4
1
by ZO = 21222324 21
= 212223x4y
. . . . . . z7
= X,X,X3X,,
. . . . . . zlS
= x1x2x3x4
~
~
X
~
X
~
~
X
242
WI.2
METHODS OF SYNTHESIZING AUTOMATA
of
(01020304)
fi
a
xi, xu:, x?] ,
z j = D,
1, ..., 15.
(A)
Fig. 6.6
6.6.
vI.21
243
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
m N = m.2".
(6.10)
N = 64.
Fig. 6.7
In
by
EXAMPLE 4. by 0
6.71.
1
by 100100100... A The dotted lines on the diagram, linking any two nodes the tree, indicate that these nodes are indistinguishable, i.e. they are nodes coincident branches.
244
[VI.2
METHODS OF SYNTHESIZING AUTOMATA
TABLE VI. 1
qo, q1
by
q2 (q1q2): qo = 00,
q1 = 01,
q 2 = 10.
VI.1. ql(t)= 1
VI.1,
q , ( t ) = 1, ql(t+
by
6.7,
v ( t )9 2 ( t ) W v Yz(t)x(t),
z ( t , = cpl(4 cp2 (4 x ( t )
9 l ( t + 1) = Y l ( t ) f ( t ) q 2 ( t 1) = z ( t ) .
+
3
(B)
J by
z(t)=
(t,[X(t)
= T [40 L sD1 ( t
qz(t
+
+
= [Cpl
v cpPz(01 [ X ( t ) V 9 2 W l = p [. ( t ) , 9 2 011Z?L p 1%(4, T [ 9 2
( t ) v 491 b
=
= z(t).
2
( t )v = &p{X(t)J[Y2(t)llY
5
1
(C)
VI.21
245
SYNTHESIS OF A SINGLE-OUl'PUT LOGICAL NET
Fig. 6.8
6.8 D.
by ~ ( t= ) Q[Xl(t),x,(t), 'p1 ( t
+ 1) = yl
~ p (2t
+ 1) =
(t
+ l) =
[Xl
~2 [XI
:
...,xj(t) ,.-.7x,(t), cpi(t) , . . . , ' ~ j ( t ) , . . . , ' ~ , ( t ) ] ,
*.-) ( t ) , *. .)xi ( t ) ,
Xm((t), 'PI ( t ) 7
(t)7
.7
Xm ( t ) , ~1
(t),
."7 Se.3
qj(t)7
*.'3
qs(t)]
. . . . . . . . . . . . . . . . . . . . . . .
'Ps
[Xl(t)7 . . .
M. (6.12) p. 'Pi@+ l ) = @ j ( t ) P j [ X , ( t ) ,
V q j ( t ) Yj [XI
7Xi(t),...7Xm(t)7(P~(t)7..'7(Pj(E)7'.'7~s(t)].
qj(t)
7
'Pj ( t ) , * * .)' ~ (st ) ]
9
1
(6.11) (6.12)
qj(t+ 1) (1.1)
..., Xi(t) ,...
...7x,(t) ,ql(t)7...,'~j-,(t),O,qj+,(t),...,qs(t)l V ( t ) , *.., xi(t), --., x,(t), 91( t ) , *-.) 'Pj- i ( t ) , 1, cPj+ 1 (t), ' ' ' 3 V s ( t ) ] . (6.13) (3.21)
M. L. Tsetlin, Non-primitive circuits, Collection "Problemy kibernetiki", no. 1, Fizmatgiz, Moscow 1958.
246
Ivz.2
METHODS OF SYNTHESIZING AUTOMATA
111, p. 136), u ; ( t ) = Yj[X1(t) ,...,x i(t),...,X,(t),cp,(t),... ...,cpj-
2);
( t ) = !Pj[XI( t ) , ...)X i ( t ) ,
l(t),O ,cpj+l(t),...,cps(t)],
... x m ( t ) , cpl (t),... ...)Vj-1 (t),1, V j + 1 ( t ) , .*.,c~s(t)] )
cpj(t+l),
i
(6.14)
(3.21). switching functions by uj ( t )
(6.14) (6.13)
= qj ( t )pj [XI ( t ) , ...)xi ( t ) , ...,x m (t),~1
(3.25), ( t ) , ...
. . . , ( ~ j -l ( t ) , O , ( ~ j + l ( t ) , . . . ',~ s ( t ) ] V
V
( ~ j ( t ) Y j [ ~ , *..,xi(t), (t), ...,X m ( t ) , r p j - , ( t ) , l , c p j + l ( t ) ,
..., c~s(t)].
(6.15)
(6.14)
(6.15)
K, by
s=log,K z(f)
5. EXAMPLE by
~ ( t )
t;
x, 2)
by
1
odd xd
by
6.9 Zd&,
Xdf,,
Xdxsr
(cpPA1, cpAZ), (cplcp2). qo=oo,
ql=lo,
q,=11,
qi
q3=01.
VI.21
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
VI.2 .in x,= 1, xd= 1,
Fig. 6.9
247
.
248
ql(t+ 'pz ( t
WI.2
METHODS OF SYNTHFSIZING AUTOMATA
(D)
@l(t)@2(f)xd(t)7
+ 1) = 'pz ( t )'pl (t>v 'pz (02 s (4 v cp2 (t> ' p l ( 0 xs ( t )
3
z(t)=
(F) (D)
by
xs(t)
(6.14)
(3.27)
u; (l) = (P2 (t> xd (t> u';(t) = @ Z ( t ) Xd(t), 9
v
by 1 'pl(t
+ l>= =
[V2 ( t > ,xd
9 1 (t)
[FZ (l>,xd
@1
(t).
4 ( t ) = 'pl ( t ) X , ( t ) ,
v .&),
4(t) =
'p2 ( t
v%
+ 1) =
(0,
= p [@I
'p2
(4 v [@l (t>v 2,
v p ['pl ( t ) , xs
'pz (t>
by
2
6.10 on
6.1 € In
on
'p,(t)
x,). xy'(t) =
1, 2
cp2 (t>= cpz ( t ) .
VI.21
249
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
- (t)
x;
L_
_ _ _ _ _ _ _J Fig. 6.10
XP'(t) = Xd(t)(py-l)(t)
=
Tz [zd((t),(p(2n-')(t)].
EXAMPLE 6. : u = u1u2u3... u,
w
= w1w2w3 ... w,.
t:
: u (1)u (2) ... u ( t
- 2). (t - l ) u
w(l)w(2) ... w ( t - 2 ) w ( t y
(2),
p
a
(t),
(1)
l)w(t)
(2)
250
IVI.2
METHODS OF SYNTHESIZING AUTOMATA
/.I I I
I
I I
I I I
I
Fig. 6.1 1
a, b, c, d, (x1x2) :
d=ll.
c=lO,
a=00,
(z1z2)
p=01,
y=oo.
6.12b
3
VI.21
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
((pI(p2): 40
= 00,
41 = 01 4 2 = 10. 9
251
252
WI.2
m T H O D S OF SYNTHESIZING AUTOMATA
m(t)
__
0 0 1 1 0 0 1 1 0 0 1
0 0 0 0 0 0 0 0 1 1 1
0
1 0 1 0 1 0 I 0
' I
1 0 1
'pl(t)= 1
(p2(t)= 1
V1.3,
9 2
(t
+ 1) = rp2 ( t >21( t )XZ ( 4 v (71(0'pz (t>f l ( 4 v Cpl(t> 'pz (9x2 (4.
(A*)
'pj
rpz (t> [Xl(t) v 2 2 = T2 CXl(t)> 22 0';(4 = T [fl x2 ( t > ] >.
0; ( t ) =
= T2 {'pz ( t ) ,
0; =
0';=
T{T~[xi(t),2z(t)]}-
3
I
I
(C*)
v1.21
253
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
6.13.
(A*), (B*)
by transition matrices is
by 5,
b jir tt)
Fig. 6.13
EXAMPLE 72.
(0,1, 2, 3,
2’ - I),
. . a ,
(u,
(XI, XZ)
w).
:
xl(t)=xz(t)=l
x1( t ) = 1
xl(t)=xz(t)=O,
x2(t)=0,
no. j
no.
M. L. Tsetlin, Non-primitive circuits, Collection “Problemy kibernetiki“, no. 1, Fizmatgiz, Moscow 1958. This example was borrowed from the above article by M. L. Tsetlin.
254
IVI.2
METHODS OF SYNTHFSIZING AUTOMATA
2');
no.
no.j
x 2 ( t ) = 1,
3) j- 1 4)
1; 2'-
1=2'-
u= 1 is
0,
1
w= 1
In
u = w =0.
by
S=2. (q),
VI.4. VI.5.
+
q2(t
pl (t 1 )
+ 1):
TABLE V1.4 .
_.__
O
0
I
-
01
10
11
10 00,11 01
01
1
00
00,11
10
01
01 10
00,ll 01 -
10
-
10 00.11
11 ._
~~
00 01 10
11 - .~
00, 01, 10, 11
I
01, 10, 11 00, 01, 10 00, 01, 10, 11
00 -
-
11 -
255
SYNTHFSIS OF A SINGLE-OUTPUT LOGICAL NET
by :
I
= X l ( 9 22 ( t ) 4pz (4 v 21 (4 x2 ( 4 9 2 (0 > ( 4 = 21 (4 XZ (t> v x1 (t>2 2 (t>
Ul(t> 02
*
ZIA(t)=~l(t>(Pl(t)=Xl(t>~Z(t>~l(t)(PZ(t>
v 21 (4 XZ ( 4
9 1 (t>i r z
(t>
9
by
1 1
(B**)
w**>
z l B ( t > ~ u l ~ t ~ ~ l ( t ) ~ x l ~ t ~ 2 Z ~ t ~ ~ l ~ t > ~ Z ~ t > ~
v 21 ( 0 x 2 ( 4 rp1 (t>i r z ( t ) .
ZZA ( t )
= u2 (t> 4pz ( t ) =
zZB
(t) = = 2 1 (4 x2 ( t ) i r z (4 v
= f l ( t > XZ (t>4pz (t>
v X l ( t > % (4 v2
= u2 ( t ) @Z
(B**)
x1 (t>2 2 (t> cpz
7
(4.
(C**)
ul(t)= Z2A(r)Xl(t)V
ZZB(t)XZ(t)*
(t> = x 1 ( t )%Z ( t ) 431 (lVZ > (l)= Z I A ( t > x1 (t> ) (l>= 2l (t> x2 (t> @l(t>@Z (t> = Z I B (t>x 2 ( t ) . \ a 9
by by %(t)
= X l ( t > 2 Z ( t ) v21(t>xz(t)9 uj(t> = X I ( t > * Z ( t ) q l ( t > ' ~ z ( t.*. ) qj- 1 ( t ) V
V 2 l ( t ) ~ z ( t ) c p i ( t ) ( ~...cpj~(t) l(t), ZAj(t) zBj(t)
= Uj(t>'Pj(t)7 = uj(t>'Pj(t)3
u j + l ( t )= Z A j ( t ) X l ( t )
(l> = zAS
( t ) = zBS 6.14
zBj(t)xZ(t)7
( t ) x1 ( t ) (t>x2 ( t ) . 7
(j=2,3,...,S)
256
METHODS OF SYNTHESIZING AUTOMATA
I*'
lVI.2
VI.21
251
SYNTHESIS OF A SINGLE-OUTPUTLOGICAL NET
As
3, on 0=2,
z(t + 2)=f[xl(t),x,(t),...,xrn(t)] =
V
=
....urn
I
~ ( o ~ , o ~ J . . . , G ~ 1, ) GV G i H ( x i ( t ) , O ) V ~ i H ( I , x i ( t ) ). i= 1
Ulr~2,
no
EXAMPLE 8. 5,
on
by
2.
5,
z(t+2)=q(t+2)=q(W
of
[Xl(t)~Z(t)VRl(t)xz(t)]~(t),
M. L. Tsetlin and L. M. Shekhtman, Two-cycle ferrite circuits and an algebraic method of synthesizing them, Collection “Problemy kibernetiki” no. 2, Fizmatgiz, Moscow 1959.
258
IvI.2
METHODS OF SYNTHESIZING AUTOMATA
TABLE VI. 6
0
00,11
01, 10
1
-
00, 01, 10, 11
I
z(t+2)=q(t+2)=q(t)Vxl(t)~,(t)V~,(t)x,(t)-
(A')
by
(t)
: z(t
6.15 xl(t)
xz(t)
of G
H
VI.21
259
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
by
by by
(3.39)
1
G(x,x) = 0 ,
G[L(H(x,y) V F)] = G[(G(l,x)
(6.17)
v G(y,O)),F],
G[G(x,y),F] = G[l,(H(I,x)VH(y,O)VF)],
(6.18) (6.18’)
F x 2 , ..., xm)
on on
v...,
UlK2,
i
G F,P
an
v
n
v [oiH(l,xi) v GiH(xi,0)]
i= 1
P do on xi, x2, ..., x,. f ( x , , x2, x ~ ) = R ~ x ~ x ~ V X ~ X ~ X ~ , f(Xi,Xz,X3)
= G[l,(H(Xi,O)
1
(6.19)
= G(F,P),
v H(1,xz) v H(1,x3))] v
v G[1,(H(1,x,)V
H(1,x,)VH(1,x,))].
(6.19)
f (xi, xz,x 3 ) = G [I, ( H (1, x2) V H ( h ) ) ] is
on on
n
V G [ F , ( P V H(l,xi))] = G i= 1
n =2, G[F,(PVH(1,xi))] VG[F,(PVH(1,xz))]= = G[F,(P V H ( l , x l V
:
xZ))].
(6.21)
260
m.2
METHODS OF SYNTHESIZING AUTOMATA
f =x1xzx3Vx2x3x4,
f
v H ( 1 , x z ) V H(l,x3))] V V G[l,(H(1,x2) V H(17x3) V H(1,x4))],
= G[l,(H(l,xi)
on
f
(6.21) = G[L(H(1,x2)
v H(kx3) v H(1,XI v Xq))].
EXAMPLE 9.
(3.25)
(3.26) ;
u ( t ) ’ P ( t ) = G(l,[H(l,u)VH(cp,o)]), ( t ) cp ( t ) = G (1, [H (u,O> (1,Cp)l)
v
cp ( t
+
=
G {1 [ H 2
9
).
.
v H (cp, O ) ] } v G { 1, [ H (u, 0) v ff (1 cp)]) 9
. by
Six (6.18’),
VG[G(cp,u),O].
(A”)
5
(B“)
z ( t + 2 ) = u ( t ) c p ( t ) = G{1,[H(1,cp)VH(1,u)]>.
(A”) 6.16
is
8
EXAMPLE 10.
a=2. 6.17.
by a, b, c, d by
(x1xz):
a=00,
b=Ol,
c=10,
d=ll
vI.21
261
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
Fig. 6.16
and binary symbolsq, =0, q1= 1 are associated with the letters in the alphabet of states. We denote the output function by z, and construct the table the operator generated by the adder (Table VI.7) according to the tree shown in the diagram. Table VI.7 we obtain the canonical equations in the
(A) (
z (t
0
1
I1
1
+ 2)
"I)
262
(A”‘)
q(t
WI.2
METHODS OF SYNTHESIZING AUTOMATA
+ 2)=
(B”’)
H[17(H(1?x1)
H(19x2))]
VH[1,(H(17xZ)
vH(lYq))]?
H[l?(H(l>xl)
H(l>q))]
v (C”’)
z(~+~)=G[~,(H(~,O)VH(~~,O)VH(~,~~))]V V G [ l , ( H ( q , O ) V H(1,xi) V H ( x 2 , 0 ) ) ] V
V G[l,(H(1yq) V H(xi,O) V H(xz,O))] V
v
v H(l,x,) v H(1YXZ))l.
6.18
(C”’) z(t+2)
13 q(t+2)
by
,
7 i, Fig 6.18.
. ?*
VI.21
263
SYNTHESIS OF A SINGLE-OUTPUT LOGICAL NET
111,
by z(t =
:
+ 2) = f
v...,
(t),xz(t),
= m
f(gi,gZ,...,om)R)
v
!\
v
i= 1
urn
Ul.UZ.
(6.22)
(6.22)
R
by
As by (6.22) on 1,
R [R R [ ( R (x,0)
vR
Vi R Y)
01 = R R Yi) = R [x, V
(6.23) (6.24) (6.25)
Y
5
i
v R (1, S)),01 = R [ ( R (x, v S ) ) ,01
(6.26) (6.27)
9
=R(S,O), S
by by
a
(A”’) :
q ( t + 2) =
v R [ ( R (4,O) v R z ( t + 2) = v v v v R [ ( R ( q , O )v
v R [ ( R (4,O) v R (XZ, v v v v v R(x,,0)),0]. 03
03
7
(E”’)
1 L. M. Shekhtman, An algebraic method of synthesizing choke circuits of semiconductor diodes, Collection “Voprosy elektroniki” series 12, no. 18, 1959. 2 Only a few of the transformation formulae in L. M. Shekhtman’sarticle are quoted here.
264
WI.3
METHODS OF SYNTHESIZING AUTOMATA
R on
by by (t
by
+
[(R (x1. v q)),01 v R [ ( R ( 4 , x1 v x2>), 01 v v R [ ( R ( x z , x , v 4)),0]
=R
x2
(H”)
6.19 (E”’) VI.3. As
logical
of a 1,
n
by
n
VI.31
265
SYNTHESIS OF A MULTI-OUTPUT LOGICAL NET
n by
by :
no 41
no 2122x3
22
21~2x3.
on
19
on
z2
z1 z1
rn
1954.
266
[VI.3
METHODS OF SYNTHESIZING AUTOMATA
z2. =f
2;
; (XI,
~
2 .*., , Xrn, ~ 1 )
z1
z2
z; 2"
f; ( ~ 1 ,
~
2
-..>Xrn, ~ 7
1 = ) f 2 ( ~ 1 ,~
2
...)X m ) . 9
iz; Fig. 6.20
z1
by
z2,
z;
V
Z; =
a,,az,....a m
f i ( ~ ~ , ~..., ~ ~2 , ,) ( Z ~ ~ X ~ " X ~ ~ . . : X ~ ~ ) ,
a= 1
o=O
zl,
do
z1 by
z2
z2
zl.
least
on. a
by
VI.31
267
SYNTHESIS OF A MULTI-OUTPUT LOGICAL NET
EXAMPLE 1. by
:
V 2122x3 V 21x223 ~2 = 2 1 ~ 2 x 3V ~ 1 x 2 2 3 V~122x3. ZI = 2 1 2 2 2 3
z1 z2.
7
by (6.30), 2; Z;
=i 1 2 1 ~ 2 V ~ 3. T 1 ~ 1 2 2 2 3V .flx122x3.
z1
z;
v 2,) = T ( x 1 ) & P ( x 2 , x 3 ) ,)
z 1 = 2,(z2
z2=.T1(21
T(z,)&P(x,,x,).
6.21
(A)
j
(A),
6 z1
z2)
9 several cascade method,
by
on by
multi-output pyramid net,
Fig. 6.21 1 G. Povarov, A method synthesizing contact circuits for computing and control, “Avtomatika i telemekhanika” (1957) 18, no. 2.
268
[VI.3
METHODS OF SYNTHESIZING AUTOMATA
by =
f
..,x,) .
(XI, x2, .
xl,
v Xlf(1,X2,X3,...,X,).
=21f(o,X2,X3,...,Xm)
(6.31)
x2,xg, ..., x ,) do
x2, x3, ..., x), xI
do
x2,
x2
by
pyramid circuit. m on
by M Z ( ~-
l)(m -
= $m!*
m
on
EXAMPLE 2. = x1.f2.f3.f4
v
21x2.f3X4
v xlx223.f4 v x1x2x3x4
9
xl,
z
= 21 (22233.4
v x2.f3x4) v x1 (x22324 v x2x3x4) = = 21.f(1)(x2,x3,x4)
*
See the reference on p. 265.
v X1f(2)(X2,X3,X4),
VI.31
269
SYNTHESIS OF A MULTI-OUTPUT WGICAL NET
1
c
Fig. 6.22
x 3 , x4)
x 3 , x4)
222324
v x223x4
x22324
v x2x3x4.
(')
j ( " ( ~ 2 ,~
f ( 3 ) ( ~ x4) 2,
3~ , 4 =) 2 3 (2224
V ~ 2 x 4= ) ~
3 f ' ( ~x Z ), x q ) ,
Z2Z4V ~ 2 x 4 ,
f ( 2 ) ( X z ,x 3 , x 4 ) = x 2 (2.324
v x 3 x 4 ) = x ~ f ( ~ ) ( Xx34, ) ,
R3R4Vx3x4.
no no
f 3 ) ( x 2 ,x 4 ) = (2, V
V z4) =
~ 4 ) ( ~ 2
= p ( x 2 , 24) & p ( 2 2 , f(4)(X3,x4) .f(')(x2?x3,x4)
x4)
9
= P(x3,24) & P ( 2 3 4 4 ) ,
= T(X3)f(3),
f ( 2 ) ( x 2 , x 3 , x 4= ) ~(2~)f(~).
z z = T { P [ 2 , , f " ) ]& P [ X , , ~ ( ~ ) ] } .
(B')
27 0
tVI.3
METHODS OF SYNTHESIZING AUTOMATA
6.22.
15
20
z
3. by TABLE VI. 8 x1
X2
0 0 0 0
0 1 0
0
1
0
1
0 1 0 1
I 1 1
1 1 1
0 0
1
z:
16
by
x1
V
z = 21
V
=
(A")
= 2223
(A") =2 , p
j"' 9
=
v Xlf(" V
= P(x,,f'")&P[f,,
=P
6.23).
:
T p ) ] ,
xj) & P (
~ 2 ~ 2* 3 )
(B")
(C")
VI.31
271
SYNTHESIS OF A MULTI-OUTPUT LOGICAL NET
by
EXAMPLE 4. z3=f 3 ( x 1 ,x2, x , ) ,
z1
( x l , x2, x3),z2 = f 2 ( x 1 ,x 2 , x,)
V 21x3 V ~ 2 x 3 , = xi12 V ~ 1 2 V 3 2223, = 2 1 x 2 ~ 3V ~ 1 2 2 x 3V ~ 1 = 21x2
~2
~3
V
x 2 ~ 23 1 2 2 2 3 .
of x 1
G.
.
.
Fig. 6.23
x2
x,;
VI.9.
z2, z,
on
on xl.
1-3
y), y), y), i2), y), f $’). xz
4-9, x3)
VI.9), Our
VI.10.
212
METHODS OF SYNTHESIZING AUTOMATA
TABLEVI. 9
I
1
wI.3
VI.31
273
SYNTH SIS OF A MULTI-OUTPUT LOGICAL NET
TABLEVI. 10 ~
-
~
z2
OL
z3
= x3
no
VI.10
:
by
Vf,C”)(x, v p) =P(x,,j~Z))&P(a,,~~l)), v j i Z ) ) ( X 1 v p)= P ( x l , f p ) & P ( 2 1 ,fy), 2 3 =(2, Vf:”’)(X, v p > =P ( x , , ~ j f z ’ ) & P ( ~ l , j j f l ) ) . 2, = (a, 22 =
(a,
274
WI.3
METHODS OF SYNTHESIZING AUTOMATA
Ji’”
= 83, = ~2 (xZ,
~3),
fy)= 8 v Z2 = P(xz,x3),
fil)= X Z V p = P ( ~ 2 , 2 3 ) , f;” = ~ 2 =8 TZ ( R , , z ~ ) , J?:” = (x2 v B)(Z, v a) = P(Z2,P3)&P(xz,x3),
p
=
T(fj1)). 6.24
A 25 EXAMPLE 5.
is on
10
4 ( t + 1) = X l ( O X 2 ( t )
2.
v 4 (0x1 ( t ) v 4 (t>% z(t).
(p.
TABLE VI. 11 - ._.______
0 0 0 0 0 0 0 0 1 1 1 1
1 1 1 1
(A*) re-
V1.31
275
SYNTHESIS OF A MULTI-OUTPUT LOGICAL NET
Q‘T
Fig. 6.25
z(t)
z ( t ) = q(t + 1 ) [ 4 ( t ) v
Xl(t>
v .z] v 4 ( t ) x , ( t ) x z ( t ) .
(B*)
(B*)
4 ( t + 1) = P [xl (0,x2(t>l& p [4 ( t i , x1 4 ( t + 1) = T [ 4 ( t +
& p [4
)
(flyx2
K*)
~ ( t=) P{Q,P[4(t)yxi(t)]I &P{Q~P[xi(t)~xz(t)]) 9 Q = P [ q ( t + l ) , q ( t ) ] & P [ q ( t + 1)yxi(t)]&P[q(t+ 1 > , X z < t > I -
(C*)
6.25.
21
by
z(t) 10
v Xl(t)52(t)l v v 4 ( t ) [a, ( t )2 2 (4 v X l ( t > XZ 011 = d t ) E (4 v 4 (t1-f
z(t> = 4 ( t ) [ % ( t ) x z ( t )
(t)
(E*)
9
(0 = 21 ( t ) 22 (9 v x1 ( t ) x2 (4 by :
261),
q(t + 1) = q ( t ) J ( t ) v f , (t)a,(t).
(F*) g(t
+ 1) as (G*)
216
WI.3
METHODS OF SYNTHESIZING AUTOMATA
A
(E*), (F")
6.26.
(G*),
14
on
c
Z(t)
Fig. 6.26
on
2
(a=4) 10,pp. 260-262).
A
6.6 p. 242). EXAMPLE 6.
VT.31
277
SYNTHESIS OF A MULTI-OUTPUT LOGICAL NET
:
zo = Z4pb, z1 = x,pb, z2 = Z4p;;, z3 = x4pg, z4 = z4p;, zg = x,p; , z6 = 24pT 27 = x,p’;, ... , Pb = 23P0
p;
7
= I3P17
pg = x3PO p’; = x3P17 9
. . . . . . . . .
Po = 2122, p1 = Z1Xz,... .
4 pairs
of input channels
~
Fig. 6.27
on
6.27,
56
N = 2 m f 2-8 by
m by
zo = ZZ
= POP;
z4 =
9
z1 = POP; Y
Y
z3 =
POP;
Y
9
z5 = PZP;
9
. . . . . . . . .
278
[VJ.4
METHODS OF SYNTHESIZING AUTOMATA
p o = 21x2, pb = 2 3 2 4 , p1 = 21x2 , p ; = 23x4, p2
= x,%,
p3 = X l X 2 ,
Pi = x324 Pj = x3x4.
9
rectangulur circuit, is
6.28,
48 rn
N=rn2*m+2m’1
VI.4. Synthesis of a ”rat in the maze” logical net by 25 on
by E. by on
way
of
VI.41
279
A “RAT IN THE MAZE” LOGICAL NET
by by
S S
no
S,
no
on
on by
(Y),
3)
(G), by
on
S
on
B
on
by
Y
G
D, 120”
120”
!,
S
t,
D, f
!,
280
M.4
METHODS OF SYNTHESIZING AUTOMATA
A. by Sign detected by “rat”.
Prescribed behaviour (or route).
2.
1. 2.
by
by by 3.
3. by
4.
4.
by
by by 5.
5. on
no by
S.
by
-
qo
ql.
qo
q1 by 1)
ql.
qo,
W.41
281
A ‘‘RAT IN THE MAZE” LOGICAL NET
qo
“G” by
Y
R
Fig. 6.29
“Y” qo
“G”
“G”
282
METHODS OF SYNTHESIZING AUTOMATA
tVI.4
qo.
“Y” ql. 6.29, x
. -
qo A
by V1.12). qo, q l , q, q ! ,qOR,qOG,q l R ,
qo, q,,
q, ;
q! qOR,
the TABLE V1. 12
j/
I/ il
.-
Red
VI.41
283
A “RAT IN THE MAZE” LOGICAL NET
qOG,
qlR, qly,
VI. 13. VI. 14.
(xlxz)
(z1z2) ((~i(~z(~3)
VI.15. VI.16). REMARK. zI(t),z2(t), ‘pl (t
+
10-14
+ I),
qZ(t
q3(t
+
VI. 16 :
(0( 2 2 (4 [(Pz (4 v 473 v (Pl(t>> v (20 v X l ( 4 {(PI (t> c(P2 (4 v (P3 v XZ 22 ( t ) = ( t ) {[(Pl(t> (Pz (0 (P3 ( t ) v X l ( t > l v (8) v X l ( 0 [(PZ (t>(P3 (4 v (P1 9 1 ( t + 1) = XZ ( t > [ X l ( t > v v ((9 v X I (t> [(Pl(t> v (P2 (4 v (P3 (Pz ( t + 1) = 21 (0 Z.{ (t> [(Pl (t> v (P2 (0 v (P3 v (P1 (t>(Pz (t>(P3 v (3) v (4Z.[ (4 v (P1 (t> v (Pz 0)4% (0 v 4% (4 (P3 (4 (P3 ( t + 1) = 21 (4{XZ ( t ) [(Pz (t>9 3 ( t ) v (P1 v v (0 {(Pz (t>(P3 (4 v 91(4 [(PZ (4 v (P3 011 v XZ (41 . ((-9 21 (t> = 21
9
9
9
WI.4
METHODS OF SYNTHESIZING AUTOMATA
o o o o o o o g o o o - - -h 3 a,
w
c
h
w
c1 v1
*
3
a * 0 'cl
0 0 0 0 0
0s-
--
0 0 0 0 0 0
0s-
-
-
0
-s-s
0 0
5
u"
3 -
ogo
0
5 0
* a *
3 Y
0 - 0 - 0 - 0 - 0 3 0 - 0 - 0 -
a,
5
0 0 0 0 0 0 0 0 - - - - - - - 3
0
*
a c3
__
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
VI.41
A "RAT IN
MAZE" LOGICAL NET
P)
-0
8
* 1 a *
5
0 0 - - 0 0 - - 0 0 - - o o - -
0000----oooo----
d 8
0 0 0 0 0 0 0 0 - 3 - - - - - -
Y
1
a U
~
~~
d d d d d d d d m m m m m m m m
285
286
METHODS OF SYNTHESIZING AUTOMATA
[VI.4
(a)-(@), (Yi)-(E) on
t are
wc f:4’
= ay
fi3’ = PY
fJ5’
, 2
= cp2P
v (Pza,
f:“’= cp2a v (P2V1
f
VI.41
A “RAT IN THE
287
LOGICAL NET
where :
The formulae thus obtained can be transformed to combinations of elementary valve circuit operators based on the following relations: (t) = =
7’2
f;‘“‘l> 2
‘pl (t + 1) = K, (T2 [XI, .I?“], Tz [.%> ~$2’1> 2 q2(t + I) = K, {T, [XI, fk”‘], [XI, .fi2’l> 9 b3(t + 1) = Kz Further: p= (x2 v j;3’)(z2 v $5,) = P(x,,f:3ydyX.2,44; because we obtain: We leave it to the reader to derive the formulae which represent the remaining functions in terms of valve circuit operators. Their final form is: rP1 = T(R),
=
= = = =
=
= =
= =
=
= =
V
= =
=
=
X3 =
=
V
V
=
(a,
=
=
= v = = = = T2f;(4) = =
The valve circuit which simulates the formulae we have stated contains 84 input channels, not counting the grids required to generate the negation functions of the variables x1, x2.
CHAPTER
VII
AS YM P TO TI C ES TI M ATI O N F O R TH E CO M P LEXI TY O F LO GI CAL N ETS
VII.l. Fundamental concepts. Formulation of the limiting theorem The present chapter will examine in its most general form the problem of constructing an optimum logical net to realise a given bounded-determinate operator. The specific feature of this investigation is that its aim is to discover the asymptotic laws which become evident at high values of the parameters of the bounded-determinate operators (weighting, and number of input or output letters). This approach to the matter was first developed by ShannonI, as applied to the realisation of truth operators in contact circuits; it was he who obtained the first substantial results in this direction. The final solution to the problem formulated by Shannon was obtained by Lupanov, who generalized the problem to cover other types of circuit, including logical (truth) nets2. Section 2 is devoted to this body of questions. The following sections contain the material necessary to extend the Shannon-Lupanov limiting theorem to cover logical nets with memory. In the present section, we shall introduce a number of concepts and notations required for our subsequent exposition, and we shall state the fundamental theorems. These theorems will be proved and the designs which correspond to them illustrated in the following sections. In this section we shall confine ourselves to a few preliminary explanations. We shall examine all kinds of logical nets3 formed of a given arbitrary though fixed - system of elements {%JI,>, the sole stipulation being that it be complete; we shall term such a system of elements the basis. In order to state the problem of finding an optimum net to realise a given operator, an
1 C. E. Shannon, The synthesis of two-terminal switching circuits, BSTJ (1949) 82, p. 59-98. 2 0. B. Lupanov, A method of circuit synthesis, Izv. vyssh. uchebn. zaved, Radiofizika (1958) no. 1. 3 Henceforth we shall sometimes omit the word “logical”. 288
VII.11
289
FUNDAMENTAL CONCEPTS
by complexity, cost, reliability Q index of simplicity 8 by
L(Q), cost {%Xi} by
Q.
by by L(8)
least 8
of
L(8).
of optimum net minimum number of control grids. on
evaluate
the complexity unrestricted,
:
%Xi L(%X,)> 0
i = 1,2, ...,p),
p
11.
of
connections total number
8. on 1.
:
290
pIII.1
ASYMPTOTIC ESTIMATION
{m,} L , , L,, ...)Lj, ...
L,
2.
Lj
SLj Lj,
is
0 by Lj,
SLj by
0 8.
S,,,
S,,, SLB,..., 0;
S, L
(cL)cL,
by
c
do
body
k
A=2k= 1, 2, 3, ...),
rn (m=
(n=
2, 3, ...), 2, 3, ...).
n
m, n, k
VII. 1
29 1
FUNDAMESTAL CONCEWS
8m,n,k
N (m, n, k). by
( t ) = qi [XI ( t > 7 . * . y x m ( t ) , 41 (t>, *..)qii(t)] q j ( t ) = + j [xi ( t ) , ..., x m (t>,4 1 (t>7 ... qii(t)I zi
7
<1= 2k< 2k.
9
7
)
(7.1)
5
E.
L(m,n,k) Om,n,k
L, L.
L ( m 7 n , k )= OE Om,n,k *
1
do
L(m, n, k).
L(8)
L(m, n7 k). L(m, n, k ) by
L(m7n, k ) by
L(m, n, k ) on
m+k n.
by
is
z(t) x ( t ) , q(t). actuaZZy used
<
is
n
n 6 m+k.
(7.3)
m+k.
292
WII.1
ASYMPTOTIC ESTIMATION
L(m, n, k)
L ( m , n, k)
m, n, k
ern,,&
1)
by on
1
As 2 1
bound
speczfi cost L(YJl,) i s
L(YJl,)/(n-
fm, n
by by p
p
L(m, n, k ) .
.-.
-
We would recall that two quantities p , v are termed usymptoricully equal in some process (F v), if in this process lim ( f i / v ) = 1 ;this means that the relative error produced by replacing one of these quantities by the other tends to zero.
vII.11
293
FUNDAMENTAL CONCEPTS
LIMITING THEOREM. I f m +k+ L
11,
00
n/(m +k)+O, then
and
- ne m2m.2k + +k ___
(2" - 1).2k
(7.5)
ke
:
(7.6)
1)-2k(n-t k ) + n * 2 k .
k =0
n = 1, (7.7)
:
L ( m ) = L(m,1,0)
-
2"
(7.8)
@-.
by
L(m)
m by
L(m)
z.
by (7.8). by
by
L(m, n, k ) on by
1
As
2
L(m, n, k )
We would recall that N (m,n, k ) is the number of all the operators in Om, L. C. E. Shannon, The synthesis of two-terminal switching circuits, BSTJ (1949) 28, pp. 59-98. 0. B. Lupanov, The possibility of synthesizing circuits from arbitrary elements, Trudy matem. in-ta im. Steklova (1958) 51. 0. B. Lupanov, A method of circuit synthesis, Izv. vyssh. uchebn. zaved. Radiolizika (1958) no. 1. A. Trakhtenbrot, Asymptotic evaluation of the complexity of logical nets with memory, DAN SSSR (1959) 127, no. 2. 1
294
[VII.1
ASYMPTOTIC ESTIMATION
m, n, k . by two theorems: theorem of the lower bound, on
of the upper bound
theorem
O E Om,&
E>O
(2" - 1)2k m+k
2"-2k :
theorem of the lower bound. OF For anyJixed E >0, the ratio of the number of operators of the class 6m,n,k, for which is true, to the number of all the operators of om,n,k tends to as m + k tends to injinity and nl(m + k ) tends to zero.
8n,,n,k. OF For whatever E >0, there exists a v >0 such that, if m + k > v, for any operator 6 E Om,n,k the following inequality is valid:
(7.10)
m+k
m+k+oo),
0 E 6,n,n,k A (7.9, E
L ( m , n, k ) ,
Om,n,k
L(8) O,n,n,k
A FUNCTION
295
THE ALGEBRA OF LOGIC
bound
is almost
optimumfor almost all of these operators.
precisely optimum realisation procedures
all operators.
by by
by bound. bound
VII.2. Synthesis of a net to realise a function in the algebra of logic; upper bounds for the complexity of a net
..., m-+co. For LA(m) stronger
A,
by
A
m
L L. A LB(m)
A m,.
m,
Lupanov’s method,
on “
V
x1 &x,,
x1V x2
2.
L,, L v , L,. of
1
no. 2, 2
-
1959.
296
pII.2
ASYMPTOTIC FSTIUATION
1, 2, 3
4
&-
by 5
6, systems 1.
( m-
x
perfect disjunctive normal form f ( X I , xz,...) x,), m f , , Zz, ..., 2")) xy' xy ... xLm)
L,.(f)< mL,
+ ( m - 1)2"L, + 2"L, m+co
L , . ( f ) < m2"L,[1
1:
+
(7.11)
by on
1).
by (7.11')
2. The use of a universal decoder.
on
1
xy' xy ... x",". universal decoder m
x l rxz,..., x,.
~~
We recall the notations: o (j?) signifies that a/j? 0, = O ( b ) signifies that, some constants m,M (0 < rn < M ) , m < a/B < M is true. particular, o ( 1 ) denotes a quantity which tends to zero in this process.
a = a
+.
v1r.21
297
A FUNCTION IN THE ALGEBRA OF LOGIC
m-,
+
<
+
=
co :
+
..., :
+ L"] [l + 0(1)].
L,.(nz) <
'. A
3.
by
j: I):
f (x17
* ' * 9
YIY
.'.Y
yfl) = Vx;'
'.' xp'f
xyl
... xp'.
(Ol?
YlY
''.>Y n )
l-+co, n+co.
f ( o , , ..., o , y , , [I
..., y,,)
:
on 1
n.
n 1
-3
m),
C. E. Shannon, The synthesis of two-terminal switching circuits, BSTJ (1949) 28,
pp. 59-98.
298
[VII.2
ASYMPTOTIC ESTIMATION
m],
[
m -Ylg
ni
[2L,
+ L , ] + Tni (
m-3
nz)(L,
+ L,)
1+
+ L , ] [t +
L3"(ni)< -[2L, m
(7.16)
by C .
1
2
by
of
1 = nz -
(nz - 3
In).
Lupanov's method 2,
4.
by &-, V -
f
m X17X2,
...,x,?,
01, v2,
...' U P ,
Y2,
-..,Y v >
Yl7
il+ p
+ v =rn)
f by
VIT. 1 ; by
y , , ..., yv,
xg,x4,x5 .
''
.
xl,..., x,, ul, ..., up,
by 1.10,
(cf.
~-
0. no. 1.
xl,x 2
vII.21
299
A FUNCTION IN THE ALGEBRA OF LOGIC
TABLEVII. 1 XI
XA
v1
my2
. . . 7”
1
5 P
t’P
I
~
0
o . . .0
AI AZ
4142
. . . . . . . . f ( r l , . . ., T A , 01,
.. . q v
. . .,ow, qi,
. . .,4 9 )
1,1, ... 1
x
no x,, ...,
u, y
u, y ) , f ( z , u, y), stage A , , A,, ..., A,, A,, by S ) . f ( x , u, y ) on A,
z
zl,
z
..., ,z,
on.
S
2’ 0,
y)
on
(7.17)
REMARK 1.
zl,..., z, o,,..., o,, x,, ..., x,, u,, ..., up. A ..., ,z, ol,..., op,y,, ..., y,)
S S At.
300
fvII.2
ASYMPTOTIC ESTIMATION
i
0,y)<2',
< ((2'/S) +
px
x
second stage u, y )
v
ji(X,U,Y) =
x?, ..., X?ji(zl, ..., Z I , O , Y ) .
(7.18)
II,...,ZTL
:
P
f(x,u,y)=
V V
X'fi(z,u,y)*
i = 1 TI,...,I>.
REMARK 2.
f i ( z , u, y )
+
p x 2 I < ((2"/S) 1) 2a.
Thirdstage.
v, y ) , v: ,.(if
0,
y) =
v 0 19 . . .
v?
...V P f
(z,
01,
..., 0&,Y ) .
.aw
f
7. I)
...... ..
;j
L Fig. 7.1
(i\l
VII.21
301
A FUNCTION IN THE ALGEBRA OF LOGIC
y , u, x
A,
v+ co
+
L(A) <
A*
I + co,
p-+ co
0(1)], L(C)<
+
L(B) <
+
y),
CT,
((2'/S) +
x
1). 1).
S
L(A*)<
(: )
v;'u;2... V P f (z,
cr1,
2'SLV
...),'rc
y1,
...)y , )
by A* :
G ) G
fi(z,
0,
f i ( z , v, y ) ,
+ 1) 2*
by
Y). :
L(G)<
6-+ ) 1
+ 1) 2*
G.
+ 1)
x;l
L(H) <
... xyJ(zl, ..., z, v, y ) f
:(-+ )
1 2*(L, + L " ) .
m+
00
302
WII.2
ASYMPTOTIC ESTIMATION
v, p, A, S :
) I
v=
=
m- v-p.)
G: m
L (A), L (B),
W*), L (D), L (H),
(C),
o
W) <-[I + m by 5.
general case,
by
G
ul, v 2 , ..., vp. G
1.14)
) = F { y o [ui ...)u p , f i ( 7 7 0, -..)0, Y ) ] . . . > O ~ ~ ~ Y ) ] ~y 2. -r .- >1 [u,, . . . , u p , f i ( z , 1 , . . . , 1 7 1 , ~ ) ] ,
fi (7,u, Y ~1 [ ~ l .,. . , u p ’ f i ( z , O ,
9
9
X2r(U1,
..., up), ..., XN-I(V1,
...,up)>,
F cp(x,,
cp,
r
..., x,), cp,
L,,
L v / ( r - I), F 1
See Chap. I, Sec. 4.
by
v11.21
303
A FUNCTION IN THE ALGEBRA OF LCIGIC
cp.
<(2"/(r-
l),
L,
F,
cp,
(L+ l)., = Q.2" + L,.
L, < r - 1 no
1-, &-
by
1-, &LV.
L,, L ,
-
-
C , A*,
A, As
D
G',
G,
(7.23).
u, y),
Yi[ u l , ..., v,,, ,fi(t,(rl ..., cp,y ) ] , p2'x 2', r
xi
Yj
.2
x i ( u , , ..., up), p 1
+
2,
p
[1 +o( l ) ] L,.
2" :
< p*2s2p2p+'[l G'
+ u, y ) ,
by u, y )
D'. (7.23)).
(7.21) A, p, v, S, m-, co L(G')
In the same way as the number of functions generated by block D. by virtue of constructing F from q (xi, ..., xr), it is clear that the number of variables in F exceeds 2u by not more than r.
304
WII.2
ASYMPTOTIC ESTIMATION
REMARK. cp = L v / ( 2 - 1) = LV ; (7.22).
cp,
(7.24)
6.
L(m, n) L
=
L(m, n, 0),
by
0E
L. by
OEOrn,,,
n
m
0
zi =fi(xl, ..., xm),
n =I,
2, ...,n, by Q.2"
+
q e ) < n-[1m
(7.25)
VI), by by
(7.25)
on
hgh
by by
i6n).
m, n,
on
(7.25)
k=O)
n)/m)=O, (7.4),
1.
bound
m, L(8) 3
2"an m
-
El ,
(7.25')
1 (E>O i s
L(m,n)
-
n *m2 m
).
(7.25")
VII.31
305
A SIMPLE REALISATION
VII.3. Operators which admit a simple realisation no
m
n.
by
1,
3,
of 1. x ( l ) , x(2), 2(2), ..., z(p).
p
..., x ( p )
l} p
x l , x2,..., xp on
z1,
...)zp.4
= O[x,,
..., x p ] ,
(zl,
by
..., zp)=
zi
1 S. V. Yablonskii, A family classes of functions which permit simple generation by a circuit, UMN (1957) 12. M. A. Gavrilov, The theory of relay-contact circuits, Izd. AN SSSR, MoscowLeningrad 1950. G. Povarov, A new method of synthesizing symmetrical switching circuits, DAN USSR (1955) 2, pp. 115-117. 4 This is the case, for example, when a parallel adder is used instead of a serial adder.
306
pII.3
ASYMPTOTIC ESTIMATION
x i + l ,x i + 2 ,..., x,: z1
=Vl(XJ?
z2
= (P2(Xl?X2),
Z”
= 4op(X1,Xz,
. . . . . .
...J ”).
q l ,q2,..., qp
qn+
I=
..., qp p 2 ,n = p -
q l ,..., q,,
n
n
n on n
on I=p-n):
+ 1, n
..., p
p2
q l ,..., qp.
p-00
:
@.2”+1 L ( 0 ) < -* P
+0
3
do
(zl,..., on
n=m), zi
..., xm] on
xj, S(i)
S(i).
no
2.
p). 2k, :
by Z(t) = @
qj(t + l)=
[X(f),41(t),...,4k(t)],
vj[X(t),41(t),...,qk(t)].
VII.31
307
A SIMPLE REALISATION
...) x,]
..., z,)=O[x,,
p =3 ;
k =1
:
(Ql7 1\
(9 = Q, (9Y4 4 ( t + 1) = q X ( t M t ) ] Y
Fig. 1.2
by
is t: z, = @[X,, q z = y [ X i , 411 9
ZZ = Q,CXz9q213 q3
= ~ [ X ZqY 219
z3 1..
= @Cx3,q31Y
.
by C 7.3 on p. by
p
7.2 C x p7
C
308
wII.3
ASYMPTOTIC ESTIMATION
4
=
= '2
'2)
. . . . . . . . . . . (z1 z2
...
x2, ...,
= 0,
,
1,2, ..., p, ...
on, is
p.
0
1
EXAMPLES A.
z= x + y x, y
p
(Sl,S,, ..., S,)
z,, .... z,)
(zl,
: z1 = x 1 0
9
= x2 0
z2
s, 0 Y,,
. . . . . . . . . . . . . . . z , = x, 0s, 0 Y , s, = 0 , sz = x1 ( Y , v 9
v
. . . . . . . . . . . . . .
v
v
=
C<2(L, + L , + L v ) . z = x + 1,
zi =
0 Si,
S i + l= x i s i ,
S,
=1
=
1,2, ...,p).
C < L , +La,
x,,x,,-
x
...
z(t)= Z(t)'Y(t)
z,,=fn(xl, .... x,,,yl, .... y,,), x, y
v [Z(t)O Y(t)]'Z(f
- l),
1,
VLI.31
309
A SIMPLE REALISATION
zi= X i ‘ yi
v [Xi0 yi3
‘Zi-
,
1
C < L , + 2 L , + L ” +L,.
C. on
...). 2
A
x2,..., x,, y,, yz, ..., y,), <xl ... x,)
(yl
A“
3.
by
... y,)” s
8
A””
A”.
8
R 8,
8 on 2. by
on A”
8 on on 4
3
K,, KB,K,,, KBt,K,,. on
by
A” on
on f(x,,
A”,
A/A”
..., xs), L(f) <
2” S
+
A“ =
S
[1 +
3
310
vII.3
ASYMPTOTIC ESTIMATION
A""
<
S
L-1+
lxtr ..., xI, K,, K,, K , on Ka f ( x , , ..., x,). A" (2"'<<<2'), Ixl ... x,l
f ( x , , ..., ,x,
K, x1 ... x, q1
2"xA L ( f )<
2"l. m+k
... qr,
xm+'
q t , ..., 4'). lqt ... qrl
+
E K ~ , m f ( x l , ..., x ,,
K,,
x1 ... x .,
xt ... x,qt
... qr,
+ &+
00.
q l , ..., qr). I,+' lql ... qEl
(xs... xml#O. m+k"-+co.
K,', K,. on
... zk zs, ...)T,J
17,
'sl
by
Ks, K ,
...)x,, q t , ..., q L ) = f ( x l ,..., x,
... 71'
... 4'1 < A
q 1 ... qk
zt
K,,
... z').
K,, K,, E K,,, K,). Also 8 T, ... zg lql ... qkl > A L(f')< L(0) by K,.
on qt
... qr
q1 _..qr
L(J')+L(d);
K,; K,
K,,
no
-1.31
311
A SIMPLE REALISAllON
A7=1' x 2' 1.
2' 2s- 1 Q 1 Q 1, = 1'.2' Q 2". A'Q2".
p=s-v;
f
f (ol,..., o,, x , + ~..., , x,+')=O L(f) <
...,fl,lr,X,+l,
lo1 ... o,l
21'.
...,XIC+J]+ LP,) + A'@&
+ L"), xl,...,x,,
D, (7.12))L(D,)=O(2') v=s-3
p=3
1' x 2' = A v
N
s+co,
I,
4. Q Q' c Q T (4;
p+m.
k
2a;
m +k"
2" x 1 x1 ... x, q1 ... qa;, ... 4); E Q'.
(ql
... qE)E Q',
8
qS(t) = yj(xl(f),
.--7xm(t),ql(t),
j = 1,2, ..., l),
T on ?m+X
L(8) < QR"-
m+k"
...,qn(t))
m +k+ co) :
+ 0(1)]. &),
on
1
It is henceforth assumed that v
by
< s.
312
mI.3
ASYMPTOTIC ESTIMATION
z1 ... zi, Q',
A
q1 ... qk
Q'
... zk). Kb
0 :
(*)
co,k =
= 3.
by KBt.)
x1 ... x,
of
... x,.
xl, ..., x,
...,x,=o,
8
Oun+
x1 .._x, q1 ... qi,
... q;
q;
Q'.
(**)
8u,+l.,,bm
x,
+
,'
, ... x, ,)'
el,..., O,,
of
..., OP 8,
8,,+ :
,...,
,..., qk),
j = 172,...,k7
8,;
V,(x,+ ...,
1
on
c,+
.cm,
8,.
durn+
Y j ( j = 1,2, ..., k),
8 = P
=
v
1
...,qk).
VII.31
313
A SIMPLE REALJSATION
Block H,. on+
... nm
u1
ri .....
... ulgP
v(v
......
.....
v;
......
Y ;
~v,
F VP
%
OUn+ m--R
Wl)<
112
- 7l
+ 4111
V,(X,+~, .... xm),
Block H,.
u,,
V,,
*
.... V,, .... V,.
.... ulgp
H,,
JwL< ) Block xl, .... xn,q,, .... qn
+ YJ);
L(U) < Yj,
Block V,,
F
Y y Y
V
kxp
L(F) = k p [ L ,
+L,].
by
314
[VII.3
ASYMPTOTIC ESTIMATION
+&).
L(H,) <
2
[l
~
+ ~ ( l ) =] e
K,, K,, K,, Kpj
2"-1
A-
m
[l
+ ~(l)],
K,.
-
bound, upon.
-
by
5. A
by DEFINITION. A
(z,,
..., z,)
_.., x.]
= 0 [xi,
z=p(x)1
m =n =k .
by b - ' ( v ) v
p ( x ) 3 v.
x
/?-
x).
p - ' (1
x = 7 , p(x) = 10.
zk zk = [x 3 1
If we interpret
...,z,)
of
z),
. and (XI, ..., x,) as binary notations of the numbers z, x .
315
A SIMPLE REALISATTON
~____
0 0
1 1 1
0 0 1
1 1 1 1
0 0 0 0
1 1 1 1
1 1 1
0 0 0
0 0 0
0 0 1
1
1
1 1 1 1
0 0
0
1 1 1
0 0
0 0 0 0
1 1
1 1
1 1
0 0 0
0 0
0
0
0
0
1
1 0
3 4 5 6
0 0
0 1
0
0
0
1
01
01
I
0
1
1
1
8 9
1 1
0 0
0 0
0
10
1 1 1 1
0 0
1 1
0
1 1
0 0
1 1
1 1
0
0
:I
1
11
12 13
15
0 0 0
1 :
1
1
1
1
0
... ...
zk-'=
v v v
< x < fi-y100 ... v < ... v ... < x ... v ... < x ... v ... < x].
<XI,
...
1 (7.29) by
VII.2, z4 = [7 <
XI,
z3 = [3
< x < 71 V
< XI. j
z.
x2p-'(v)
316
wII.3
ASYMPTOTIC ESTIMATION
x
=
- 1<
< C Sk),
x > p- (v)
+ L , + L” + L , ) .
cj <
by
-j
k-j-tco) 2k
EL-j
LJ + Lr-
k,
0 L(Q) <
2k
= L(0)
k
7 [1 + 0(1)],
(7.30)
0 E ak,k,o,
-
L(O)
k2k k
Q -=
e 2k
(7.25”)).
REMARK. z4
by xl, x2, x3
z3,
VII.2,
x4
zl, z2, z3
z4,
by
As
+
0. B. Lupanov has recently obtained a better estimate: L ( 0 ) < 2p(2”/k) [l o (l)]; see 0. B. Lupanov, On the principle local coding and the realisation functions of certain classes by circuits of functional elements, DAN SSSR (1961) 140, no. 2.
VII.41
317
CHOICE OF A CODE
VII.4. Choice of a code 1. by by
by tetrads) 0, 1, 2, ..., 9
by 8
A,
K
8
A A)
A).
on
on
2k-
by by
+
L ( 8 ) < @.2k[1 0(1)].
(7.31)
by
A K 2. We
K=2k.
6 by A
by
EXAMPLES.
b A on
by
318
wII.4
ASYMPTOTIC ESTIMATION
TABLE VII. 3
II
Input
output
i
Input
output
i
1
j
i
k
0
P k I m n
P
0
I In
n
TABLE VII. 4 output
Input
6, 5
Input
i
I
j
k
rn b
I
I
IIZ
0
n 0
m P
P
0
3
b
3
output
k).
To
1) 2)
on
+ 1. 7.5b
VII.41
319
CHOICE OF A CODE
e
i, p
3.
2, 1 , O . A is
(b)
Fig. 1.5
by by 3 . The numbering of nodes.
Ki
320
wII.4
ASYMPTOTIC ESTIMATION
i=O, 1, ..., S ) . 0, 1, 2, ..., K O KO+ 1, ..., K , + K , - I ,
KO, on.
on 0 on
I
on.
K,+K,+
...+Ki,K o+K l+ ...+ Ki+
0, 1, 2, ..., Ko+K, +... +Ki-1 i i. ..., K o + K 1 + . . . + K i + K i + l - l
+ 1;
i+ 1)
a, b a'
a', 6'
b',
b.
a by 7.5.
8
4. KO,KO+
8
..., K 8 on
by 0,
KO
>KO 8, 0, 1, ..., K - I .
on
v I,, I,, ..., 1,.
0 = 6 o , G 1 , ..., Dv-1 , G , x
0, 1, 2, ...,KO- 1 on
= KO
li,
< x< di. by 8,
VI1.41
321
CHOICE OF A CODE
ui-l < x < a i . +2Zi- 1, ...)ciPl 2) T ( x )=x + 1. Ii=3, oi=26, C T ~ - ~
1)
ci-l +Zi-ly T(x)=x-(Zi-l); ci- = 5,
x
( X - G ~ - ~
Zi,l
7.6 8(24)=25 8
5. The synthesis and estimate. z=O(x)
A, ByC
7.7).
Fig. 7.6
Y
1
A
I
Fig. 7.7
BLOCKA. K=2k
xlxz
... x,, 8
:
Zi- 1 (i= 1, 2, ..., v); 2)
8,. by
A; 1
by
a b indicates that a is exactly divisible by b.
322
wII.4
ASYMPTOTIC ESTIMATION
i = 1,2, ..., v),
qi(xl,
S(x1, ...,xk), t(x17
.*.yXk)
by (pi(x),S(x), t ( x ) ) ; = [ai-,
(pi(.)
< x
+ 1): 41,
&[(x - CT-1
S(X)= [x < av & qi(x)]
3
i$v
t ( x ) = [av
< XI.
oi<x
v
by li,
x 3 ai
L(A) < v-[C,k
v
ai > x)
(pi,t , S
+ C2k2+
+ 2L,]. Fi(x)= [x-Zi
i= 1,2, ..., v ; O,(x)
w
2v
4v
+ 11,
x+l.
O,, L(B) < v.C,k
2k + C , k + 2~-[l k
k
+~(l)].
8,
A v : = (V
+ 2)*k[L, + L,].
k-+
+
L(8) C2vk2
is
v
1,
1 + 2 + 3 + ... + V = +(l + V ) V
2 2k
< 2k,
+
L(A) k.
VI1.51
323
SYNTHESIS OF NETS H A V I N G A MEMORY
< 23(k'1)
8
L(8) < C k 2 * 2 f ( k + 1 ) .
VII.5. Synthesis of nets having a memory
8
bound p
8m,n,k,
Nm,n,k,
1.
by
As
(xl, x2, ..., x,,,)
by
(zl,
... qa,
by
z2, ..., zn).
k 2'. :
z i ( t ) = ~ i ( x ~ ( t,...,x ) ~(~),4~(t),...,~E(t)),~
qj(t
+ 1) = ~ j ( x 1 ( t ) , . . . , x m ( t ) , q l ( t ) ..-,q~(t)),j ,
i= 1 , 2 , ..., n ; j = 1, 2,
-
..., k . L
by
324
WII.5
ASYMPTOTIC ESTIMATION
n
+ k"
rn +f
f Ld).
rn
+k"+
00,
LLd,
(7.35)
rn +k+00).
m+cc
k=k", bound
Qi, !Pi
by
by
8,
1
R x1 ... x,
by Q' 40,41,
...?
4n-1
-
z1 ... z,
VIIS]
by
325
SYNTHESIS OF NETS HAVING A MEMORY
p = 2'-
p
A) qn,qA+1, ..., q2n- 1,
qr, Y(0, ..., 0, q )
Q' 4). qz.
by
qn, ..., q2fi-1
by
Y
Qi Qi(X1,
..., X m , 4 j )
= Qi(X1,
y ( x i , - . - , x m , q j ) = YJ (
..., X m , 4 J , )
~ 1 ,
-.a>
xm,
4,)
7
5
..., 2'--1. 4,
2E
k"
Q, Y(0,...,0,q )
by
Q
0(2' (Ig i ) / k ) ; (11)
(11) qn, qnfl,
...,
qj ( j 2 1) by
q,, by
8,
4j(t
zi(t) = Qji(xl(t), **.,xrn(t),41 (t), qe(t)) ) = Y j ( x l ( t ) ,..., X . r n ( t ) , q l ( t ) , . . . , q E ( t ) ) , ) 9
+
0 {!PjIj6',
(@JiGn,
k"
+ L { Y ~ ]+~&. , ~
q e )<
Q i ( i = 1,2,
A.
Qi(x1, -*.,Xm,41,
lql ... qEl
...,n), by (zl
is . . . , 4 ~= ) Qi(X1,
...,Xm,tl,
... T ~ ) ..*PA),
326
yvII.5
ASYMPTOTIC ESTIMATION
Kfl,
Qi
p. 310):
{Yj}js~ in 2E>
on
2n
+k"+
B,.
00,
{ Y j }j s L K,
4
3). (7.40)
(2"-1)~2~
(7.38)
(7.39)
(7.40),
ELd
(7.38'). 2
2'+ca
m+k"+co,
x 1 = x 2 = ...=x,=O,
..., x,, q l , ..., qE) 0 Yj(xl, ..., x, q,, ..., qn)
Y;(x,,
1, 2, ..., k,
: Yj(X,,
... x,, 41, ..., qn) = 2-,TT., ... fmYj(O,...,o,q1, ..., qr) v )
v ! q X 1 , . . . , X , , q l , ..., a).
L { Y j } j < n L{YIJI}j
+
Ky.
+
p. 310). L{YIJI}j<ji j<E
2 4 co, (p. 325)
k
N
k (in k"+co,
< Re.
(2" - 1)n
in + R
g+
00).
"1
+ 0(1)].
+ LL).
VII.51
327
SYNTHESIS OF NETS HAVING A MEMORY
o(kp(2"-
L{!Pj}jQli < As
k-
k.
f e e -
A/(m+E)),
(2* .[1 m+R
+ 0(1)].
(7.41)
(7.38)
(7.39)
(7.41).
8'38
no
8
a
8'
8
bound is
APPENDIX
As body
book
book ;
by
11, 4-6,
5,
IV,
VI-VII no
A
0.
2.
body on A. on
by 1 2
In the present book, as applied to logical nets. See the References the end this book.
328
A.
329
APPENDIX
by
0. B.
by by
Sec. 3). B. by A.
A
in
11, 5.5,
1-4
1-3,
V).
by by
by by
:
3, 21;
on 3),
A. A.
V.
IV,
4)
by IV,
by
S. C .
by
V. M. on The bounded-determinate operator is sometimes termed an automaton operator. These terms are employed (but not alike by all authors) to indicate particular features of the canonical equations. V. M. Glushkov, An algorithm for synthesizing abstract automata, Ukr. matern. zhurnal(l960) 12, pp. 147-156. 1 2
330
APPENDIX
by
A.
A.
A A. 5
V,
A
by by
V.
E.
V.
by
L.
on
on on by V.
by body
See also the article by Elgot[651.
is
REFERENCES
by
1.
J. 1956.
no. 34), 2.
11.
(1960) 21, (1960) 21, 3. 3.
3, 3,
1959.
Yu.
4.
1958.
by
5.
no. 3,
1960.
6. 1956.
by
7. 1950. 8.
(1954) 15, 6.
D. no. 3, D. Yu. V., V. M.,
9. 10. 11. 12. 13.
i 14.
1961. 1947.
by
(1961) 141, 5. (1961) 16, 5.
(1961) 3. I., 51,
by
1958. 15.
I., 1958.
16. (1960) 15, 3. S. 1962. 18. S.
17.
11. 331
332
REFERENCES
19. by 1949. of
for
20.
132, 2. 21.
130, 6. for
22.
141, 6. A.,
23. 24.
2.
of
A.
for
4. 25.
10, 5. 26.
0.
27.
0.
28.
0.
A. A
51,
by
1958.
of
1. by
140, 2.
29. 5,
1961.
A
30.
1956. 31.
by 1952.
32.
1, 1960. on
33. 34. von
11. 35. by
1959.
36.
18, 2. 37. 1955. 38. 1954. 4,
39. 1960. 5,
40. 1961. 41.
112, 6. of
42. by
118, 4.
333
REFERENCES
A.,
43.
51,
1958.
44. 127, 2.
45. 138, 2.
46. 140, 2. A.,
47.
1948. 1,
48. 1958. 49.
2,
1959. 50.
51. 99, 6.
by
52. by 1959.
53. 12, no. 18, 1959.
54. 5,
1961.
55. 51,
by
1958.
56. 1959.
2,
by
57. 1.
58. 6, 1.
und 59. 60.
41, 4.
G. 261
34.
61. 1956. 62. 2.
63. p. by
64. 5, 2.
65. 98, 1.
334
REFERENCES
J. B.
66.
(1959) 6, 1. 67. (1958) 5.
on
68.
(1960)
96, 3.
D.
69.
(1954)
257, 3. 70.
D. A., (1957) 4, 1.
71. (1957) 4, 1. 72. (1956) 6, 2, 3. 73.
(1956) 5, 10. 74. (1952) 58, 8. 75.
J. (1959) 3. 76. 77.
A., 1954. E.,
2,
BSTJ (1949) 28, 1 .
INDEX
A
97 -
267
16
109
-, 16 -, 13
-, 109 64,83
-, 27
-,140, 142,
-, 21,42
257
-, 85 -, 241, 268
65
-, 76 -
K,, KO,K,, Kd’, K,,, 309 L (m,n, k),
76
-, 158, 178 29
68 160, 189
-, 162, 189
Om,n. of
1
27 290
12, 317 14 214
-, 4 -, 6, 102, 157, 174 -,1
-, 21 5 14, 23, 31, 34, 35 -, 31 31 -, 32 -, 23 97
- (see also 82 -, 3, 64,83 -, 2 -, 6, 8, 102, 182 -, 3
B D
85 241,242,276, 296 68 -, 76
C
110 50
14, 22, 31 -, 31
-, 59 -
60
31
-, 35 -,32 -, 22
76 79, 159 335
336
INDEX
74, 164, 165, 167
-, 166, 187
-
14
(see
(see
-
14
E
-
5, 64, 90, 233
-,288
-, 233
3, 82, 101, 288
-, 102, 288 -, 86
-, 97 -, 88
-, 146 -, 5 -, 83
N
-, 100 210,211 18
14 on
42, 84 165
0 64
-, 74, 75 Om, L -, 290 -, 70, 172 -, 68 -, 76 -, 75, 164, 165
F 126, 137 97
-, 68
-, 314 -, 68 -, 64,76, 98
32, 56 -
-
-, 74
122 84
-, 74 -, 68
-, 75
-, 37
-
-, 84 107 215 -, 21 6, 21 8
74
290
-
P 105, 109 169 49, 50, 204
L
-, I6 -
Q
50
16
50
337
INDEX
-, 60 -, 60
-
-
51 51
-
160 198, 253
71, 186
-, 76 -, 74, 192 -, 186 -, 188
R
-, 186 -
-, 72
71
-, 71 66, 105, 116 3 212
64, 76, 98 103, 150,
-
-
-, 71
176
154 103
-
226 215 215 226
-
V
-, 52, 59 on
-, 42, 84 -, 52, 59 -, 84 -, 16
S 2, 64
-, 76 -, 75
84
W
T -
-, 32, 56
75
-, 290
