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M 34
Ma~ewaruwaceroniig (C~OPHIIK CTa-
~ e i i nepe~on . c anm.).M., U ~ H ~ H U ~1974. B , 64 C. (Hoeoe B XHSHH, HayKe, TeXHWKe. Cep~fl %MaeMOTwa, rr~fieptre~a~aa, 7. kf3AaeTcfl exeMecasHo c 1d7r.) B nepsofi c r a m e6opnn~aonncan ueron @ ~ ~ M H ~ OHBnpudnn~HHR XeHHOrO O C T H R I p a 3 ~ ~ 6 $ q a n n~ xBKTHPeCKHX aanaP SKOHOMnKn. ncnxonorp * ~ ~ T B W $ $ ~ I $ e gdnacrei?, a ~ompblxo c n o ~ ~ y w ponb nrph5 hoBi?fieIi~e, xapdtM?ptoe nnr rrrns~x cyqecrs. BT0paq crarbrr nocssqe~ia npaKTsPecKolay npsuenennm reopstr arp 06J?aCTH MBTeMaTHKH. WCUlenYK)lI(efi CHTYBUHA, B KOTOPUX CTanKneamrcr unrepecbl pa3nn~nblxnmnefi, oprann3aqnR n T. a.
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HeMIIeTKHX B b i C ~ a 3 b t ~ a H UaHBnOI'H4HO fi 3aAaHUIO o ~ H ¶ H Q B @YHP UHU aaBn~uefinap B u n a (x, f (x)), x s ~ a s t t ~ uaprye MeHTa, f (x) C o o T B e T C T B m e e 3HaYeHHe @ ~ H K U U U . BMWTQ ~ a 6 . nO ~~~ ~ I ~ H Y@KYIH K L I N ~ MOWO OnpeAeJImb aamp~waWCKU (T. e. C nOMOl4bIO I I p o r p a ~ ~ b IB) TOqHO T a K lfCB C Q@ HerIeTKylC M O u b I o H e q W K O r O aJIrOpUTMa MOXHO OJlp@eJrUTk @~HKUUK).TOx e c a M o e OTHOCHTCR u K O n p e n e n e H u x o MHOXWTB, ~ e s g r f u xm~orrre~uB H npyrax a h e ~ m s . no C B MCyTU ~ H e q e ~ K U f iBJIIDPUTM 161 n p e A ~ a B A I f e TCO@% ynopqoreasym n o c n e A o B a T e n b p o c T k UHCT Y K Q H ~ (no~ Q ~ ~ HnpoQaHM!2 CI AJIa B ~ ~ H C J I U T ~M~a ~w UHblf,' H O ~ HeKWphl. N3 QQTQPbIX M O r P C O A e P X a T b CUMBOJIH q e S q K u X M H O X ~ T P , H$IIpaM@p: ecnti y eenqm, TP WMMW y m e ~ b w ~ mx b ecnu y He O y e n b eenum u tie OWN^ m o , mo o w ~ pi? b IWM N W YBeAUYUmb X ~ C J I U y MMO, TO cmn; WJIU tiem, mo yMnuuumb x H a 2.
-
rae
-
nw4-
To, ao B a n r O p w r M e A o n y c K a m R BHCTPYKUHB H O ~ O T u n a , ~ O ~ B O J I H npu6JInxe~no ~T O n u c m a T b c ero noM O U ~ W came pa3H006pa3nb1e c~oxuiwR B A ~ H H R . Baqno $YAY(IU HeqeTKHMU n0 C B O ~ %RPHP@4et T a K B e Q p w a g U f l TO, MOrYT 6 ~ nWHOCTbJ0 b W e K B a T H b I UarlSM ~ Q C T ~ W ~ 3aH O ~ ~ nasn. Hesff~uea n r o p u ~ ~ TaKQg'o bl w r y T aaTb S ~ K TuBHare cnaco6rJ npn6naxe~~oro onsawin u e ~ e q g g@YHKNEB, O I ' ~ ~ B U S ~ H H~ ~ H~ ,K U H O H N ~ O B ~ HCHCTeMbf, IIR CTP~T~SH~ w T. A. B nanb~elrrrenrMIA oCTaHeBwMcn H a H e K o m p w x n3 OCHOBHHX CB~Z~CTBJ I U H ~ B U C T H ~ ~ C B H nepenreaawx, X HeveTKwn BMC K ~ S ~ ~ B U~ He4WKUX H U ~ aJifopUTheoB. QH@KO Mbi H e 6y.W nwraTMI-s n p w a B w T b H a m nonxaq ero npwwxesua B KaKOM-TO OKOHsaTeJl bHOM BUAe. %y CTaTbrQ CJI6?&yeT PflCCMaPkfe a ~ bn p w e B c e r o K a K p p e m a p w T e J r b n o e o614ee o n H c a H w e w m a , senammema mxo40~OT TP~IJU~FUTO?HYTU H.c~po~ % T H B HaYqHoM a~anuse,41ew~a;n p u & u x e ~ ~ anpwpma n KOTOpOrO o'rpaXae'C HeOnpeAeJleHHOCTb n O B g e H k S ~ ~ 0 5 , ~ s e ? y n 0 3 ~ a ~MyO X ~ Tc-raTb O C H O B O ~h~ ~ e pee a n H c T f i s w K o r a m a ~ q s sr y M a n s c T u q e c K n x cHcTeM.
pwa
u
a
M3 n O c n e A y I O m A x pa3neno~~ Y A ~B HT ~ H O , s ~ oT e o p m u s e c K w e o C H o s a H m H a w e r o nonxona H a c a M o M Awe s n o n H e ~ o s HbI A MMaTeMaTAqHbI f I O AYXY. fieJl0 B TOM, ¶TO UCTOgHHKOM H e o n p e n e n e H H o c T u B H a w e M nonxone n s n R e T c R H e n e x a u l a f l B a-0 OCHOBe T e O p A R , a cnoco6~HCnOJIb3OBaHUR JIJraHrBACTAW?CKUX IIepeMeHHbIX H H ~ ~ ~ KaJIrOpUTMOB U X B @OPMYJINPOB~HHU H P e U e H H A P e a J I b H b I X 3aAas. , ~ ~ ~ ~ c T B H T ~ J I ~ H OB C TKAI X , AOM cnysae c T e n e H b T O ~ H W T H p e t l I e H u R ~ o x m6b1~bc o r n a c o B a H a C T ~ ~ ~ O B ~ H N F 3aAaWi IMM U T O s H o C T b I o A M e I o ~ U X C R AIHHbIX. n o n o 6 ~ a f i~ H ~ K O CCOCTaBJIReT T ~ OAHY U 3 B a X H b I X W P T P a C CMaTpHBaeMOIW MeTOAa.
B
F A e CUMBOA U H f e r P A P O B a H A R 0 6 0 3 ~ a ~ aOeI I~ e P aUBK) 0 b a e ~ U HeHHR OfiHOTOqegHbIX H e W T K H X M H O X W T B PA (Y)/Y. Ecnu HOCATeJIb d COCTOAT A 3 K O H e s H O r O WiCJIa 3JIeMeHTOB, TO HHT e r p H p O B a H H e B (2.2) MOlKHO 3 a M e H U T b CYMMUPOBaHUeM:
+ +
A = PIIYI unu
n
A=
II. H E K ~ T O P ~CBOBCTBA IE HEllETKMX MHOXECTB A n r l T o r 0 s~06b1 CAeJIaTb H a W e w 3 n o ~ e ~ B n~o neH e CaMOCTORTeJlbHbIM, MbI A3JIOXNM T e C B O ~ C T BH~e s e T K U X MHOXeCTB, KOT O p b l e ~ O H ~ A O ~ H T CBR 8 a J I b ~ e f i ~ ~(60d'Iee e ~ . n o ~ p o 6 ~ o06e c y x n e H t i e n o n o x e ~ ~ iT ie o p A u s e s e T K u x MHOXWTB, memW A X OTHOUIeHHe K A ~ H H O TeMe, ~ COfiepXHTCR B pa6o~ax
(2.3)
k I ~ n
2
(2.4)
R/Yi,
i-1
,' . . .,
-
r A e sucno pi ( i =r 1 n) c T e n e a b npuHaAJIemocTu 3 J I e M e H T a yi MHOXeCTBY A. ~ ~ M W U Mq ,T 0 3 H a K IIJIIOc B (2.3) 0 6 0 3 ~ a s aO~~ % ~ A U H ~ H aH ~ H, e a p s @ ~ e ~ u s e c ~CYMMUPOoe BaHAe. nOJIHOe MHOXeCTBO = ljn) MOlKHO n p e A C T a B A T I i n p O C T 0 B BAA€! CYMMbI
KoH%?~Ho~
(Y1,. . .,
U
[I-171.)
XOTR, C T P O r O rOBOpH, B b I P a X e H U R 3aIiACaTb B BAAe
.
(2.5) li (2.6)
CJIenOBanO
6b1
.- .
MHOX~CTBOM H ~ ~ ~ I B ~ ~MHOX~CTBO, T C R H O C H T ~ J ~KOTOPO~O ~ CoCTOAT A 3 ~ A H H C T B ~ H H OT ~0 9~K H . Ecnu d OAHOTouesHOe H e 9 e T K O e MHOXeCTBO, HOCUTWeM KOTOPOPO RBJlReTCR T O P K a Y, TO M H ~ Y A ~ n M HCaTb
-
A
-
ply,
(2.1)
T o r ~ aHerIeTKOe
IIOAMHOXWTBO
MHOXeCTBa
U,~ O n H C b I B a e MAO e
nOHRTAeM UeCK0AblC0, MOXHO 3 a I I A C a T b B B A A e ( C W M B O ~= ~ 0 6 0 3 ~ a s a ep~a B e H c T B o no onpe~ejre~uro)
a ~npmepa l ~ npeanonoxm,
wo
X = {Toi, AUK), Y = {LIxoH, AXUM), TorAa 6 u ~ a p ~ oHeYeTKOf2 e OTHOuIeHUe C X O ~ C ~ @Me>Kny U 3JIe MeHTaMIhMHOXeCTB X U Y MOXHO 3anllCaTb B B U A e C X ~ C ~ S= O
-
TOM, AxoH)+O,~/(TC)M, AXAM) I+~,?/(AHK, AxoH)+O.~/(AHK, AXUM).
~ O M H M O 3TOr0, JaHHOe
OTHOlueHHe MOXHO
hp&!~$t)d~b
B Bli&3 MU?lpU@l O~OuLeilWr
U
A - H ~ W K OM ~ HOXeCT@Onp&?Qp&&dii,TO MOXHO HanHCaTb nposop~1~~" =cpell~eflo~ m a ~ o / A x u ~ cun~uo/b6
+
+
+ ~ o / , h ~ .
H e v e ~ ~ ncTeneHu e npunamexamn
MMO,
epedue
+
N ~ZUA~UO
R&RRICYCCR UpH 3TOM HeqeTKHMM RqBMHQXeCTBaMU IIOJIHOrO MHoxecTsa V, onpeAensreMoro cneAyiounM o6pasork
V = O + 0 , 1 + 0 , 2 + . . - + 0,9+1.
h
ROS=
J
Xx.2
(9.221 W y . ( P ~ ( X . Y ) A I ? ~ ( ~ , ~ ) ~ ~$),
(2.19) 1 Bblpallte~ne (2.22) onpenenner MaxcnMnnHoe apaa3~cnenneR H 8. Anano~qno OnpeAenReTcR H r+iaKcamanbHoe npomsenenne c TOR pasHHU~R, 9TO /\ 3aMeHReTTR apH MeTHVeCKHM npOH3BeAeBHeM. f i a ~ e e uonpo6~omi nponraene~" pacc$orpes. B pa6ore [2].
aVb = max.(a, b) =
a, n p u - a > b b, npu a 4 b,
(2.23)
Ilepeceqe~ueA
M
B 0 6 0 3 ~ a s a e ~Acn~ B u
onpeAenReTcR
pa-
BeHCTBOM
A n B &S((pA (P)A U
C h p E i ~ a ~ HHe, e COlo3bI U H WlU, YKJIOHYliBbIe nOHRTHR O w l i b , BeCbM, 6 0 / 1 b ~ , ~ e l i b u l eH R p y r U e TePMHHbl, KOTOPbIe BXOART B Onp-eHue s ~ a s e ~ u iniu H r B k i c T u s e c K u x n e p e M e H H b I x , M o r Y T PaCCMaTPHBaTbCR K a K CkiMBOJIbI P a 3 J I U q H b I X ~ I I e p a u H f i , O n p e A e J r e H H b I X H a H e q e T K H X nOAMHOXeCTBaX I/. Haubnee CyIQeCTBeHHbIe H 3 3THX 0nepaJlkifi 6 y f l y ~OTMeqeHEd H U X e . ~ o m m e ~Au e0 6 0 3 ~ a s a msepes ~ A H onpeneJIReTcR.KaK
1
AOnOJIHeHUR COOTBeTcTByeT OTpHUaHHiO. T ~ K H M X CUMBOn H e q e T K O r O MHOXeCTBa, TO f&? X CJIayeT U H T e p n p e T I l p O B a T b K a K ( ~ 0 u H e er O B O P R , AefiCTBYeT H a H e q e T K H e MHOXeCTBa, T O r A a K a K H e m A ~ ~ ~ C T B Y ~ T H a HX CHMBOiIbI; IIOHUMaR 9 T 0 , Mbl 6 y R e M HCnOJIb3OBaTb CHMBWH ki fit? B ~ ~ N M O ~ ~ M ~ H N M O ) . 0 6 ' b e A H ~ e ~ N HeqeTKMX e MHOXeCTB A U B 0 6 0 3 ~ a q a e ~ ~ ~ A $- B H OnPeAeJIReTCR PaBeHCTBOM
OnepmEiR
06pa30~,€!CJIH
1
-
1~
1
PB
WIY.
(2.29)
+ + +
(2.34) u unu v = 0,813 + 114 115 0,616 (2.35) u u v 0,713 0,516. n p o u 3 ~ e d e t l u eA N B 0 6 0 3 ~ a s a e ~ AB c ~ki o n p e A e J I R e T c R K a K
-
A
AB
=Ju PA (Y) PB ( Y ~ / Y -
(2.36)
kropoe r r p w a w a m co0oB xifner~oe ~ ~ o ~ ~ e c m aenaloro,
K W M ~OCHOBKMX
TOJIbKO ¶TO OrIPeAeJIeHHbIX ~ I E p a i t H f i , H M ~ A~Apyrue R onepaitmi, ~ o ~ o ncnanwywrcx p ~ e B npenCTaBJIeHWU JIHHrBKCTN¶eCKHX H W I X ~ ~ , ~ ~ J ~ ~ H HHOe C~ Tm~ p~ b~ I. e ff3 HEX 6 y f i y ~KPaTKO OnpeAeJIeHH (&met3 n o ~ p 0 06cy~fie6 ~ ~ H u e whx onepau~iiMOXHO ah^ B [151). OIXepauA5I KotlyeNmpUgWZHUR O l ' I p w 8 a R PaBeHCTBOM A
CON (A) =Aa.
(2.44)
46eca p c 3 y f i b T a W SlgHEaRHeHHR
B
F K
OBH09JIeMl2HTHOMY YHO-
TepMHHaX f( p e 3 y J I b T a T IIpHMeHeHEIR A A a m R paeexcrBoM
F
K HeqeTKOhrlY MHO-
fig
TeJIbHO BeJIHKO.
Onepauws pacmmwua
o n p e n e n n e T c R KaK C,
DIL (A) = A''=. Aefic~~ae ~ 0 f iOn$pa&HM p a w l 4 KOHUCHTPHPOBBHUR.
CJIeAyeT h H W p n p e ' f W p 6 ~ t n b K a K O & ~ U H ~ H U ~ ?c e % e R e r s a qeTKUX MHOXeCTB (9)K (Y), Y e U. T ~ K U06pa30~, M (2,48) a H a n o r n s H o UHTerpaJrbHOMy n p e n c T a 6 l n e H u m n u ~ e f i ~ o roneo p F p a C (y), UrpaK)uH?4 POJIb k i ~ l l ~ J I b C HXapaKTePHCTMKW. 0~ K a q e c T B e u n f i l o c T p a u m i (2.48) n p e A n o n o x u M , YTO U,A w K (II)O n o e A e m m g . TaK:
B
K
~pOTAB0IIQJIO)KIiO A ~ ~ ~ C T B A HO)n e -
one pa nu^ rcompacmrjo~u~rne~cu~urcayuu onpeaeJrcteTcR KaK
*a
O n e p a U U R oTJrU¶aWCR OT KOHUeHTPUPOBaHHR TeM, 9TO O H a y B M A ¶ H B a e T 3 H a W H H l I PA @), K 0 7 0 p b I e 6 0 J I b I U e 0,5, t4 y M e H b u r a e T Te, KOTOpbIe MeHbIUe 0,5. T ~ K H06paso~, M KOHT p a C T H a R N H T ~ H C H @ A K ~ I ~ AYMeHblllaeT R H W e T K W T b A (=pa Heqt?TKOCTH MHOXeCTBa, n o a o 6 ~ a ~ 3HTpOnHH, O n p W e J l e H a
116 1). Onepaws y e e ~ u m u f itmmnkmnu B r J p a m a e T c a B rrpespam e H H H o6moro (sw~aro) m i o x m a B H e q m o e HJIH B yse-
B
JIkI9eHHH C T e n e H A H e T K O C T H H e W K O N ) MHOXeCTBa. P a y f i b T a T IIpHMeHeHHR ~ o fOin e p a a H H K A 6 y o 6 ~0 3 ~ ~a ¶ a ~ b F~ (c At ) HJIH T ~ XK 3 0 3 H ¶aeT (( X I I P H ~ J I H ~ H T ~ J I ~PH;o~ B3>, H~ I3 TCl B p e M R K a K X. =!k@H@'JkI4 <X e C T b HeW?TKoe MHOxecmo qncen, n e x a u ~ xB O K ~ ~ C T H O C T Uwcna 3)). Onepam~
2.
-
-
T o r ~ aP€!3YJIbT%T ~ P H M ~ ? H ~ H WFR K' A ktMe&F BHA F (A; K) 0,8fl/l.+ 0,412) 0,6 (112 0,411 0,413) = 0,811 + 0,3212 + 0,612 0,2411 0,2413 * (2.52) = 0,811 0,612 0,2413.
+
+
+ +
+
+
+
AJIR
MaJIbHOM tf3MKe. HalIlMX QeJIefi R3MK L MOXHO PaCCMaTpHB a T b K a K COOTBeTCTBHe M e X A Y MHOXeCrBOM TePMNHOB W 06JIac~bmP a C C y W e H M R (&a TO9Ka 3peHBR O n H C a H a 60~1ee A e T a n b H o B I4 1 H [5 AJIR n p o c T o T b I M ~ npennonaraew, I ~FTO T o n p e A e n e H H o e MHOX~CTBO). M H O ~ O C w i T a T b , TO TO COOTBeTCTBHe X a p a K T e p H 3 y ~ T C R He9eTKMM Ha3MBaIoWHM O m HOuleHlleN N U3 T B U, KOTOPOe CBR3bIBaeT C KaXAMM TepMMHOM X B H K ~ X A ~ I M / ~ ~ ' ~ ~ KY TBO M C T e n e H b PN (x, Y) IIpMMeHMMOCTH X K .u. H a n p M ~ e p ,eCJIH ~ = M o n o d O i i M y =23 rOAa, TO PN ( A O A ~ ~ O23) ~ , MOXeT 6b1Tb 0,9. T ~ P M HMOXH 6b1Tb BJIeMeHTapHbIM, H a n p B M e p , X = BMCOKU~, MJIM COCTaBHbIM, KOrAa OH RBJIReTCR CG9JIeHeHMeM 3JIeMeHTapHMX TePMHHOB, H a n p m e p , x = o w ~ bsblcorcub klyxuutla. @EIKCEIPOB~HHO~O X @YHKUNR nPMHaAJIeXHOCTM P ~ ( XY) , OlTpeAeJIReT HeXTKOe IIOAMHOXeCTBO M (x) MHOXeCTBB U, @YHKUUR lTpMHa&AeXHOCTU KOTOpOrO 3 a A a e T c H KaK
T
U
I;
-
u
T
-
AJIR
n
PM
(x)
(Y)
xET,
= P N (x, Y), (2.53)
YEU.
3TO HeqeTKOe nOAMHOXeCTB0 RBJIReTCR no OIIpeAeJIeHHH, 3HUWHUe&L X. T ~ K H M 06pa30~, 3HaqeHEle T e MHHa 2 eCTb HeqeTKOe nOAMHOXeCTB0 M (x) M3 JIR KOTO O r 0 X CJIYXMT CHMBOJIQM. XOTR X H h f (x) HBJIRIOTCR pa3JIM9HbIMM 0 6 3 e ~ ~ a MW (X eCTb BJIeMeHT B TO BpeMR K a K (x) HeseTKOe IIOAMHOXeCTBO U), MbI ~ Y A ~ IIHCaTb M X BMeCTO M (x), 38 MCKJIIoqeHMeM TeX C J I y s a e B , KOrAa MMeeTCR H ~ ~ ~ X O A N M O C T ~ NX Pa3JIMYaTb. MJIJIlOCTpaUHU IIpeAIIOJIOXHM, 9TO 3 H a W H u e T e p M H H a A l 0 ~ o d O d OllpeAeJIeHO nOCpeACTBOM
d
T,
M
-
AJIR
I1,
ecnw y <25
K ~ yCTaHOBJIeH0 K
B p a 3 A e J I e 11, 3 H a q e H H R J I M H ~ B H C T M ~ € C K O ~ ~ CHMBOJIaMN HeWTKWX IIOAMHOXeCTB U , KOTOPbIe llpeACTaBJIRIQT ~060fi @ p a 3 b I HJIH IIpeAJIOXeHHR B eCTeCTBeHHOM MJIM @ ~ M ~ J I ~ H o M R3bIKe. Hanpu~ep,eCJIH U e C T b ~ a 6 o pUeJIbIX W C e J I
I i e p e ~ e ~RBJIRKITCR ~~fi
H 803pUCf)Z eCTb JIMHrBMCTH9eCKaR IIepeMeHHaR, 0 6 0 3 ~ a s e H H a g X, TO 3Ha9eHHRMM X M O r y T 6b1Tb &0JZod0d, N e M O ~ o d O i i , o w b wondod, H e ouem wondo2i, cmapab, w c m a p b i i i , H e o w u b cmapud H T. A.
B o 6 u e ~cnysae s ~ a s e n~ a~~er ~ u c ~ ~ q ence~poe i~i e ~ ~ o i i
...
e C T b C W T ~ B H OTePMWH ~~ X = X1 X2 X,, K O T O P ~ Il'If3eACTaB~~ JlReT ~0608 COYeTaHMe 3JIeMeHTapHMX TePMHHOB XI, X2 X,. 3 T H 3JIeMeHTaPHMe TePMNHbI MQXHO p a 3 6 M T b H a Y e T M p e K a Teropm: 1) p e p B H 9 H M e Te-I, K O T O P W IIBnRIOTCfI CNMBOJIaMN CneUWaJIbHbIX He9eTKNX nOAMHOXeCTB O ~ J I ~ C T ~H ~ c ~ ~ X A ~ H U R ( ~ a n p ~ ~ wondoii ep, u c m a p b l d B n p e A M A y m M npw~epe); M COIo3bI U, UAU; 2) OTpHUaHHe 3) I C e o n D e d e n e H H o c m u T M n a O W H ~ , MHOZO, cna60, 6onee unu ,ueNee (XOTR 6onee unu wewe c o A e p x u T B ce6e T p a cnosa, OH0 ~ ~ c c M ~ T ~ H K aB K ~3 J~I eT~ ~e ~R~ a p H b lTf~iP M N H ) H T. A.; 4) MUpKef?_bl, TaKHe, K a K BBOAHUe CJIOBa. & H O B H ~ H np06JIe~a PI, KOTOPaR B03HMKaeT B CBR3H C UCnOJIb30BaHBeM JIUHrBBCTB9eCKHX IlePeMeHHbIX, 3 a K J I K N a e T c R B cne,aymryeM: nywb ~ a 2 0 - 3 ~ a ~Ke ~~ ~ O e ~3 nOe ~ e ~ TSLPHOM) ' TepMWHa X i , i = 1, n, B CoCTaBHOM TepMWHe x = x, x,, K O T O P M ~ n p e A c T a B a R e T co608 s ~ a s e ~ w n ueH ~BWCTH¶~CKOM-~~~~M~T H ~HPO~~V ,C T C RBbIqHCJlHTb 3 H a q e H H e X B c M M c n e (2.53). %a s a ~ a s aR s n a e T c R q a c T H m c n y g a e M ocH O B H O ~ 3aAarIIi B K o J I K ¶ ~ C T B ~ H H O H ~ ~~ ~ I I ~ ~ A W I CeMaH~ H H O ~ ~ TWKe [4], a MMeHHO, 3 a A a q H BM4BCJIeHUR 3Ha9eHMR COCTBBHOrO TePMHHa. PI eCTb W C T H ~ c~~yqafi I ~ ~ n p e ~ b ~ q y 3Z4Wni, ~ e 8
. ..
. . .,
. . .,
MK, r A e h = o w r t o , a u = e b l c o ~ u l iw n 0 6 e K . ~ @ U H N M ~ RTOqKY 3PeHUR, OnUCaHHyfO B p a 6 O T e 1151, H e o n eAWIeHHOCTb h MOW30 PaCCMaTpUBaTb KaK O I I e p a T O e , KOTOPb! nepeBOAliT HeYeTKOe MHOXeCTBa M npeftCTaBnflmuee 3 ~ a s e ~ uu,e B H e s e T K o e M H O X ~ B DM (hu). yxe rosopumoca BbIrue, H e o n p e A e n e H H o c T u B M n o n H a w r + Y H K U ~ K ) r e H e p a s s H 6 o n b m o r o M H o x e c m a s ~ a v e ~ u AJIR f i - nuHrBucTusec~ofin e p e ~ e ~ ~u3 o i i~e6onbmro nePBuqHMx TepMUHOB. Hanpu~ep,u c n o n b s o B a H u e r + HeonpeAeJTeHHaGTu o w i l b B C o w r a H u u c r t e m , u u nepaugsbul TepMnHohi & m ~ u t i M ~ I MOXeM r e H e p U p O B a T b H e W T K U e MHOXeCTBa OWMb B M C O K U ~ , w own6 s w o 1 c u l i u T. A. AJIR o n p e A e J I e H u x ~ e o n p e d e ~ z e i c m mhu KaK O I I e p a T O p a ynof5~0 UCIYOJIb30BaTb H e K O p b l e OCHOBHbIe o n e p a r f s k , o n p e A e J I e H H b I e B pa3~e~1e 11, UCO&HHO oneparlnu K o H s e H a p a u u u , p a m a m e H m u yaenuqeaaa w e t l e l r a m u . B n o c n e A y a u e ~MM y K a m e M , MK a o MOXHO caena~bAJIH ecT ~ C T B ~ H HHeOnpeAeJIeHHOCTH O~~ OYertb U MCKYCCTBeHHbIX H e o n P C A W ~ H H W T tlA/OC ~ ~ ~ If .UUiiyC. X ~ ~ ~ K T ~ ~ H TaK#X CTUK HeOIIpeM A ~ J I ~ H H O C TKaK ~ ~ ~6 0, / l b t M , M H b U , Y N W , c#%, 8p&, &tlO/t-, if8 MOXHO ~ a f i ~Bl ip a 6 O T e [15 1. XOTR B IIOBCeAHeBHOM EICIIOJrb3OBaHUU HWl'lpefteJIeHHOCTb W e H b H e HMeeT qc?TKO OIIpefteJIeHHOrO 3HBqeHHff, B C P H O C T N OHa A e f i C ~ ~ y KaK e ~ yCUJIUT@JIb, r e H e p U p y R nOAMHOIKeCTBo T o r 0 MHOXeCTBP, K KOTOPOMY O H a I I P N M e H R m R . ~ P O C T ~ R onepaqua, K o T o p a a u M e a m CBO~~CTBO, O n e p a q H R : KOHQeffTpaQUti (CM. (2.44). %IHaBOBUT H a MbICAb 0 TOM, YTO OWHb X, X TePMNH, AOJfXHO 6bRb O n p e A e J R H O KaK K B a A p a T X, T. e.
19K
(US,
p pa
K~K
T ~ K Uo6paso~, M ecnu
c T e n e H b n p u ~ a q n e x ~ o : ~A u x o ~ aK K n a c c y cmapax n l o d e d e c T b 0,8, TO ero c r e n e H b npuaannexHOCTN K Kn.accy o q e i t b c m a p b l x ntoded e c T b 0,64.Hpyrofi npo: CTO# V U M e p :
U=1+2+3+4+5
-
-
o u e n b W e f f b M a / l e i r b l ~ u l i = 1/1
+ 0,412 + 0,113.
x = oreub ice m n o , I - I a n p ~ ~ ~ ecnu ep,
(CM.
puc. 2)
(3.6)
U
(3.10)
(3.11)
TeJrbHO n p O C ~ 0 f i3 a ~ a ~ e fCOBepIIJeHHO i, C X O A H O ~C~ 3 a ~ a Y e f iBbIs u c J I e H u R 3HaYeHUR ~ ~ J I ~ B C K O I B ' OM p a X e H U R . B KaqeCTBe np0C T O ~ u n n I o c T p a u u N p a c c M o T p a M B b I s u c n e a u e 3~ase~ui-i coCTaBHOrO T e p M U H a
~ ~ M ~ T U YTO M ,
He ouelzb morlzo = 1(ouelzb moulzo) = 1( m o v l z ~ )(3.1 ~ 4)
rAe
x = tie orelzb manenbtcudi, n e p ~ u s u ~ Ti ei p M u H ~anelzbtculiO n p e A e a e H
(4.1 ) KaK
1a/tt?lzb~~d = 1\11 $0,812 $0,6/3 + 0,414 + 0,215 (4.2) C 06JIacTbm paccyme~~fi U=1+2+3+4+5. (4.3) B ~ ~ A C T(3.8) B H IIpUMeHeHUe ~ 0 n e p a ~ ~O 2W H b K MHOXeCTBY iUa&?lzbtc~dA a e T ouelzb mnelzb~usi= 111 + 0,6412 + 0,3613 0,1614 0,0415 (4.4) N, YqHTbIBaR (2.26), IIOJIyqaeM ue ouelzb manelzbrcuG = 1(oclelzb manelzbtcudi) = = 0,3612 + 0,64/3 0,8414 + 0,9615 0,412 + 0,613 0,814 115. (4.5) B KaqeCTBe HWKOJIbKO donee CJIOlKHOrO n p U M e p a P a C -
+
BCJ'I~ACTBH~(3.15) U AeCTBO
(3.16) MbI
nnmc nnmc x
UMeeM ~ P U ~ ~ I U ~ E I T ~T JO IX~- H O ~
= mulzyc
olcelzb
X.
(3.17)
-
+
-
+
+
+
CMOTPUM C O C T ~ B H O ~TePMUH ~ 13
ebiculedi cmenelzu = ~ u l z y colcelzb ouelzb, e sbiclued cmenelzu = n n m n n m ouelzb.
(3.18) (3.19)
oueub ~&eubtcu~i u He oueub 6onbluo1Z. B TOM cnyqae oueub 60/2buoZi= (6onb~10G)~ = = 0,0411 0,1612 0,3613 0,6414 115 owlzb OYeub 6onbluoG = ( 6 0 n b w l i ~ ) ~ 0,113 + 0,414 + 115 . h..menb owlzb60nbfuod 111 1/2+ 0,9/3+0,6/1 x
= tie
+
- -
IV. BblYMCnEHME 3HAqEHMFf IIMHrBMCTMLIECKOFl IlEPEMEHHOFl
E c n ~MbI
3HaeM, K a K BbIYUCJIRTb 3 H a q e H U e COCTaBHOrO TepMUH a BUAa hu, BMqHCJleHUe 3Ha¶eHUR 60~1eeCJlOXHOrO COCTaBHOrO T e P h l P H a , K O T O P ~ IM ~O ~ X W BKJIIOqaTb TePMUHbI He, UAU, U B AOnOdlHeHUe K TePMUHaM BUAa hu, CTaHOBUTCR CPaBHN-
24
-
+
+
+
+
+
(4.7)
(4.8) (4.9) (4.10)
cnedoeamenbiro, o w fie OUelzb M A ~ N ~ K 'U ~H e O W7 Hb 6) 0 ~ b ~ d = (0,412 0,613 0,814 1/51n (111 112 0,913 (4.11) + 0,614) 0,412 0,613 0,614. ~ H ~ ¶ ~ H ~U UR H ~ B H ~ T H ~ ~~ C~ K P ~O M~ ~~ H H O0 6A 0, 3 ~ a q e ~ ~ o i i ~xdcmtw,A ~ I ~~ TP H M ~AP P Y ~ Op o A a . B 3TOM c ~ ~ y q aMU e n p e A n o n a r a e M , %TO 06nac~bp a c c y ~ ~ q e ~ ecrb uii U,
+
-
+
+
+
+
+
+
re
U
3JIeMeHTbI npefiCTaBJIRlOT BepORTHOCTH. n p e A n o ~ r o m U M , ~ITO~ bXOTUM l B ~ I ~ U C J I U3T ~ ~ a ~ e ~ n e
rAe
x = 8 s w w d cmeneuu tie noxoxe, (4.13) s eucu(eG cmeneuu onpeneneuo K a K '(cM. (3.18)) s s b l c u r e d cmeneuu =muu yc o v e u b o w u b (4.14)
T~XHHK KOTOPYIO ~, MM WCIIOJIb3QBaJIW AJIR BMqWCJieHHR 3 H a q e I i W 6 COCTaBHOI'O TepMWHa, eCTb s a ~ ~ ~c b n y~ q aff ii 6 O J I e e 06uero nO&XOfia, ~ o ~ o p bOfIWCaH ~ f i B [4 H [5 P ~ C C M ~ ~ T P W B ~ @ M M ~IIOAXOfi MOXHO IIPHMeHRTb K BbIVHCJIeHUIO ~ H ~ ¶ ~ H H B J I U H ~ B I ~ C T H ~ W KnOe~ p ~ e ~ e ~ ~ oIIpU f i , YCJIOBAW, q T 0 COCTaBHHe TePMHHM, IlpenCTaBJIRIOlUUe X U 3 H a q e H H R , MOrYT 6 b l ~ breH(?PNPOBaHbI J I U I ~ I ~ H H O KOHTeKCTa ~~ ~ ~ ~ M M ~ T U K O &K . asCfBe U J I J I K I C T p a ~ U PaCCMoTpUM JIuHI'BUCTIlYeCKyIO H e p e M e H H y H X, s ~ a s e ~~mopoil u~ n p e A c T a s n e H N c n u c K o M : ~ e u w u d ,ue Y a ~ e t i b ~ ~ t 6. iO, A b ~ u , d , Ne 6 0 ~ b ~ t . iOWHb , M O X e f l b K ~ 8 , Ize W U b & a / l e ~ b K ~ d ,~ e f t b UAU ~ ~ Ned W t l b OWHb 60nbtuoG, MCUeNbK U U~ (60~bUU)t.i i&lU H e Y c u t e ~ b K u G ) , OiieHb OWUb Y O X e l t b K ~ d u ue W N ~ w u b 6 0 ~ b m du T. A. P a c c ~ a ~ p u s a e3 ~H ~ a see H w M o r y r G H T ~ reaepnpoeisar J I u I I I ~ H H o ~KOHTeKCTa ~ ~ P ~ M M ~ T U 6 K O=~ ~ S, P), B K O T O P O ~MHOXeCTBO ~ KOHesHbIX CJIOB BKJIIOqaeT 3JIeMeHT a p H b I e TePMUHbI N U l e ~ b ~ U f i60dbUU)d, , He, U, U R a - O W H b Ii 'J'. &.; Ha! OKOHqaTeJIbHbIe CJIOBa ~ ~ O ~ H ~ Y ~S,W A, T CB, R C, E; p E 4 y J I b T U p y I O U a R CUCTeMa WMeeT BNA
1
I.
B
VT
w e u b oveub
tie noxoxe = (ue noxo~~ce)~ *
a~1/0+1/0,1+1/0,2+0,4/03+0,2/0,4.
s sucused cmeneuu ue noxoxe = muuyc
ouenb
(VT, vN,
D,
(4.18)
S-tA
C-tD
B-tC
D -+ M J ~ ~ N E +6 0 n b ~ t . i
ouenb ne no-
CJ'IeAyC!T 3aMeTBTb, qTf) IIpU BM¶HCJIeHWU 3 H a W H H R COCTaBHOrO T e p M u H a B n p e A m y w x npmepax MH ucnonmoaanw 0 6 ~ s HbIe n p a U J I a IIpeAmeCTBOBaHWR, A ~ ~ ~ c T B Y I O UnH pW ~ npw6pa~ O B ~ H H6Uy n e ~ c ~ u s ~xp a x e H u f i C . ~ o 6 a a n e ~H aeeo n~p e a e J r e H ~ O c ~ e f 3TW i n e a B W J I a npeAIIIeCTBOBaHkfR MOXHO BblPaBUTb CJIeAyIolUUM o6pa30~:
B-ti~e
c
C
~ K U ~
(S). Kaw~oe npasuno sisona 3
B (4.20) nopoxaae-r o T H o r u e H u e M e m y HeqeTKWMU MHOXWTBBMU, 0 6 0 3 ~ a 9 e ~ ~ b I COOTBeThf~ CTBeHHO TePMBHajIbHhIM W HeTePMHHaJIbHbIM CUMBOJI~MW. C a y sae (4.20) r r n oTHorueHwR T a K o B u (MU O n y c K a e M npaanna BblBOAa, KOTOPbie H e OKa3bIBaH)T BJIHRHU R H a COOTBeTCTBYIOLUWe MHOXC~CTB~):
B
S+S u/lu A * S L = S R + A R A+A u B*AL=ARnBR
B TOM CMbICJIe, ¶TO BbICKa3bIBaHIIR A 3 B (A BJIe¶eT B) II A \;lB(rie A UAU B) HMeIOT IIAeHTUsHbIe ~ a 6 J I H l i b UCTUHHOCTH. l
1
H U X H HIIHAeKCpi ~ L
H
R
U C n O J I b 3 y I o T c R 3 A e C b AJIR TOrO, g~06b1
pasnusa~b CHMBOJI~I, C T o R L q u e B ne~ofi u npa~ofi Y a c T R x . Y~06b1BbI¶HCJIHTb 3 H a q e H H R COCTaBHOrO T e p M U H a X, H e 0 6 x 0 ~ ~n p~O B0e C T H C H H T ~ K C H Y ~ C K Ha H~a~J I U 3 X B T e p M H H a X 0 I I p e ~ e ~ r e ~ I'PaMMaTAKH ~ofi G. T o r ~ a ,3 H a R CHHTaKCH¶eCKOe A e p e B O X , M O X H O IICIIOdIb30BaTb COOTHOIUeHUR (4.21) AJlR BbIBOAa CUCTeMM y p a B H e H I I f i (B ~peyr0JIb~0fi @ p ~ e ) , PeIIIeHMeM KOTOPOB RBJIReTCR 3 H a W H H e X . H a n p u ~ e pB, C J l y W l e , K O r A a C O C T ~ B H OT~e~P M H H
TO
eCJIU 6 0 ~ b t U 0 l i , T O r A a AlCUbl2i, W J I H Cic0Ab3KUd,
T O r A a 0nGZHblb;
OHM RBJIRKTCR COKPaLqeHHRMU ~ p e ~ J I 0 X e ~ U f i l
ecnw x - ~ O J I ~ I U O T~ o~r,A a y - ~ a n ~ f i , ecna A o p o r a C K O ~ ~ ~ KT ~o rRA,a e s ~ aonmria.
B CYlIlHOCTU, I l p e A J I O X e H H R W O r 0 B H A a OnHCbIBaIoT OTHOIUeHUH M e X A Y ABYMR HeOnpeAeJIeHHbIMU IIepeMeHHbIMH. 30 03HasaeT, TO H e o n p e A e a e H H o e B ~ I C K ~ ~ M B c~ nHe AHy ~e T c K o p e e OnpeAeJlRTb K a K H e s e T K O e OTHOIUeHHe B CMbICJIe (2.19), a H e K a K CBR3KY B CMbICJIe ' (5.1). 3 ~ e c b uenecoo6pa3~0 onpenenmb c ~ a s a n a dempmoso
cornacymca c (4.11).
B O ~ M O X H O C T ~OnpeAeJIRTb CMbICJI 3 H a ¶ e H U f i , n p U H H M a e M b I X J I I I H ~ B U C T M ~ ~ C K IOI~e~ p e ~ e ~ ~ o f H i ,~ O ~ X O A U M HJIR ~ aHaJIU3a H e s e T K H X B ~ I c K ~ ~ ~BIH B A a ~((ecJrH H u ~A~ T O r A a Bw, H a n p H M e p , <(eCJIU X H e O W H b iuCUbl6, T O r A a l j O W H b O W H b 6onbluoli)). 0 6 TOM r o B o p m c R B C n e A y m u e M pasnene.
-
60JIee o~LUNM, Ba)KHMM B H a I U e M ITOAXOAe RBJIReTcrR ~0npedefieHHoe ~ b ~ c ~ a 3 b ~ K e ae c~ mu eA , m r A a Bw, K O P ~ K O A 3 B, B KOTOPOM A ( ~ H T ~ U ~ A ~ U H TB ) ( K O H C ~ K B ~ H T) He¶eTKHe M H o x e c T B a , a H e n p o n o 3 u q ~ o ~ a ~ r b ~ bn1e pe e M e H H b I e . BOT TNnH¶HbIe IIPIlMePbI TaKHX B ~ I C K ~ ~ ~ I B ~ H U ~ ~ :
-
n p o u 3 6 e d e ~ f U ? A B y X H e s e T K H X MHOXeCTB. ~ Y C A T ~ He'IeTKOe IlOAMHOXeCTBO 06nac.r~P a C C y X A e H H R U n y C T b B - HePeTKOe IlOAMHOXeCTBO ~pyrofi,~ o o 6 u erOBOP R, O ~ J I ~ C T PHa C C y X A e H u c l V. T o r ~ aA e K a p T o B o n p o ~ 3 s e ~ A e ~M ~B, e 0603~asaeMoe A B, o n p e n e n n e T c s l c n e A y m L q u M o6pa30~:
u
x
V. HEYETKME YCJlOBHblE IlPEAJlOXEHMA M COCTABHOE IlPABMnO BblBOAA
r e U x V o 3 ~ a s a e A~e K a p T o B o n p o u s ~ e ~ e ~Hsees e T K u x MHOXWTB U H V, T. e.:
-
~ ~ M W U Mq ,T 0 K O r A a d H B H e H e s e T K H e , (5.2) npeo6pasoB b I s a e T C R B 06b19~0eO n p e A e J I e H H e A e K a p T O B a FIpOU3BeAeHHR MHOXeCTB. M H ~ I M HCJIOBIMH, (5.2) OLlHaqaeT, ¶TO A X B HerIeTKOe MHOXeCTBO YnOPRAO¶eHHbIX nap (u,o), 24 (I CO C T e n e H b H ) ITpUHaAJIeXHOCTH (u, V) II ( A X B), 3 a a B a e P d o f i @ p ~ y J I o f i PA (u) pB (v). B 3TOM CMbICJe A X B e C T b He¶eTKOe OTHOIlIeHUe U K
E
v,
6U,
A
V.
Ecna BblCKa3HEaHHe U W J I H A, TOrLla BB paCCMaTpHBaTb mcKmL3bleaHHe cresnu A , Torna 8 , mase CB c Heonpew neeam C, mrna noJryumR pa~nus~haemiTegnpeTawH TOM) npeftJIomeHHa B sasucu~oc~w OT n p e m o x e ~ ~ fm i ~oCK~WWKO C. B qacTHocTu, ecnH M ~ Inpennoqomtn, sn, C t.. V, Torfta uecnu A, Torna B (snu A *B)P nepexonmrT B Kate
ecna A, TorAa B, maue
c
A
=
A x B ' + ( l AX.^), (5.9)
.
L7
Boo6qe, ecnu A . ., A, - HeqeTKue no,qMHoxecTsa U, B1, , . ., Bn - HWeTKUe IIOnMHOXeCTBa H 3 V, TOrAa ecnu A,, TorAa B,, uaaue, emu A,, TorAa B, A
. ..
+
wiase, ecau A,, TorAa 8,=A, x B, A,x Ba+ *.. A,xB,. (5.10) ~ ~ M ~ T TO H I(5.10) V I , nepexofiw B (5.9), ecna n p w o w w a e CeCJlW A TOrAa B, UHaW CB NHTepnpeTHpOBBTb KaK a H X H A, TorAa B, mrase ecnu 7A, TorAa CB.Cne~ywTame OTM~THTL, sT0, IIPHMeHRR (5.9) IIOBTOPHO, MM JlOJIYWiM
+
+
ecnu A, Torna (ecnil B, Toraa C, mase D) Lmase E = A X B X C + A X ~ B X D + ~ A X(6.11) E,
ecnH A, Torna B, ,mase C = (emn A, Torna B) n (5.1 4) n ( ~ W Hi A, m r ~ ac), KOTOpW AJIR He'IeTKUX MHOXC?CTB A, B EI C CnpaBeflJIHBU nHWb I I ~ u ~ J I H X ~ H H%O 3 . RIBOpHT 0 TOM, rIT0 B COOTBeTCTBHH C (5.15) OIIpeAWHHR BbICKa3MBaHHR UeCJIH A, TOrAa B, HHaqe Ch) H e m s A, m a BB c noMoqfrm (5.9) H (5.12) He COBC~M nonX O AJIR ~ HeqeTKHX A, B H C. CJIenyeT TaKXeOTMeTHTb, qT0 y H 3) A 3 B CnpaBeAJIUBbI AJIR ~ C f i H 1) U = V, 2) X Bcex ~ J I ~ M ~ H T OU, B mma u3 (5.12) HMeeM, TO
KOHKPeTHO, THnHYHaR npo6~1e~a, C ~ O T O p o f i CTaJIKUBaKXCR npu pa3pa6oT~eT a m x a n r o p w T M o B , COCTONT B cne~yro~tte~, M M ~ ~ T c ,He¶eTKOe R OTHOIlIeHNe, C K a X e M , : U -+ V, KOTOpOe OnpeaeJrReTCR HerIeTKAMH BbICKa3bIBaHHRMA. npo6Jre~a COCTONT B TOM, 9 ~ 0 AJIR 6 ~ JI1060r0 HeYeTKOrO IIOAMHOXeCTBa X f U OnpeneJINTb HeYeTKOe l I O ~ M H O X e C T B 0Y V, KOTOPOe X NHAyUHpyeT B V. H a n p u ~ e p ,lIyCTb MMeWrCR CJleAYIOLUUe A B a npeAaoxeHuR :
R
6
1) x - oilem wan~u', 2) eCJIH X - &$ MUJblu',
TOrAa
y - 60/lb~10u',
NHarIe
y
tie o w ~ b6 0 n b ~ l o d i , ~3 KOTOPIJX B T o p o e onpenenxw c o r n a c H o
(5.9) H e o n p e A e n e H H o e OTHOUeHlle R. B O ~ H H K BOnpOC: ~ ~ T KaKOBa 6 y ~ mBeJIUsNH a y, &JIN X OrIeHb M ~ J I ?& B ~ TH a 3TOT BOnpOC AaeTCR CJleAyIoU A M nPaBAJIOM BbIBOAa, KOTOPOe MOIKHO P a C C M a T p A B a T b K a K 0 6 0 6 u e ~ ~N e~ B ~ C T H O n~pOa B I f J I a modus ponens. c o c m ~ o enpasuno s b ~ o d a : e c n A - HeonpeAeneHHoe OTHOUeHHe ++ V A X - HeOnpeAeJIeHHOe nO,U,MHOXeCTBO U, T o r A a H e o n p e A e n e H H o e n o A M H o x e c T B o y c V, K o T o p o e AHAYUNpyeTCR nOAMHOXeCTBOM X, AaeTCR K O M ~ O ~ A U(CM. A ~ ~(2.22)) ~ A X, T. e.
U
R
R
Y = [0,36 0,4 0,6 0,811 (5.19) C ~ e n a e H~e c K o n b K o s a ~ e s a ~ ~OTHOCHT~JI~HO fi (5.16). B o - ~ ~ P B ~CJIeAyeT IX, OTMeTATb, YTO KOrAa R = A B a X = A , MbI nOJIyqElM y = Ao(A+B) = B, (5.20) YTO CTPOrO BMnOJIHReTCB, KOrAa A, B N C - 0 6 b 1 ~ ~ b 1 e MHOX e C T B a , H ~ ~ H ~ J I A x ~ H HKOrAa o, A , B N C - H e s e T K K e MHOXe-
CTBa. MM~HHO B 3TOM CMbICJIe COCTaBHOe n p a B N J I 0 BbIBOAa M O X HO PaCCMaTpRBaTb K a K n p k i 6 ~ 1 A X e 0 ~ ~6 ~0 e6 u e ~ n~p e a B H A a modus P O n e n S ( ~ ~ M ~ T K YTO M , BCJIeACTBNe TOrO, KaKIlM o6pa30~ A 3 B o n p e A e J r e H o B (5.1&),rIeM 6 0 a b m e x crmusawcn OT A , TeM M e H e e ~ ~ T Ko nOp e A e n e H y). Bo-BTOPIJX,B b I P a X e H N e (5.16) aHaJIC$rNYHO ~ a p r t i ~ a J I b ~ 0 f i BePORTHoCTH B TepMllHaX (PYHKUHHY C J I O B H O ~BePORTHOCTU, ~ T.e.
(5.21) rAe
9s = B ~ { P X-xi} r j = Bep { Y = y,]
-
X
N
P f j Bep {Y = yjlX = x,}, Y - c n y s a f i ~ ~n eep e M e H H b r e co ~ H ~ Y ~ H N R xl, M N x2,
...
li Y1, Yz, 0 3 H a Y a e T , rIT0
me .ua/rMu' N 6 0 n b u o u ' OnpeAeJIeHhI no (4.2) M (4.7) COOTB~TCTBeHHO. T o r ~ anOACTaBJlRR , B (5.9) ~ a f l b l dBMeCTO A, 6 0 ~ 6 ~ 1 0 6 BMeCTO B H N e OW3fb ~ O R ~ U CBMeCTO O ~ MbI nOJIYqAM MaTp ~ u yOTHOW~HMR R A J I ~ H e o n p e A e n e H H o r o B ~ I C K ~ ~ I J B ~ H U ~ ((eCJIll Ma/rblli, TOrAa ~ O R ~ U L O ~HHaYe ', IZe OWN6 6 0 . f l b ~ ~ ) b . P e 3 y J I b ~ aKOMnO3HuHN ~ R C X = O W 6 &Ulbllf: eCTb
C,
.. .
COOTBeTCTBeHHO. OAH~KO 3 T a aHaJI0rUR H e (5.16) e C T b OTHOIlIeHAe M e X A y BepORTHOCTRMH. B - T P ~ T ~ MCJIeAyeT X, OTMeTUTb, YTO H3-3a HCnOJIb30BaHHR MaKCHMNHHOrO npOH3BeAeHAR M a T p H q . B (5.16) OTHOlueHHe MeH
R
R
33
PaHee,
HeS€!TKHe BblCKi%lblBaHHH H r P W OCHOBHYiO POJIb B OnUCaHHU W p e a J I U 3 a u U He¶eTKUX aJIrODUTMOB. 06 3TOM n o f i p l m pesb B c n e A y l o u e M p a s A e a e .
VI. HEYETKME AJlfOPMTMbl
Hanpumep, ecnu x - w w d mu cpea~ua,u R = A 3 B 0 3 ~ a s a eecnu ~ x H e YM u tte 6 0 n b t u o l i , TOrAa y - w t i b Yadblli, TO MOXHO
nUCaTb:
o
nwtbul) =
(rce wanwb u rce
6 o ~ b w o i b o w r c bma~ud)
unu cped~ud(He mmwd u rce ~ O A ~ U L O ~ J O W ? Hm~ a d ) . b c n ~ ~ sea ~e e s a ~ u es :a m o O C O ~ H ~ B ~m T ~ B, n p a K T U 9eCKUX npUMeHeHURX HeseTKUX B M C K ~ ~ M B ~ AJIB H H ~ OIIUCI~ HUSI CJIOXHUX NJIH nnoxo o n p e A u e H H b t x O T H O I J ~ ~ H H B ~M~~ I ~ C JIeHurr (5.91, (5.10) U (5.16) ~ Y A Y T ,B a d B U e N B O P R , BbInOJIH e m B 60nbuoii CTenLtHu I ~ P H ~ J I U ~ -~BH o aH e eOTOM, . AononH U T ~ J I ~ HUCTOgHBK ~I~~ HeT04HM3TU CBSI3aH C n p e ~ h a ~ n e He9eTKOI'O MHOXeCTBa KaK 3HaYeHUH JIWH~BEICTB~~WKOR nepe~ e H 4 5 0 f i . ~ ~ ~ ~ ~ ~ o J I oHX a nU p UM M e, p , lIT0 OTHOtlreHHe M e X A y HeseTKUMU IIepeMeHHbIMU X U lj O t I U C M B a W l I HeOnpeAeJIeHHUM B b f C & a s m a H u e M awns Manbffi, m r A a bonbruoi& mase, e c n f i c p e a z l u l i , T o r A a cpedtiu21, U H a s e , ecna bo/lbllcoti, T o r A a o w b ~mafi&. 0 6 ~ 4 MM ~ 0n p u n u c m a e M n e p e ~ e ~ ~xopf ai 3 n u s ~ s enuarBUCTUWCKUe 3 H a W H H R U. BbI'fUCJIReM COOTBeTCTBYIOIIlUe 3 H a 9 e HHR y, ucnonb3y~(5.16). T o r ~ a n, p u 6 J I u x a f I 3 ~ a s e ~ unepek M ~ H H O JIUHrBHCTUrIeCKUMU ~ ~ CWMBOJIIMU, MbI IIPNXOAAM K
cne~ymwefi ~a6nuye:
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d o d b ~ d ~e m d W H ~ marlMa cpea~ud W H b JULtbLd OWHb p l e H b
-u He W e H b 60~bUOfi
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rpy60 rOBOPR, H ~ W K U ~aJIMPUTM eCTb YnOPRAO'IeHHOe MHOXeCTBO HeseTKUX U H C T ~ ~ K U U B KOTOPbIe , IIpU UX peaJIH3aUUH nawr npu6rruxe~~oe D e w e H u e n p o 6 n e ~ ~B. ~ o i iunu u ~ o i i @ p ~ eH e Y m A e aJIrOpUTMU IIgUCYTCTgYroT BO MHOrHX Y@UIHX PaCCMOTpeHURX. T ~ K MbI , UCnOJIb3YeM H e g e T K U e aJIWPHTM6I C03HaTeJIbHO UJIU ~ W C O ~ H ~ T ~ ~KOrAa ~ H O ,UAeM ffeWKOM, BOAUM MILUNHY, U U e M npeAMeT, 3aBR3MBaeM y3en, FOTOBUM oben, N m e M HOMep B T~JI€!@HHo~ K H W e U T. A. E0JIee T O N , no C y U e C T B y , 3TU aJlrOPUTMM IIIUPOKO UCnOJIb3)WTCR B np0rpltMMHPOBaHWU, UCCneAOBaHHH ~ I I e p a U ~ f i nCUXOJIOI'UU, , Hay K e y n p a B J I e H U H U M ~ A U U U H C K O ~AUarHWTUKe. ~ ~OHRTUR HQseTKOrO MHOXeCTBa U, B YBCTHOCTU, HeqeTKOrO BhU3Ka3biBaHUR CO3AaeT 6asuc AJIR HCnOJIb3OBaHHR HeUeTKHX aJIrOPUTMOB 6 0 J I e e CUCTeMaTUseCKUM U, CJreAOBaTejrbHO, 6onee ~ H T H B H M Motipaso~,s e M STO 6mo BO~MOXHO pasee. T ~ K , HeqeTKHe aJIrOpMTMM MOrYT CTaTb BaXHbIM UHCTPYMeHTOhA AJIR ~~PH~JIHHS~HU aHaJIU3a O~O CUCTeM U npOUeCCOB nPHHRTHII peUIeHU8, KOTOPble CJIUIIIKOM C W X H W &JIR npHMeHeHUR 0 6 ~ s ~ 0 f i M ~ T ~ M ~ T H ~ TeXHHKH. ~ C K O ~ ~ Q > ~ ~ M ~ JXapaKTepUCTHKy I ~ H ~ w IIOHRTUR HeYeTKOfO aJIr0PUTMa MOXHO A a T b C nOMOwbK) IIOHRTUR Ht?¶eTKHX MaIIIMH T b i o p U ~ r aHJIU H€?WTKOI'O MaPKOBCKOI'O aJITOpHTMa 16 1- 18 I. B TOM pasame r n a B H a R ymb oBeyme~uaCOCTOUT B TOM, Y T Q ~ MCBR3aTb IIOHRTUe HeqLTKOI'O a J I M p N T M a C nOHffTURMU, BBeAeHHbIMU B npeAMAYUUX pa3AeJIaX, U llpOUJIJIK>cTpUpOBaTb nPWTbIMU n p U M e p a M U HeKOTQpMe npHMeHeHUR TaKHX a J I r 0 P UTMOB M H C T ~ ~ K U UBU HeYeTKUX aJIrOpUTMaX MOXHO pa3AeJIHTb Ha ~ p K unacca: 1) Hamawo4ue n p e d ~ o x e w ~H, a n p u M e p ,
I He W H b 6 0 ~ b U l d p l e ~ bw e n b 6 0 ~ b t u o d W e H b OqeHb d 0 n w t 0 d hi& UAU cpedHulS
x w 5 = ,uamli x = 60AbuloGi x = 6 0 ~ X
b ~ Ul itit? oqefib ~ a l i 2) Hewmrcue 861cKaau3aHux, H a n p u M e p , aecqu x - wanwli, T o r n a y - 6 o n b t ~ 0 f i , 60JIb~ofi8,
uHaqe
y - He
-
WJIH X I ' I O J I O X H T ~ J I ~ H ~TI O ~ r~A, a YMeHbIIIHTb Y ile3HaI1Uml?/lbHOB, 6(eCJIH X Mi1020 6 0 / l b ~5, TOrAa CTOnB, <WJIH X OWHb Maflbla, TOrAa UATM K 7)). ~ ~ M ~ T H9TO M , B TaKHX IIPeAJIOXeHHRX a U 6 0 l l p e ~ ~ y YJIeH O T H O I I I ~ H A ~nu60 ~, I I ~ ~ A I I I w T B ~ I O I ~ JHI H~ 6~0, 06a MOrJ'T 6 b I ~ bCHMBOJIaMU He9eTKHX MHOXeCTB. 3) B e 3 y c n m ~ b l e a r c m u s i l b l e npedno%ce~u%, H a n p u M e p , YMHOXHTb X H a X YMeHbIIIHTb X CAeZKa C T e p e T b HeCKORbKO n e p B b I X IIORBUBIIIHXCR eAHHHQ HATH K 7 nerIaTb x CTOn OTMWMM,9TO HeKOTOPbIe U 3 3THX IIHCTPYKUH~~ RBJIRKlTCR HeYeTKRMH, a HeKcYrOpbIe - H W . K O M ~ M H M P O B ~ Ha3HaYaIOLUee T~ A HeqeTKOe BbICKa3blBaHHR HYXHO B COOTBeTCTBHH C KOMnO~HUUOHHbIM npaBHJIOM (5.16). HanpFi~ep,eCJIU B H ~ K O T O P ~MOMeHT I~~ BbInOJIHeHUR He9eTKOrO anropuma M ~ IB c T p e g a e M M H C T ~ ~ K U H U 1) x = O W H ~u a / l e ~ b r c u C 2) (ecnu x u a n e n b ~ u l i , T o r g a y 60nbwoa, m a Y e y H e OWHb ~ O R ~ ~ U O & > , r A e ~ a n e ~ b r c u ZHi 6 0 n b w o C o n p e A e n R r o T c R c o r n a c H o (4.2) N (4.7), TOrAa pe3YJlbTaTOM BbIIlOJIHeHHR M H C T P Y K U M ~1) ~ H 2) 6 y n e ~s ~ a s e ~ y, s eA a H H o e c o r n a c H o (5.19), T. e.
-
.
~ ~ f i
TO p e 3 y J I b T a T BbInOnHeHHR AOJIXeH 6mb eAHHCTBeHHbIhi 9HCn o M B H H T e p B a n e M e m y 3 11 8. Ha ~ a ~ oO cf Hio B e ~ b r 6 a p a e ~ c ~ T a K O e ~HcJIo? K ~ K y K a 3 a H O B I6 1, Pa3YMHO AOnYCTHTb, YTO pe3yJlbTaTOM BbInOJIHeHHR 6 y ~ 3JIeMeHT e ~ HeYeTKOrO MHOXeCTBa, E i ~ e I 0 U H f iH a H B b I C I U ~CTeIIeHb npHHaAJIeXHOCTH 3 T m y MHOXWTBY. ECJIH~ a ~ 03fleMeHT fi H e enHHCTBeH, K a K B (6.5), T O r A a MOXHO CAeJIaTb ~ J I y s a f i ~ b I fMJIH i n ~ 0 ~ 3 ~ 0 J I b ~ ~bbI1f 6i 0 p CpeAH 3JIeMeHTOB, HMeIOmHX HIHBMCLUYH) C T e n e H b IIpHHaAJIeXHOCTH MHOXeCTBY. MOXHO T a K X e BBeCTH H ~ K H~ p~H ~ e p U f i , COrJIaCHO KoTOPOMY J I H H ~ ~ ~ Y HnoO P ~ O Y M B a I O T C R3JIeMeHTbI HeW T K O r O MHOXeCTBa, HMemIIJ,He HaHBbICIUyIo C T e n e H b IIpUHliAJIeXHOCTH 3TOMy MHOXeCTBY, H, TaKHM o6pa30~,0 6 p a 3 y e ~ c R ~ A N H C T B ~ H H ~~aIH~6~0 J I b L t I H f i 3JIeMeHT. Hanpa~ep, B cnysae (6.5), eCJIH BBeCTH K P U T ~ P I I ~MHHHMH3aQHH ~ 4HCJIa LUarOB, TO H 3 BCeX 3JIeMeHTOB C ~ a k f ~ b I ~ I U CTeneHbIO efi npHHaAJIeXHOCTU MHOXeCTBY 6 y ~ m~ b 1 6 p a ~ H0 a H M e H b I l r e e YHCnO 5. A ~ a J I o r H q ~ b I BOnpOC fi B 0 3 H H K a e T B CBTYaQHU, K O r n a KCI I b I ~ y e ~ b AOJIXeH ~fi oTBeTHTb ((AaP HJIH ((HeTP H a . ~ e 9 e ~ ~ BOllfi npoc H a n p H ~ e p , npe~nono%u~, YTO ~ c n b I T y e ~ onpenno~y x e H a IIHCTP YKQHR
.
-
y = 0,3611
+ 0,412 + 0,6413 + 0,814 + 115.
(6.1)
B K O T O P O ~&an0 ~ o n p e A e n n e T c R no (4.2). T e n e p b npwnonoxm, YTO x = 3, n p u Y e M TO sacno u M e e T c T e n e H b n p u H a n J r e x H o CTH MHOXeCTBY MCUbLd, P a B H y I O 0,6. ~ o J I x ~ H JIH 9eJIOBeK BbInOJIHUTb KCTOIIB HJIH UHATH H a h? Mu ~ Y A ~AOnyCKaTb, M YTO B ~ O A O ~ H ~CHTYIQHRX IX 9eJIOBeK CAeJIaeT ~ b 1 6 0 p ((60JIee H C T A H H ~ I ~ ~H B a, l l p H M e p , ((X l)taROP IIpeAnOYTHTeJrbHee UX H e MCUO)), T a K K a K B H ~ I & M C J I y Y a e C T e n e H b HCTHHHOCTU n p e A n o x e H m g3 M~JIOB pama 0,6, YTO 60~1bmeCTeneHH HCTHHHOCTH YTBepXAeHHR ((3 H e MGXO)). Ecnu 06e aJIbTepHaTUBbI u M e m donee s n H M e H e e p a B H b I e ~ H ~ Y ~ HHCTHHHOCTH, H R ~1,160~ M O X e T 6 b l ~ bCAeJIaH npOU3BOJIbHO. Y A O ~ C T B hlbI ~ 6 y ~ H a 3 b I B a T b 3TO l I p a B H J I 0 ~bI60paO A H O ~U~ 3 AByX aJrbTepHaTHB npasuno~n x l e o 6 n a a a t o w C m b m e v i z a m u s b l . O W H ~B a X H O IIOHRTb, 9TO TOJIbKO 'lTO PaCCMOTPeHHbIe BOnPOCbI B03HIPKaIOT JlHLUb B TCX CHTYaUNRX, B KOTOPbIX T ~ ~ ~ J ~ TmC 0R 6, ~ P e 3 Y B b T a T l3bInOJIHeHHR H ~ ~ ~ T K MHCTPYKo R QHH RBJIRJICR eAHHCTBeHHbIM 3neMeHTOM ( ~ a n p ~ ~ YucJIoM), ep, a H e HeqeTKHM MHOXeCTBOh!. T ~ K U 06pa30~, M ecnu pa3peILIa€!TCfl, 4~06bfp e 3 y J I b T a T BbInOJIHeHMR (6.6) HeYeTKUM, T O r A a AJIR X = 3 MbI nOJIYYUM He9eTKOe MHOXWTBO BHAa O,~/CTOII f ~ , ~ / u & T uK 7,
-
-
-
AJIR
B a x ~ oOTMeTHTb,
YTO K a K B (6.1), T a K H B (6.4) Pe3YJIbTaTOM BbInOJIHeHAR HHCTPYKUHU RBJIReTCR H e 9 e T K O e MHOXeCTBO, a H e f?,QHHCTBeHHOe qHCJI0. O A H ~ KKOrAa O 9eJIOBeK A a e T H e 9 e T KyIO HHCTPYKUHIO, TaKyIO, K a K ((CAeJIaTb HeCKORbicO IUarOBP, rne Hecrconbuo o n p e f i e n e a o TaK:
e ~
omasam, m
Kmpoe
onepaquH
A o n m U BH~O~HHTT~CR ~ ~ ~ ~ J I J I ~ J HI eUm~H M o ~ npUCYTCTBye'l' 0 COCTaBHOM IIpaBHJIt? BbIBOAa H Hrpal?T =HOBHYiO POJIb B IlOHHMaHBW IleYeTKHX BdlrOpWMOB B HX BHIICUtHeHHR JIIOAbMH H MaIIIHHaMU. Aaaee MH n o K a x e M H ~ C K O J I ~ K On p w e p o B H e g e T K u x anwPUTMoB B CBeTe K O H U ~ I I ~ HPaCCMOTPeHHbIX ~~, B IjPefiMJlyI4HX naparpaaax. C n e ~ y ne o~n x e p r c H y T b , YTO mu n p a M e p w unnmCTPHP)a(YF J I U U b OCHWHble aCIIeKTH HeWTKNX 8JIrOPUTMOB u H e A e M o H c r p u p y I o T HX -KTUBHOCT~ B p e m e H u u nparcTs~ ~ C K Hnpo&ne~. X HeseT~ue WIrOpNTMbI ~ ~ 0 6K J~I ~0~ C C U @ H ~ COOTU~OB~T~ BeTCTBeHHO o ~ J I ~ c T R M NX IIpNMeHeHUR: WIrOPHTMU Onp,l@JIeMUR U UA~HTH$BKBIJ.UN; ~ U X ~ B N O P B C T U ~ ~ C K 8flIWPHTMbl; H@ a n r o p u T m np UHRTNR p e m e ~ fi. n ( G e ~ y wO T M ~ T U T ~ , r ~ ~ o aJIlVpUTM OIIpeAeJreHHOrO T u n a MOWeT BKJriOWTb B ce6a aJXr0PHTMH A p Y r N X THIIOB K I K IIOAaJII'OpBTMbl, H a n p ~ w p ,anr0PUTM OnpeAejIeHHR MOXeT B K J I I O ' ~ ~ TB ~ c e 6 ~IIOAaJII'OPHTMbl OTHorUeHUR U nPHHRTUR ~ ~ U ~ H U RMU . ) HaqHeM C I I p H M e p a a n r o p n T M a onpeAeJIeHaR.
napannenbao. Aonylue~ue T a K o r o
O A H H~3
OCHOBHMX 06JIaCTefi IIPUMeHeHUR HeYf?TKUX #!U(rOPUTMOB o n p e A e n m w e C ~ O X H ~ I X nnom , o n p e a e f i e n w x nna HeYeTKHX I I o H R T U ~ ~B TepMUHaX %flee IlpOCTMX HJIH MeHee HelIeTKUX ~ O H R T W BOT ~ ~ . nPElMk?PH TaKHX HeqeTKBX ~ O H R T H ~ ~ : x a p a K T e p r J n o s e p K a ; M e p H CJIOXHOCTH; M e p b I ~ O X Q X ~ H ; HWffpeAeJIeHHbIe 60JIe3~ii(IPTPHT, ~ P T ~ P H O C K J I & P ~N~ ) T. 4. ~ O C K O J I ~ KHeYeTKOe Y IIOHHTHe MOXHO PaCCMaTPUBaTb KaK CWMBOJI HeYeTKOrO MHOXeCTBa, i t e ~ m r ~a/ppUm& u~ QfZpf2&A~H#E RBJIRWCR B A ~ ~ ~ C T B W T W ~ H O KOHeYHbIM CTN MHOXWTBOM B03MOXHO HeYeTKHX HHCTPYKUH~~,KOTOPbIe O I I P W R I O T He. W T K O e MHOXeCTBO B TePMUHaX APYrHX HWeTKHX MHOX€?CTB H O )A a w F (u, B03MOXH0, CaMOrO C&R, T. e. ~ ~ K ~ ~ C H B UJIH npouenypy A n s B H s u c n e H u R c T e n e H u ~ ~ U H ~ A J I ~ X H O C KT WUA o r o s n e M e H T a O ~ J I ~ C T Hpaccyxae~uiio n p e m s e M o M y MHCB nOcneAHeM cnysae a n r o p u T M onpeAeneutisr urpam ponb ampurnma udeurnu@wyuu, T. e. a n r o p H T M a , KOTOPM~~ J'CTaHaBJIUBaeT, IlPHHaAJIeXUT UnU HeT 3JIeMeHT MHOXeCTBY, HJIH, donee e e ; - YCTaHaBJIHBaeT er0 CTeIIeHb IIpHHWXJIeNHOCTU. l l p u ~ e p T a K o r o a n r o p u T M a A a e T c s npouenypofi (CM. 15 I) B m n c n e H u R c T e n e H u n p u H a A n e m m T u pana ( n p e ~ -
-
XWLTBY.
J I O X ~ H U H ) B HeYeTKOM R3bIKe, IIQPOlKJ(eHH0M A N L ~ ~ ~ H K BW O -~ TeKcTa r p a k l ~ a T H ~ 0 f i . B KatIeCTBe OYeHb IIpOCTorO I I p U M e p a HeWTKOrO W Q p H T M a O n W W I e H U R PaCCMeTpUM HeWTKOt? IlOHRTUe W. BHOB~ c n e a y e T n o m e p K H y T b , YTO ceepxnpwme o n p w e a e H H e nplrrB e A e a o H w x e numb A ~ Ru n n r o c T p a y H H u H e n p e T e H A y e T urr ToYH4E OIIpeAeJIeHUe IIOHRTUR 06U.4. ~ & ~ C T P Y K ~ H H COCTBBJIRIQ, wue anropum OBAJI, 3qecb n e p e s u c J r e H M . CHMBOJIT B s ~ u x H H C T P ~ K L W ~ 06031iasae~ X u c c n e n y e ~ b ~ 0f 6i ' b e K ~ . Tepmu~61HrntWIb1ff u t n m b s y m f i o6parueuarr K n o a a r r r o p u T a s y ~btm KfibItf, KQTOP& iIpeAHa3H51YeH AJIR O n P W d l B H R , 5 i s n R e q - i nu T B ~ ~ K R Huna M HW. MHCTPYKWK)BHAR WJIH A, TO BI CAeAyeT UHTepIIpeTUpOBaTb, KaK at!CJIH A, TQ B, a H a s e n e p e i i ~K~ c n e ~ y ~ ~ uUeHf Ci T ~ ~ K U H U I .
ma
AJI~OPHTMCTBAJl:
1) W n U T H e 3aMKHyT0, TO T H e W; CTOII. 2) ecan T ca~onepece-iica, TO T H e o a p ~ ;m n . 3) ecnw T H e BbInYKJIO, TO T H e m;CTOII. 4) WJIA T He H M ABYX ~ &nee ktnw MeHee o p T o r o H a n b H w x oceB cu~~iSTpm, TO T ge own; c T o n . 5) ecnu donbrrra~O C ~T H e RBJIII~TCR ropa3~0bee n n u H H O ~ , W M MeHbILlaR OCb, TO W ; CTOn.
6 T -O d-pumu
T
H e OBa/l; C T m .
B ~ I ~ ~ Y K J I B~ IOCHOBHOM R: m noAanrnPHTM BKJXIOqaeT B ~ e 6 RnPOBePKY T O N , COXpaHReT J'IU KptiB v 3 H a 3 H a K I l p H &BUXeHUU BAOJIb T B n e p B O H a W M b H 0 BH~ ~ ~ H H HaIIpWAeHHH. O M 1) x = a ( ~ e ~ o ~ Ho aps aan b~H a R Tosmta T). 2) ~ b 1 6 p a ~Hba i l PaBJIeHMe ABUXeHNR BAOJIb T. 3) t ..r H a n p a B n e x u e ~aca-renbHofi K T B x. 4) X' = X 1 ( A B H X ~ H U ~ U 3 X B COCeAHmK) TOYKY). 5) t' H a n p a s n e H u e ~ a c a ~ e n b ~Ko iT i B xr. 6 ) awyron my t1 H t. 7) x
-- +
--
xr.
H a n p a m e H u e ~ a c a w b ~ oKf i T B X . 9) x' x +l. 10) t' w H a n p a a n e H w e ~ a c a ~ e ~ r bK~ oTk B x'. 11) fi yrm ~ e m q y ti w t. 12) eWH $ H e UMeeT T ~ K OXe~ 3HaK, YTO H a, TO T
8) 1
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He 8trUlYMIIO; BO3BpaT. 13) e C n U X' a, T o 6bMYICllO; B03BPaT. 14) nepeRTu K 7). I f p ~C n e ~n y e T. OTM~TUTL, TO n e p B H e T ~ HHCTPYKH LLUH B a n r O p H T M e OBAJI o n p e A e n e H H u e . rIn, we K a c a e r c x
.-
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KOMaHA 4) 11 5), TO OHA BKJIIOqalOT O l l p e A e J I e H U R T a K H X nOHRK a K a6o~teeUJIH fieflet? O p T O r O H a J I b H b D H 3Na%UfTleRbiu, A J I H H H W , KOTOpbIe XOTR U RBJlRlOTCR H e s e T K H M U , B C e Xe MeH e e CJIOXHM u 6onee ~ O H R T H M , q e M n o H m e man.% HJIJM)C T p H p y e T OCHOBHYH) (PYHK~UIOHeYeTKHX aJIM)PHTMOB O n p e AeJreHUR, a UMeHHO, C B e A e H U e HOBbIX HJIH CJIOXHbIX HeYeTKUX I I O H R T W ~ ~K 60~1een P o C T b I M UJIU 60JIee nOHRTHbIM HeYeTKHM IIOHFITUHM. B 6 0 J I e e p a 3 p a 6 0 T a ~ ~ B0a ~p H a H T e a J r r O p U T M a OBAJI OTBeTaMH H a 4) H 5) MOrJIH 6b1 6 b l ~ bC T e n e H H BbInOJIHeHUH Y c J I o B H ~ ~ B 3TUX UHCTPYKUEIRX. K o H ~ Y H ~ I Mp e 3 y J I b T a T O M a J I r 0 p H T M a T O r A a 6 ~ n a6bl C T e n e H b IIpUHllAJIeXHOCTH HeYeTKOMy M H O X W T B Y OBaJIOB. B 3~0fiCBR3H CJIeAOBaJIO 6b1 OTMeTHTb, YTO B CHJIY (5.15) ZUIrOpFiTM OBAJI, K a K yCTaHOBJIeH0, n p k i 6 J I H 3 U ~ e J I b ~ 3KBEI0 BaJIeHTeH BbIPaXeHKH)
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man = sanz~tiymatln tie ca~onepecetcatoquZicRn s b ~ n y r c n b l d n donee unu v e H e e o p m o z o i l a n b t i b l e mu cunznzempuu (6.7) n 6 o n b l u a x O C ~~ u a ~ m e n b uanuwuee o mu, KOTOpOe O n p e A e J I R e T H e s e T K O e MHOXeCTBO 08UR K ~ IKI e p e C e Y e H U e 0 6 b 1 4 ~ b 1U~ HeYeTKUX MHOXeCTB, CUMBOJIH KOTOPbIX CTORT B n p a ~ 0 f i W C T H (6.7). O A H ~ KOAHO O CYIILeCTBeHHOe o T n k i w e C ~ C T O H TB TOM, TO WOT a n r o p m M H e T O J I ~ K O onpeA e J I R e T I I p a B y I O W C T b (6.7), HO T a K X e YCTaHaBJIHBaeT n o p H &OK, B KOTOPOM C J i e A y e T COBepIUUTb 3 T U Bbl9HCJIeHUR.
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U 3 H a 3 B a H U R , H e 4 e T K U e a3IrOPUTMbI n O p O X A e H U R C J I y X a T A J I S n O p O X A e H U R , a H e AJIR O n p e A e J r e H H R HeYeTKUX MHOX~CTB. B O ~ M O X H n pHH ~ J I o X e H U R 3THX aJIrOgUTMOB: IlOJrym H a e o6pa3vo~nosepea H A p y r o r o p o A a obpa340~;pesenw IIpllrOTOBJIeHHR IIUUki; COYHHeHHe MY3bIKH, n p e ~ J I 0 X e ~ H k B eCTeCTBeHHOM R3bIKe, p e 9 H . B K a s e C T B e n p O C ~ 0 f iH J I J I l o c T p a Q H U aJIrOpHTMOB ITOpOXAeHUB p a c c M o T p a M H a n H c a H H e ~ Y K B MP, r A e ' s b I c o T a h H WHOs a m e b n p e A c T a B n r r m co6ofi n a p a M e T p M a n r o p H T M a . nPOCTOTbI P 6 y BOCnpOH3BOAMTbCR ~ ~ KPK ~ Y H K T N P H ~ I ~ ~ o6paseu C BOCeMbH) TOYKaMH n0 ~ e p T E i ~ a J I b ~ JIUHHH. 0fi A J I ~ o ~ u T M P (h, b): 1) i = I . 2) (i) = b ( n e p ~ a f i T O q K a H a OCHOBIHUH).
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OnUCaTeJlbHOI'O a J I r 0 PUTMa, 0 6 0 3 ~ a Y e ~ ~ 0 r 0~ O T 0 p b 1 f iBKJIIOqaeT B c e 6 ~ TPN HapaM e T p a x, y, z. 3Tm a n r o p M T M onpenemeT H e s e T K o e T e p H a p H o e O T H O ~ ~ H HB~ O ~ J I ~ C TPUa C C Y X A e H U R = 1$2 3 f 4 f 5 C TePMUHaMM ~ a / l f ? t i b l C ~ d U ~ O R ~ U U )O~n ,p e A e J l e H HHMH (4.2) a (4.7).
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f l p U ~ & 3 6 ~ 0 ~ U M o c l . 3HarleHUH u 3TUX ~ b I C ~ a 3 b U 3 a H U fMOXHO i B E J s u c n u T b c ~ O M O ~ A(5.9) O u (5.11). B TOM cnyqae WHOweaue 6 y y n e ~ n e p e c e s e H l r e M m~orue~uii, onpeAweHHHx HHCTPYKUUHMW 1) U 2). Apyrok n p u M e p H e s e T K o r o anr0pm.rhl.ao T H o m e H u r r F (x, y), ~ o ~ o pu n~n il oic T p u p y e T npyroii a c n e K T n o a o 6 ~ ~anropmx MOB, COCTOUT B C n e A y u m e M . A ~ z o p u m F (x, y): yem1) ecnw x M a n o a x c~a60y s e n u s w b , TO y 9HTCR. 2) t?CJIU X MfUO U X 3HaclUtTERbHO YBeJrUrlHTb, TO dl yI3eJrH9UTCR 3 ) E ~ U t n e R b H O . 3) emu x wnmo H x c n e r K a y e e n u c l u m b , TO y yB@JIH¶HTCP
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TO y yemaW i t b 3HaYUt?WRbNO. H B n p e n u A m e M cnyrlae, s ~ a s e ~ n H eas e T K u x yc.rio~HHX , n p e ~ ~ ~ o ~ Be ~TOM u f i anropume MOXHO B ~ U C J I H T ~ , ~ c n o n a 3 yM~~ T O A ~wI3 pa3neno~IV u V, ecns AaHM o n p e A e n e HUH IIePBUqHbIX TepMUHOB 6~?Abu10d W MZMttbtC~d, a T a K X e HeqeTKHX TePMHHOB CRU60, 3HUclUllzeRbN0, Cpedile. K a s e c m e n p o c T o r o n p u M e p a 6 ~ x e ~ u o p u c ~ u r l e c anro~oro pwma n p e n n o n o x u M , m U M ~ CRu e T e M a S c ABYMR onpeEWX?HHbIMH COCTORHURMU (CM. [3 0 6 0 3 ~ a q e ~ ~ bgl I ~U Hg8, AByMX HeYeTKUMH BXII)AHbIMU BeJIHqUHaMH N U ~ K UH~ BWCOKUG ki A B ~ M R HerleTKHMW BbIXOnHbIMH BenNqWHaMU 6 o n b w i i U Af&WG. 0 6 I a c ~ b I C ) P a C C Y m @ H H H 3 H a q e ~ t i i BXOAa t H BblxoAa cIsnrreTcR neficmmenb~a~ ocb, I I p e a n a n o x ~Ranee, ~ WO nOBeAeMUe S Maw0 I I ~ H ~ J I H X ~ H H OX Oa p a K T e p H 3 O B a T b n p w s e n e H n u M H w w e a n r o p u T M o M . OAH~KO n n R O u t i c a H H x OTHOW H N R M e X A y BXOAOM, BblXOAOM U COCTORHHeM CUCTeMhI BMeCTO B ~ I C K ~ ~ ~ I MbI B ~ HCnOJIb3YeM H U ~ ~ 06~9~ ~861Iliky 0 nepeXOAOB. 9HTCII
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B 9 ~ 0 8T ~ ~ J I UTU~ P~M U H & p o s m f i o w ero M O ~ H @ ~ K ~ U H H W H ~ u MUO c n y x a T AJIR n p ~ & n m e ~ ~oon urcoa i i H H eepomHOCTHbIX XapaKTf?pUCTHK. Hanpu~ep,aecnu BXOA M U A TeKYCOCTORHHe eCTb gl,TO C J I e A y m e e COCTOSIHHe 6f?pORmHO 6ynm g,,. A ~ a n o r u s ~ aecnu o, BXOA &?AUK u T e K y m e e COCTORH H e eCTb g p , . T O W H b MUCMf?pOIUtlMO, qTO B b l X M 6 y ~ w eeAlU€. ECJIH3 ~ a q e ~ uTee p M u H a sepomm o n D e a e n e H o c r r e n y m w M 0 6 ~ 0 1 3 0 (CM. ~ (4.16)):
5) ECAU Bbl HaXOAHTeCb OWHb f?ZO OCTaHOBNTeCb; H e p e f i ~ H K 7).
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6) n p o n o ~ ~ x a i iocleiib ~ e ~ e d ~ e f npu6nmambc~; t~o nepeii. 5). 7 ) Ecnu H a n e p e K p e c T K e H n H 8 6 1 ~ 3H ~e r o HeT M a w H H , rno npoe3xafi~e. 8) ~ O A O X A H T ~HeCKORbKO CeKyHA; n e p e f i ~ H K 7). B p ~ nJIH CJIeAyeT rOBOpMTb, 9TO p e a J I b ~ b 1 g BapMaHT 3TOrO a n r o p H T M a ~ M J I6b1 HaMHOrO CdIOXHee. B ~ X H ~BI M 3TOM n p H -
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1)
li'odan~opumu 3HAK: 1) Ecnu H e T s ~ a cccTonB ~ a H a sameii c T o p o a e , mo ecnu H ~ TM a m u H ~a n e p e K p e c T K e , mo n e p e c e K a T b H a ~ o p ~ b n o P CKOPOCTH, UHaW )KAaTb, n O K a aBTOMaiUHHbI H e OCBO~OARTnepeKPFTOK, H 3 a ~ e n e~ p e c e K a T b ero. 2) Ecnu B a a a B T o M a m H H a H e npu6nu3unacb K nepeepeCTKY, mO IIPOAOJIXaTb A B I l X e H H e C ~ 0 p ~ a J I b H 0 fCKOPOCTbK. i B T e q e H H e necmbtcux ceKyHA; n e p e i i ~K~ 2). 3) 3 a ~ d n u m bd s u ~ e ~ u e . 4 ) Ecnu BM O W H ~cnemme A H e BHAHO nonauebc~axMarum, k~ H ~ TM a m m H a n e p e K p e c T K e u n H tz06nu3ocm~OT H e r o , mo iuedne~tio npoesmaik-re n e p e K p e c T o K .
B K a s e c T s e n o c n e n H e r o npmepa p a c c M o T p w M a n r o p m M n p H H R T m p e r u e H m A n R n e p e B o A a cnenoro M c n b I T y e M o r o H I13 H ~ W J I ~ Hn03E19kIH O~~ Cmapm B K O H ~ ~ H ~ll03MLlUK) K) pnb (CM. PHC. 3) ( 0 9 e ~ bCJIOXHbIe q e T K I l e ~ J I I ' o P ~ T M ~TaKOrO I Tuna AJIR HCllOJIb3OBaHHR p 0 6 0 T a ~ W n p H M e H e H b 1 B Shakey p06OTe, IIOCTpOeHHOM r p y n I I 0 b M C K ~ C C T B ~ H H MO H ~TO ~ J I J I ~ K TB~ C T ~ H @PACKOM M C C J I ~ A O B ~ T ~ J I ~ C K OMHCTATYT~. M O n H ~ a H l i e 3TOlW po60Ta npHse.qeHo B pa6o~e1201). A n r o p F i T ~ , ~ a 3 ~ a H ~ b IOBSTACLE fi ( n p e n ~ ~ c ~ s ano e), IIpeAnOJIOXeHHlO EICnOJIb3YeTCR 9eJIOBeKOM-KOHTPOJIePOM K O T O P ~ IMOXeT ~~ H ~ ~ J I I o A ~ TK~a K , H BbIllOJIHReT eI'0 HHCTPYKUHa. 3 T a H e q e T K a R 0 6 p a ~ ~ a CBR3b R W r p a e T B a X H y I o POJIb B C03AaHHH BOSMOXHOCTM AJIR C H a n p a B H T b H K l+?AU He; CMOTpR H a HeqeTKoCTb U H C T P Y K I ~ M U ~ ~ H a O I I I H ~ K U WX BMnOJIHeHHR. A J I ~ P H T M COCTOUT k13 T p e X n0aanropu.rMOB ALIGN ( B ~ I P ~ B H U B ~ T HUG ~), (npHfiepxW~a~bcR)H STRAIGHT (npa~o). 3anasa a n r o p u T M a STRAIGHT n e p e B o & H CO C T a p T a B ~ ~ O M ~ X Y T OnYHKT ~ H ~ I1,~ 3~a T e M H 3
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n p e n m B H f f AO T e x nop, noxa H e o T K p o e T c a npsuo% nym K UWIH. BM~CTO OnHCaHUR 3TNX nOAaJII'OpUTMOB B TePMUHaX H e W2TKHX YCJIOBHbiX ' ~ o J I o x ~ H H ~ ~KaK , 3TO AeJIaJIOCb B npeAJ&QyU U X n p U M e p a X , y ~ 0 6 ~ eORUCaTb fi HX B BNAe ~JIOK-CXW, KaK n o ~ a 3 a ~Hoa puc. 4-6. B 6 n o ~ - c x e ~a ner o p u T M a ALIGN E 0 6 0 3 ~ a q a e To m a 6 ~ yBbIpaBHHBaHHSl, I4 MbI AJIX nPWTOTbI npeAlIOJrO%NM, q T 0 E H e MeHReT 3HaK. TepMHHaX HeWTKAX BHCK~~MB~H ~ JHI~o ~K - c x ~aM n r~o p m M a STRAIGHT, H a n p u M e p , IIepeBOAkiTCR B C J ' I ~ y I O l Q U e HHCTpyKSHH.
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l 7 0 a w o p u m ~ STRAIGHT: 1) Ecnu H e 6 ~ ~ 3 1 mo ~ 0 ,c n e n a T b mar; n e p e i i ~K~1). 2) Ecnu H e o w ~ 6.0~3~0, b mo c n e n a T b ~ l a ~ b l mar; li nepefi~~ K 2). 3) ECM H e OWHb OWHb 6 . 0 ~ 3 ~ m0 0 , CAeJIaTb OWilb &a;llblfi war; n e p e i i ~ K~ 3). 4) Cmn.
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