Research and Development in USSR Cybernetics

Citation preview

Research and Development in USSR Cybernetics

Research & Microfilm Publications U.S. Government Translation of

RESEARCH AND DEVELOPMENTS IN USSR CYBERNETICS

Moscow 1969

This document is prepared by Xerox Copyflo. Since the publishers have access only to those documents translated and prepared by the U.S. Joint Publications Research Service, the quality of text and illustrative · material herein is limited by the quality of the documents Received by Research & Microfilm Publications.

CCM INFORMATION CORPORATION A subsidiary of Crowell Collier and Macmillan, Inc. 909 Third Avenue, New York, N. Y. 10022

JPllS: 47,821

10 April 1969

l,

I

l 1.-

~ ·I

~ Iia

l

msa ~ c s .

1 CCffl'EWTS Cybernetic l.bdellng ot Thought · · (Krnmnm-J st Belorussii) o e, ee o •••oo • • • • • • • •••.• ----- ----.. ISi

Reading Automat.ion

PAGE · • ••• ooe ece• •

1

;i

.

(Now i Zh;J.zn') ••••••••• ~ ••••••••••••• , •••••••••• ~ ••••

·l .,'1 1 ; ~ . I

11

.! ? ·j

.

·1

Reflect.ions on the Structure ot Inductive logic

(Voproey r11oeoa,1>~··•···························•·•··

16

•

[ I • tESil • N]

"';. =.., _____., ""'"'_________________________________________

.,..-.•L - ~

--=---- - - ===-

-:::;.._~

,, ·•• • 1 ; ' • \ ; •,•

. l ..

...

.'

, .. ,.

•~~

t I

H,: I,:··, ... ,, . ..J •

~- \ ..,

.,·· '

l

_.

' t ·•

:1

·i q

_.l

· .• :

-:

;

·- •..

\ '

c..

•'.\

, • ,"

, 11 . ; .

i-; .'ti

, •'\"\.,· I'

I

f

;\; ,

~ ~

,I

1 .;:, . · \

~· .': . 1:1 ~ .

,._

f, ' "

i .:_ : ~ '!

J ..

. • I

•

•

J.. .,.

I ...

>j

•

• •

•

~- ., I '--;.

.H t .,,_•-:-ri: .. ·.. .. ,

: .. , ' . .- ,'i ·.., ·. I. F

.,

• ' ; ;_,

\.

/'., •. •;,_,,,. 't

--:- ·: .-,, _; [Article ~ Prof•••or A. Sptrlda. Doctor of . l'hllo~phlcal Science•• •. ·: Mo■cow; Mln•k• IColmlunl ■t Beloru■■IID ltua■ lan. No 11., Noveaber 1968. pp 31-37) J •

.,

j

,

.,.

~· r

,..

·• ·:;;,t

la

I

I•'••

: ,-

.. .'.. !, ,' , ·:,,. , ,·,·)

I

-

,;

• f t • •:

•:,

' :' f

'

• ~( .:

·\

a, THOOCIHT •. : £•

•.~

I,

~-~-• ·'

f"' ... ;-:., ~.. -, :: ,· ,,

• I

/ ....,..,l_',.'

f;- _f

I•, , •

l.

·: . ' ) _, ).

• I

(

• _.,,-!

'.!o~1· ....••

,• I

! • •·;

'·h.

),

. ,_;

CYB!RN!TIO tam.ItfO

•

.

-~----.1.~

.-c.:.-•.

/

I

J ,.Ir. -

< I

The creative develo,-ent of cybernetic• ha• in a thoroughgoing

•

•

...._

and entirety convincing fashion corroborated the need for the most In• tl•te contact between lt and the phllo■ophy of dialectical materlall•9 The solid connection disclosed between the pressing problems of phllo•• ophy and the problema on which cybemetlca la vorklng has demonstrated that this ■clence la at the present time proving to be an exceptionally · ■ lgnlflcant general- ■clentlflc methodological tool for lnveatl~atlng a multitude of natural• ■oclal. and spiritual procea■ea. Such a tool la, In particular. made up of the cybemetlc categories of 'Information•• · • 'control,' •organization.• ' ■elf-organization.• 'feedback.' and other concept•• a■ well as the treatments of new fields of mathematic•• which the ■e categories are ■tlmulatlng. and the very rich technical resource• '· of cybemetlc ■ whlc~ are being applied in extensive fields of science and practice of ever-increa■ lng range.

·~ I I

\,.

I

I

One of the cardinal phllo■ophlcal problem■ ln cybemetlcs· has been and reniaina the problea of artificial "intelligence." the problea of whether a machine -can think, about which the heated debates have not even yet died down, Some believe that thl ■ l ■ not an urgent proble•• other• that its urgency l ■ of the highest degree. ■till others that the probleftl la not a rightful one. a fourth group that the very attempt to model the human Intellect l ■ stupid, while even other■ ln tum cherish a fifth opinion that the preceding one la unintelligent. and ao on. Thl'a debate ls going on both ln oral conaunlcatlon and In the press. Thus. the newspaper Vechemly Minsk of 29 December 1967 publl ■hed an article by Profe■■or A. ICarlyuk ·entltled "Can a Machine Think?" This article enunciated the Idea of the qualitative, ■oclohlstorlcal ■peclftclty of man and his rea ■on, of the ancillary role of •logic" machines, and of the fundamental Irreducibility of human thinking to the logic operations of cybernetic devices.

·11

la

I I

I

[ ~ 1,\,l

1rl

In reply to this article the Zvyazda of 14 March 1968 ran an article by P. Prot•Hnl and A. Rakov captioned "Not Advice. but Charad•••" vhlch In a coaruly hard-■elllna and satirical t'llle ■poke of the first

~

f 1'

'\

I

• I •

~

~

!·

,,

- )

J

......

I .

)

r: .,-· ·:r ir··· b:s :: · ; .·

,· •{ •··· 21ww ·· :tr

t I tr

·tr

I

I ,

' I

j

l

I

I I

('

l

I

I

t I

tllua, •A. ICarlyuk ~• taken up a ■tranae po1lt1on for a 1clent11t.· He belte¥9e It offen1tve and ln1ultln1 to hU111an dignity to c0111pare the

Intellect vlth 1 thou1htl1•• autmata.• Thie 'noble 1ndt1natton' 1erve1 a• the around• for conclucllna that 'attempt, to r•place man by a machine illult be refuted tn advance.• But what do the fact, confirm? The author•• theala contradict• the whole ht1tory of the development of culture, the IIOlt important feature. of which 11 preci1ely the replacement of man by a uchlne, flr■t in tile aru of phy1tcal, and then of intellectual labor.• In P. Prota1ent and A. Rakov'1 article tt ts asserted that the probl• of mdeltng the •ntal functton1 of man on a computer 11 •quite realistic and to a 1lgnlflcant degree effected _in practice.•

!.

!

The thought occura to• that the different viewpoint• In the•• from the different understanding of such fundamental categorlee a1 •man,• 'thlnklna, • •con1clouene11 ,1 'machine,' 'Image,' and• 1ucce11lon of other■• arguaent1 t1sue

• It must be etated that philosophy has no ba1l1 for erecting any aethodologlcal barrier, either In coanltlve or In creative activity. The history of actence, particularly In recent decade,, has entirely altered our concept, of the po11lble and the Impossible. The thoughtful IIOdem philosopher Is Inclined to have a more skeptical attitude toward scientific dogma,. He takes a dlffere·n t approach even to what previously aome tended to regard a• 1omethlng Impossible. Nowadays ' the term 'lmpos1lble' 11 becoming Increasingly discredited -- If, of · course, It 11 1omethtng within the framework of objective laws which 11 under dl1cu11ton. ,I [ I t

f

f',

l

Dialectical material!• proceed• from the fact that every sort of pattem 11 concrete and qualitatively specific. Every level of the . 1tructure of matter niuat be approached with regard to Its qualitative determinacy. In thl1 the higher Includes the lover as one of Its prea• laea and at the ume tl• a1 Its own basta.

1

When scrutinizing the problea of •consclousnes1 and cybern•tlca• tt ••t be taken Into account that the problem of "man and hi• lntelll• f :, ·" aence• ta not only and not 10 much a cybemetlcal problem. Thls la f a problem of the whole c011poslte of modem sciences of nature, 1oclety, , and man. I

! '· · ·

/

The controver1y which 11 occurring over thi1 matter 11 taking r place because 1ome author■ are unrtghtfully clal•lng that the probleni / .. ,- of hlBlln intelligence, tu 1tructure, and the fea1lbillty of modeling ,·. .it ii entirely encornpa11ed within the frameword of the categories I which cybernetic• utilizes. Thia ii an unlawful claim. The h1a11an ! brain and the products of ta activity represent nn extraordlnarl ly COIi• pltcated phenomenor1 of phy1tcochemtcal, blologtcd, physiological, /-' paychologlcal, 101.tcal, ltnautatlc, cybernetic, sociological, esthetlc,

t, ·'·

l. l [.

I. r

• 2 -

•

I t;

and phlloaophlcal nature • . Thrucia . . knOllft to ■ctence have H•tnal of -alLthe- fora■ of .aotlon of 1111tter oua unit In the brain-. the'i::"" thaaelvea Into a alngle :conUnu. nal characteristic• of the con■ el of cybemetlc IIOdellng such cardl•. Ideal • •wnt • •mott- · • , c 1ouanea■ a■ •aubjectlve Image,• 'the • • ·•• conaclen ll■ lta. ce, • and ao ,on remain beyond lta

°"

• •

One of the Important probl t · • gatlon of ■an and his Intellect 8118 In ·thla aort of composite lnveatl• and Internal conditions of er 1• to uncover the social determination feats Itself, 81 well as the eatlvlty In all spheres In which It manl• . •tructure- of our Intellect and conaclouane••• In the approach to the bl · · onesidedneaa of thla type la . :0 811 of artificial thinking Intolerable that thinking arlsea from I t etlmea perpetrated • e thft aupposltlon that It la a natur 1 cha ~ raorganlc physiological brain processea, thinks just becauaa It lracherlatlc of the Individual, and that brain 1 t e brain. The ~ g belief la encountered t • hat everything depends on how the brain .la organized and what physlo- · logical proce11es go on- In lt. Hence lt ls deduced that It ts enough : to create a model of the brain for this IIOdel to begin to produce thoughta, Ideas, feelings, and effort■ of will. This assumption ls naive.

·

·

The &ht of the matter la that Intellect ls_. not simply a natural property of the brain and of man. In the last analysla the true subject of conaclouaneaa and reason la not only not the br~ln, and not even 11an as such, but aoclet~. A social organlsa thinks tn the person of man . by means of hla brain. Man thinks ln a social fashion only as the subject of history. Han leama to think by mastering the logic of utlllzlna the prapatlc. world which la undergoing creation by all preceding history, logic, and by all culture. Alwaya and everywhere man carries with himself his whole lndlvldual history and the history of mankind. The hand and the brain, nourishment and multlpllcatton, hereditary. changes, even pathological change• ar•, ln Marx's words, the result of past world history. The specifically human level of the determination of mental processes, processes of thought, consist In the social condltlons of existence.

i

i' [

I'

:I

Man I ■ a social creature representing the highest level of devel• opment of living organisms on earth, capable of producing the tools uf labor and of e11ploytna the• tn hb- action on his environment, and possess- · In&. articulate speech, consclou■ness, and weltanschauung. Han ls the subject of labor, thought, aensattona, will, beliefs, and COIIIIIUnlcatlon. He ts the subject of scrutiny of the totality of social sciences, as well as of certain dlvlslons of the natural sciences, medicine, and the technical sciences, vhlch analyze 1111n fr011 their nwn speclftc angle of contemplation. Philosophy, expressing the essenre of man and. of hls relation to the world, has been s..-oned to effect an Integral theoretl• cal lnvestlgatlon of man.

!'

I'

I'I:

i I~

i

I

t '~f-

• 3 -

'I

.....------------------------.-

~ ,1

;.

·,,.~

In pre-Marxlat and non-Marxlat atudle■ auch conatltuent charac.. certatlca of ..n aa intellect, the ability to operate with •yaab01• a nd co ·thtnk, are u■ually dlattn1utahed. It la precisely herein, and often only herein, that the dect ■ lve dl ■tlnctlon betveen man and antmal• 1• 1een. Marxt• haa deaon■trated that theH properties do tndeed comprise the char~ctert ■tlc featurea of man, but that they are derivative, not ,: ort1lnal, tn nature tn deter111tntng the ••••nee of man. The Initial I' characterl ■tlc of man la the capacity for effecting conscious tran■• , . foraatlon of reality by arttflclally created toola. "The first htatorl • cal act ••• of lndlvtduala by whtch they differ f·rom animals does not r r . • consist ln their tblnktna. but ln their beginning to produce the mean• I vhlch are neceaaary to theai for life" (Karl Marx and Friedrich Enaela, · Sochlnenlya [Works] , Vol 3 0 p 19). I

'.

Althouah conaclousneaa and aelf-conaciouaneas are essential to •an, man ls not Identical either to conaclousnesa or to aelf~consctoua$ neaa, aa ldeallata assume. As does any llvlng creature, man need his bodily nature, for the body constitutes hla only natural life. But ..n cannot be reduced either to hla spiritual prtn~lple or to his bOdlly oraantzatlon. He Is a unity both of the natural and of the social, both of the physical and of the spiritual, both of the hereditary and of that developed In life.

I.

• •

• I..

Marxl• refute• metaphyalcal and ldeallattc Idea■ of the exl ■tence ,, of Innate Idea ■ and capacities, unchangeable properties of the psychic life of man. The anatomophyalologlcal lnatlncta with which a child ts ! I. bom cannot In themselve■ lead to the rise of cc,n,plex psychic peculi• I - -; I . arttl~•• These trait ■ do not appear tn the proceas of the lndlvldual '., . . I .· . . development of a person, but are formed, for the child learns to be t ..... -, '· a person ln communication with adulta. The aaslmllatlon of aoclal, I hl ■torlcally lald•down types and forms of activity and the transformat i tion of them Into hla own active capabilltlea are the main condition I and the declalve mechani• Jn the Individual process of becoming a man. I

I

I·

.. i •·•

i, ; ~

I

I

Ii (

I

,•I • I \ I

'

The starting point of the Marxlat concept of man ls treatment · of ht• as a derivative froa aoclety, •• the result of rise and develop. sent of aoclal~labor activity, for In his whole physical and spiritual beina •• upright aalt, cerebral atructure, general facial feat~res. the shape of ht ■ hands, · his speech, emotions, and Intellect ••1 man ta -Indebted to the labor and aoclal relationships which have taken shape on that foundation.

•

Man ta not somethlna one• and forever given and comp'leted. He I• a concrete•hlatortcal being changing in the course of active trans. , \ 1·, formation of the natural and aoclal world. The f~rmatlon of the physical construction and of the aplrltual itructure of man ls the product of u-n•• Intellect, dhll aplritU1Jl capacities and inter. _ un l versa l ht s t ory • na · . •,·. e1t1, are f orae d 1r, labOr and e111bodie In lta result ■• By what signs , a ■ked Lenin, can ~e Judie of the real thought■ 1nd emotions of the real ..nt

- 4 -

1.

', ·1

":,• ·' - .

$$:XV

:: a ·· a r r r

: 1¢

·: ··

x 2 ·rt ~

1·

·er t1tf ,· :: rt1l¥ s : >:i:&rt1thr:ds s'ritittbctr de

And he anawred, "Naturau,, there can be but one •uch •Ian : ••_. aotlOft. •

•

The natural prehlatory of ..n preceded hla •oc.lal hl•tory, . 'nlla prehl•tory vaa the evolution of the anataaophyalologlcal atructure, the aeraa of labor activity amona the anthropoid ape•, the develop111ent of areaarloua relatlon1hlp• In the hlaher animal•, ~and the -evolution of audio and ·-,tor Mana of •l&nallng. But the comparatively high level of development of animal• In the anthropoid ape atage lnnedlately pre• ceding the appearance of man contained only the opportu~lty for the genesi1 of man. The decldlns condition of ' the actual transformation of the anthropoid ape Into man was labor. The start of manufacture of artificial toola of labor stanlfled the ,tart of the rise of man. Labor proved to be the determining Influence on the development of conscl0\18• nes1, on the perfection of the cerebral structure and of its cognitive capacities. "First labor, and then and together with it articulate speech were the tvo mo•t Important stimuli under whose influence the brain of the apea vaa gradually transformed into the human brain" (Friedrich !ngela, Dlalektlka prlrody [Natural Dialectics], 1955, pp 135-136)

I .

Animal• cannot produce radical change• In the condition• of their existence; they adapt to their environment and depend on It; they are fettered to a certain element-• air , water, or the dry land -- which determines their mode of life. Man himself creates the conditions of his existence by transforming -hi• natural environment. Here man differ• from the animal ''not only In that he changes the form of that whlch is given by nature; In that which ls given by nature he atmultane·>usly also realizes his own cognitive goal which aa a law determines the •nner and nature of his actions and to which he mu■t subject his will" (Karl Marx, ~pital [Capital], Vol l, p 185). Before every person entering life la spread out the world of objects and social transformation• In which la embodied and objectified the activity of preceding generations. It ta exactly this humanlfled world In which every object and proce•• Is, as tt were, charged with human meaning, aoclal function, and purpose that surrounds man~ And only through It does man enter Into connection with natur~. Assimilating thla already hU'llanlfled nature the child In various ways unites with the human easence and the existence of culture. In thh union of the person each of hla human relationships to the world•· sight, hearing, smell, ta■te, touch, thought, contemplation, emotion, desire, actlvlty, love In a work• all the organ• of his Individuality, participate • The historically forae. standard■ of right, morality, domestic life, the rules of thinking and grannar, esthetlc taste ■, and so on front the very begln,,lna fora the behavior and reason of man and make of every Individual ean a repreHntatlve of a certain mode of llfe and level of culture and psychology. "If by hla nature man 1 la a social

•·,

- 5 -

t'

I

i.Jex·?:-- tMt.i

; t

to'·,

t±: r·

•

:- ·

1 ½

a 11 1

, ·t1i tw: w..., · t

• 0 ft

tr bl

l

I'

.

'c · ··

i

... ..

g .

.

· nr

· 2t

•

'

belq,· then It foll•• that onlJ· In aocle~y can he develop hi• true nature, and the atrangth of hh nature auat not be judged by the atrength of holated. lndlvlduala, but by the strength of all of .,_ · aoclety" (Karl Marx and Friedrich !nae la, Sochlnenlya (Works], Vol 2, p 146). ·

In opposition to lndlvlduallatlc teaching• In which the Individual aan and hta unlqueneaa cme to the fore aa the prlt110rdtal given fact, aa an Identity encloaed tn ltaelf, Marxl• regard• man as something which has been conditioned by aocial relattonahlp1: every 1111n bear• all htator, vlth hluelf. The underatandlng of man as a social beln& · la deeply baaed by Marx, who wrote that "the essence of ean la not an abatract proper to an laolated individual In tt• actuality It la the totality of all aoclal -relatlonshlpa•• (Karl Marx and Friedrich !ngela, Sochlnenlya [Works], Vol 3, p 3). Thia thesis has enormous methodological stgntflcance, giving bearings for consideration of •n as a socially determined creature, not as an Isolated monad. In crtttctzlng the concepts that man ts some Isolated and self-enclosed 1110nad Karl Marx emphasized that tndtvtduals "create each other" both physt'cally and aptrltually and the development of the tndtvtdual ta stipulated by the develop111ent of all the other In• dtvtduah wt th vh0111 he h In direct or Indirect connuntcatlon • . The . perception by man, of himself as himself ls alway• mediated by his attl• tudes toward other people. "In some respects man ls reminiscent of .· a coaaodtty. Since he ls born without a mirror tn his hands and not aa a Flchtean phllr1opher ("I am I"), man first l?oks at himself,•• In a mirror, only In another mane Only by relating to the man Paul as to one like himself does the man Peter begin to relate to himself as to a man" (lCarl Marx, lCapltal [Capital], Vol 1, p 59)o Man la Included In every aspect In the context of connunlcatlon with society, even when he remains alone with himself. Even, wrote Marx, "when I aa engaged tn scientific and like activity -- acitlvlty that only In rare cases I can perform in direct Intercourse with others•· .even then I am engaged in ioctal activity because I • acting as a man. A• a social product I am not only given material for my own activity -even the language 1 tself in which the thinker work• •· but also my own existence 1s a social activity; and t.h erefore even that which I do of my own person I do of myself for society, acknowledging myself a social being" (Karl Marx and Friedrich Engels, h rannikh prolzvedenly [Fr0111 the Early Work•], Moscow 1956, p 590). · Emphasis on the social nature of man does not mean ignoring his biological aide •• either of the general or of the Individual characteristics of hi• bodily orpnizatton. Every individual ta a unique individuality In the whole make-up of hla phyaic,l and spiritual trait• and at the,_... ti1111 he bear• In himself~ universal human prln• clple, a certain g,nerlc essence. He makes ht• ,,ppearance as a personality

• 6 •

.·

I ~ /

vh!n heattatM •elf-conaclouane... undentandlng of hl• •octal relattona, and Interpretation of hl■Hlf aa a subject of hl•torlc creation. . • Man•anaturallndlvldual characterl•tlc• alao participate In the proce•• of fonilng hl•• but they remain, a• It were, neutral In re•pect to the •ubatance of activity. The mental capacltle• and properties of un are fotw!td during hi• life In •oclety and are determined by concrete social condition•. Man ral••• hl••elf to the level of personality by the force of the historic proceH, paHlng through the Immense history of hla development, beginning with the gregarious state, to the sur1111lt of IIIOdem culture. Under the conditions of life of the generic collective the lndtvldual man still does not become Independent with respect to the community. Personal Interests ■till have not been segregated fr0111 the Interest• of the collective, and the personality, as such, ls still absent. During •octal dlfferentlatlon. and the development of personal rlghta and duties the lndlvldual more and more dlstlngutshea. himself frca the collective a~d beconles a personality. The naturalistic treatment of man and his intellect, a treatment powerle•• to explain hl• constructive- creative activity, 1• overcome by the theal• that the key to the understanding of man and of hi• intellect la in the hands of aoclety which regenerates Itself through dally objectlve practical activity that tranaforms both the extemal world and man himself. The tran,fomlng action of the objective •• that ls, the historical, not the organic -- reality also embraces the higher cognitive processes, thinking, and the . Initial aense forms, and the realm of emotion and wlllo In• word, all the Pleme~ts of the structure of consciousness ln which · the principal role is played not only by the scientific. but also by the artistic method of reproducing reality. And In the artistic method of doing this cognition has coalesced with the funct-ion of creation and of spiritual-moral communication. Artistic cognition and creation are, ln the word• of Goethe. not for the world outside of man. but for the world which ts ln confol'ftllty with man. · Cognition In a certain sense Is perception not only of ••-!:-a·oundlng reality, but also of one's own relationship to this reality, and not merely one's own relationship to this reality, but also of the slg• nlflcance of what la being done for •oclety as a •mole. Since man•• activity ha• some particular •octal significance. the consciousness of this ts characterized above all by the degree to which man ts capable of realizing the social con•equencea of his activity. The greater the place occupied by manifestations of social duty In the motive• of human activity, the higher the level of realization Is. The capacity of man for thlnklns la not directly Included ln the very structure of the brain; It Is formed by the logic of objectivepractical activity through uniting with historically amassed culture, through education and Instruction, and through objective activity maklns use of procedures and meana created by society. The richness of

• 7 •

0

_,_=---~----

- - - - - - ~-,.....,._ _

~. ;L. i~

,.

t

» •I . •JI

, un•• Inner world la a conaequence of the rlchfte•• and varutlllty of hll aoclal ti••• !1111 11 .vhy effective aodellng of the con1clou1ne11, lt1 1tructure, and lt1 function• cannot reitrlct Itself •rely to reproducing the atruc• ture of the brain. !1111 requires, t aaphaalze, that the logic of the whole hl1tory of huaan thought be reproduced• a• the tndlvldual entering life ·reproduce, lt. And thl1 Man• repeating the whole path of man'• development and aupplylng It vtth all need1, Including al10 the ethical and esthetlc requirements with their natural-biological preml1e1 and 1oclal content. Therefore Academician A.N. Kolmogorov la right In saying _that an automaton capable of writing poem, on the level of the great great poet1 cannot be constructed In .any simpler vay than by modeling the whole development of the cultural life of the society In which poet1 really develop.

]l· ...

.,

t.:~ ~,.·

,.

t "

·\~•,1

~A,

•

~

k~); (.

.t f.

The probl111 of Nft and hl1 Intellect 11 not 10 11Uch a natural• 1clentl fie and cybtrn1tlc probl1111, 81 prlnclpally a profoundly 1oclal one,

{~

!· ,/

. Just a1 aatheaatlcal logic 11 unable ·by Its own means to expre11 coapletely, and even leas 10 to· explaln the nature of the real process of human thinking (nor does It claim to do thl1) •• so cybernetics cannot clala to exhaust the essence of man, his Intelligence, and his thinking. Thia demand• Involvement of the whole arsenal of modern re1earch 111ethod1 , and not only those at the disposal of cybernetic ■•

J

'

.,_

,,

,

,. ,-, ,',

.

~

.t

. 1)

::,

The above d0'!1 not at al l aean to deny'the pos1lbillty of model• Ing thought. Such tllOdel i ng l a a pr esent fact. Electronic computers are very auccesafully modeling the mechanlStll of ~ol'11181 logical reasonIng proper to man. But thl1 mechanism 11 far fr0111 exhausting the developed consciousness of modem 1oclal man, the "flexibility" of t~ought, and Its effectlvenes1 In 1olvlng the mat diverse problems -- an effective• ne11 that 11 not 1tlpulated by any previously laid-down rigid system of formal rules. This mechanlam of creative thought has still been 1ubjected to extremely little research, but It 11 obvious that lt la 1oniehov linked to the capacity of operating not with rigidly determinate and at the aame time semantically capacious Ide••• but with "vague" Ide••• with aen1ual and Intellectual Intuition, and with the capacity of effecting extensive and pithy analogies and hypotheses baaed on a ·gigantic 1tore of aclenttflc fact•~ observations, and ldeaa won by the whole history of aanklnd.

i ~

Experience ha• ahown that it 11 relatively easy to model some coaparatlvely narrowly specialized typea -of cerebral work, for example. the performance of calculating operations done by a worker tn some department of a bank. But such ftlOdellng does not encompass the 1110st general and the moat lmportant·.mechanlama of the brain'• actlvlty. The human brain la universally capable of solving assignments o( an extenalve cla•• of problem•• and at the same tt111e It can acc0111pltsh

.

I

8 -

I.

l

l· !· / , ,

~--

'l

,,

•

th• iapeclflc Individual procedure• and Nthod• of aolvlna different, .. ·. problem■ of the ac,at dlwrH dear•• of coaplexlty and type characterSatlca. The lfflportant fact 1111st be · bom• ln 111lnd __ that every peraon, , carrlea out train• of thought proper to hi• alone and often unique.••well•• general train■ of thought, vhen solving a particular practical or theoretical problem. Mlaundentandtng often arlaea 1n connection vlth the different concept■ of the ••••nee of the machine. Her• la a typical definition: In cy~rnetlc■ tnachlne I• the name given to a ay■ tem capable of perfonalng •~t• leedlng to a certain goal. Hence living being■ also, and In particular. are Nachlnes ln this sen••• The goal la Interpreted aa a ■ tat• of the ■yate111, deter■lned by a natural process or by human effort-• toward which thla aystem la regularly tending, without In the process having any conscious Intention.

•n

But In auch a motion of the ■y■ tem •• her• I use the ter■ '110tlon' In Its philosophical sen•• •• of, for excmple, a logic machine, there I• no goal In the genuine ■en•• of the vord. This term la employed here ln an expansive fashion. In philosophy the goal la a human want vhlch ls Idealized and haa found it■ object, a subjective Image of an object of activity. In whose Ideal form the result of the activity. la anticipated. A goal ensue ■ from the realization of a want ln some object and has no existence outside of wants. Want. Indeed. la the cardinal criterion of everything ai1veQ Moreover goals are formed on the bast ■ of the cOlft~lete totality of mankind'• experlenc• and are raised to the highest form■ of their 1Unlfestatlon ln the form of social and esthettc. mor11l and scientific ideals (say. the creation of a aoclety ln which the happiness of sOCM vi 11 not be butlt on the unhappiness of other ■, and so on). of . .n,

And here ve again come up against the profoundly social nature ht ■ activity. and hla rea ■on.

Can cybemetlc■ approach man and his Intelligence•• lt would a machine? Ye ■• It can. It does this on the same basis that the phyal• ology of the higher nervous system, when lnvestlgatlng the machinery of consciousness. divorces Itself from the -l 'n tenslon of consclousnesa from the essence of con■ctousnes■ and thinking itself. In exactly th; sanie way cybernetics does not study either man. or thinking. or creation ·, In the proper sense of the word; cybernetics use ■ the result of the research on these phenomena by the whole composite of sciences •• philosophy, psychology, the physiology of the higher nervous system. and other• · and strives with the aid of automata to Imitate certain aspects of the operation of the brain. The es■ence of the ■-chine aa such va ■ defined long ago by Marx. Thia definition la true atao for the present day. "The machine la

- 9 -

_.,,..

r-· . . .-.-- .- ·--•- ",.___________ a- ~ ~

r:. J

~ ....• ·-&v~ ' Mn6@ • ' r s s#:

· ·,

t

... ,

l rt: tr: ·

h

s

-,n

as~·-~

,.,,-,- r, . ~"

lt

I

J ; J, {,

F.· (;i

r~·.

,.... 'k

~-

natural •t•rlal converted Into organ• of th• hUIUII will and of It• . -•ctlve --.nifestatlon In nature. They ar• oraan• of the hwun brain which are created by human hand, the objectified fore• of knowledge.•

.,:;tI

1,,,

Thh definition h ao general •• and' at th• aa• tt• ■o prect•• ·• that It refer■ to the •chin• at any level of It• perfection, lnclualvely al10 to cybernetic automata th• goal of who1e creation i• to liberate •n from the labor which ..,•• entrusted to machine••• the "amplifier• of the Intellect." The goal of model• of thought activity 11 not, of course, to . create thinker■, poets, writer,, political and 1tate figure•·· people in general•• or to replace them, but to employ cybernetic method• for technical progre11, ·a s vell a■ to move forward . In th• understanding of . the e11ence of thinking i t ■el_f, of consciousne11.

~

t· ~

J,

l

'r~

.i

f

)

,·

The fact must be 1tre1sed that If prevloualy both the nature of thought and the nature of the human con1.clousne11 seemed rather clear, then the demands of the development of cybernetics and of technical progress are forcing both psychologists and philosophers to take deeper thought on the nature of human consciousness and thinking. They are, ·· tn particular, forced to submit the nature of the mtnd to analysts anew, as well as that most rich reals of feeling, without which not a single thought ts engendered ln man•• head. The development of 1cienc1 and practice wtll Indubitably get rid of the now existing and equally groundless extreme,-• the speculatively dogmatic, skeptical approach to the potentlalltte1 of cybernetic ■ a11ocl• ated· vlth lack of faith that certain logic operations considered the privilege of man alone can be reproduced In 111achines, as well as the · aenaatlonal-advert!slng approach to cybernetics expreased In the attempt . to Identify the machine with man, intellect, and the imitation of some of Its propertle1. P. Protasenl and A. Rakov•s article, "Not Advice, but Charade•," can serve as ~n example of the second extreme. Neither of these extreme• 11 of any help to the actual progress of cybernetics. In a few years cybernetics has achieved substantial results, both theoretical and practical. And tt baa no need of sensational clal• ltke the replacement of man with "creatively thinking" machines posseaalng con1ciousnea1 and 1elf-consclousnes1 -- claims which strive for outward effect.

I •

,.'

I' r,.:; - i! ('; I

I

·r ~

'

~

.,, :

, I

(.

f

I

/

,:, · 10,946 , CSO: 1880-S

,

II.

! t

• 10 .,,

(e

I

l

!I.

~ I

__._

'

i

•I ,_.

I

Ii

I I

l. I•

I

•i

i

.. ,·.

I

'

' I,: il ,. ',. . · :•,

I I

l

•

I

i.

I' 'I I I

.

I

! I-

' I

'

R!ADit«I AUTCMATIClf .

•

(

I

.·

..

,,

.... ,

;

. ,\

., ' • ,; 1.

.,

[Article by A. Zelentsov; Moscow, Nauka i Zhizn° , Russian, February .:1969, PP 30-32]

People became bored a.gee · ago v i t h the reading of some texts. ' It 1a uninteresting to read, sq, some report on warehouse supplies, or on the output of some component or another. But they lllUBt be · read .to generalize information and compile other SWlllll&l'ies,· tables and reports. Computations of this type are now being handled by computers • but the initial data must be fed into these machines by means of punched cards. However, pwtching takes up too much time and manual labor, sometimes . completely canceling out the effect or using computers . This is always the case when the computation i tselt is simple and there are many initial data. To solve problems of this kind, reading machines are needed which allow computer input c,f documents without the use of punched cards, reading these documents autolll6.tically. It is true that contemporary reading machines are a long vay from being able to read ·a h11ndwritten text. We must be content with machines which can read letters and numbers machine-printed in a special type style. But even in this case, the machines can .find application in the most diversified fields. It would seem to make no difference whether the text is retyped or keypunched into a card. But there is a difference. In the field of econoll\Y, there is the concept of the "primary document." And it this primary document is machine-printed, then a reading machine ~ be quite advantageous since the primary document its elf may be fed into the computer vi thout malting a copy and thus destroying the legality of the document. Therefore to compute the payroll at a plant where 30,000 workers are employed directly trom the time cards, or to plan the suppl,y of materials trom various statements, not to mention ·banking and postal operations, it is very convenient to use machines which read letters and figures. A reading macbir,~ tor machine-printed letters and figures is being developed at the Inst! 4 ute of ·C ybernetics of the Ace,lell\Y of Sciences of the

-11-

@ip $ ~

f"f t M

Thi■ MChine 1 ■ called "ChARS [Chit-.yushchiy Avtomat • ReSdrlga; Reading Machine. with Shitt Register]. Tbe ba■ ic distin1. guishing teature ot " ~ " ill ·in the vorda "ahitt register." The unusual use ot a ahitt register allOVII the machine to read 52 symbols in ordinary, I . rather than stylized, machine-printed type. Vladimir Antonovich Kovalevakiy, director ot one ot the departments ot the Institute of Cybernetics, 119¥11: "We haven't turned &VB¥ from stylized type and special machines; this 1 ■ a completely separate project, which me&DB that it's extra vork. If ve are counting on large-scale use of reading machines, then we must set our Bights on widely used printers." · The operating i:·rinc-iple of the "ChARS" consists in comparing the image ot a letter or figure with master patterns. Naturally this method limits the sphere ot application of the machine somewhat since the "ChARS" can read only one type face with a given tont of masters, e.g. the type style of' the "Optima" typewrit.er. But the master font ~ be replaced by another so that the machine ~ be converted to read the type face of the "Moskva" typewriter for instance. . . The comparison is made by computing what are called correlation coefficients, i.e. quantities which indicate the degree ot similarity between image and master.. This method in itselt is nothing new, it is used in a number of other reading machines · as well, but the symbol is comparP.d wi'th the master in the "ChARS" not once, but eighty times - ten shifts I

ft

't

·as

Ukrainian SSR. gi■troa

'

--

.,

....... ~~ _ i;

i ...... -...

, -- , - I""'"

v,-,. l':ii¼I 11'/ A ' /A

L·-

!

I

...

.

A magnified

I I

; ,f",._ •

I I I I I I I I

iJ11aee of the symbol in

the field of view of the reading liea4: 1-"Colwnn" of photodiode■ in. the reading head; 2--Motion ot the symbol during recog:iition.

Image of the symbol entered in the sbitt register.

i

~-. along the horizontal, and tor each pt .these horizontal shirts, eight shi tts 1 along the vertical. And it is this that· constitutes the special use of' the i shitt register. We have selected this method to achieve maximum reado•.1t l·t. reliability. The fact is that regardles■ of the care taken in filling out / . the documents, it is nevertheless impossible to wind up with the symbols on the documents arranged in a strictly detined W9¥ with respect to the 1 ( : · optical system ot the reading machine. Some displacements are alwf1¥s un- · f avoidable, and it ill th•?ref'ore difficult to line up the symbol and the

-12I

I·

I I

I

I

I

!I .,

•

----·•> ____ _

. , .- -· . . . blem 1• iolved by what ie cal.J.e4 ; . t reading aacbinee, thi• pro . rd.a the le:rt edge or ,,. -..ter. In .,. s ct to the edge• . In other vo then the image ot the centeriDI ~;!t ~c!:ted, then the upper edge, ::h ·a given area in· the symbymboi :: ;hi:rted so that these edgeits 11:et:a method is low since the .· 1 s o ·tt rn #ield The reliabil Y o h most liable to be diemuter pa e ~ • b eas vhic are :t # the sVW1bols are just t e ar ·"'-ere there is the g~a eat edges o~ .,i 1 just the area "" ,. r torted in various vqs. Th s s stead ot this• the "CbAI:lB per orms likelihood of smudges or breaks. In the symbol is barely obifted what is called examinati_on by shi :rts, i ~e: • and ~o on - 80 times• When and the correlation quantity is determi t und the position in which the this has been done, ve are s~e to have o this case ve get a correlation . symbol best coincides with its master. !n to master pattern "m" it means . ·I . maximum. I t the given maximum correi s~on r: ld ot view • it the maximum ia . : that the letter "m" is in the mach ne s e #i'gure · 7 . . " " th th machine · sees a ~ • · ·· that tor master pattern 7 en e "CbARS" bas still another advanth In addition to eve!7:thing else, e i the case vbere there· · tage: it can distinguish symbols .in a line even ~ t d an ordinary type- . are no spaces between them. At'ter all, symbols prin e by chines which vriter trequently touch at the edges. Therefore reading ma . distinguish symbols by ·spaces can not read an ordinary iypewri tten text.• ·· ,'. Symbols are distinguished in the Cb.ARB by means ot all the:e sam: ·1 • correlation maxima. To distinguish one letter trom another, the Cb.ARS_ 1 does not look tor a space, but rather atter noticing one maximum in simi- · larity waits tor several horizontal · shirts to see it a second maximum ap- '. pears which is greater than the preceding one. It no such maximum appeara : after a time sutticient to pass over a letter, the "ChARS" takes the ·pre- ·,, ceding estimate as final. We have turned all our ettorts to achieving maximum readout reliability. Arter all, if the "ChARS" inakes even one error in a thousand casea, it can never be used on jobs,· let us say, in . banks. 'lberetore in developiq each unit ve have tried to do everything possible to achieve maximum utilization ot the information contained in the image ot a .letter. And tor this j reason the "ChARS" nov distin~shes not merely black and white, bu:t f'our . ·i shades ot gray as veil; Imagine that some line is taint because the key l has not been struck hard enough, .or the ink · 17ibbon vas worn, or because ot a bad spot. 'lbe "CbARS" never skips a taint line and never contuses the real line or a letter or number vi th a spot because the device has a sutticiently high gray resolution. · Let us nov consider hov all this takes place in practice. A narrow metal chassis crammed vith rova ot Pertinax plates to which · electronic components are fastened, a bundle ot wires strung overhead connecting the chassis to a complicated mechanism standing on an C>rdine.ry · · . , wooden table, and on and ':'Dder the table all kinds or· instruments with wires running to the mechanism. This is bov ve saw the "ChARS " "All this vill look completely ditterent in th~ first model which is ?.ow being made at an experimental plant," explained Vladimir Antonovich All that you see in this room vill be concealed in tvo compact cabi t •· :~/~~!ng me~h:°~~m "111 be located under a transparent plastic c~:e~• 811 comp e e...., modern. A stack ot documents is placed in the reading J

4

--13-

.

,,

L. i; .

.

·:

}

--______,,_.,~ ~

...

\: , ._: '\ff1 &'itrtt?iott '. , t)Sd'I lrt» d:M

·

,.

.• . bl

If

t±t ·

b'

I

f

~

l l .

( 1)

( ' -

I . ,\ ; r:.... :

.:i~""' . .i'- -'

r..,. .__~ ~-

.: - . : ,,

rl . ,;, .-;r1.,r · :· ·. . .-,:.~'~-~ . .

-·..

L

.: .• ~

• ·•, .-;.: "'iJ:

t·. ;

! ,, . j

p., •

,- .~

1• •

! ' :. . ._( { • \ ••:. ·. :

•, ~

1-t:. \:'\

.. l[ rt

I f

f •: v

, . ' " ' l

,

(.' ;

, '

•

'I

'., ' ' , . .

,

I

" 1'

,•

• ... "- ••' 'I

;_

;

. ' • •,..~ ( ,

1 ' • I 1 --. "

,. • •

·,l , :.~·~· ,.-,._ ~J'f.:• J31J

>. ·/r•, : 1

. ~• :

I

I f

~

.·. r .... . ' .. ,·, .

'

'. . ,

' ·•.

L

..

.'••'

f

,, •

.~ •' •

'~

\,

I

, :

·• ...

r _.,

.

A.•,

1

'{•' •

't l ' ••

t ~

tM'

,i.;,v

t , .•

r·

sb

l h

t_

'I

J

( 3') C...-.wi tllllfft

r,

•

' ,t'

-Shif't t-egister; 4-lens; 5·re&dout head; 6-.set of' maater patterna t f 7-index of' maximum similarity_between . 1 symbol and maater; 8-answer unit; 9response iooooo (code f'or letter "A") •

!

!

(! ·.

•·

- .,

(8) u,111,n•

"'" ( 9 ) 100000

.

(•A ..,.... .At ·

mechanism.~ · They are all the size or a standard typing sheet which accom- •. mod.ates -about f'orty- lines with ti tty to severity symbols each. Pneumatic suction devices take the top document and teed it to the so-called trans-· , port mechanism." : - · : - 'Vladimir Antonovich stepped a little to one side and turned on ••• ! a vacuunl' sweeper. • A sheet ot paper passed to a · rotating drum and wrapped i- itself' around it. ·. · · · "We used a vacuum sweeper to solve our problem·," laughed Kovalevskiy, .t tonly in this case the air is sucked f'rom a drum in which small holes are · · .- drilled. That • s why the sheet ot paper sticks to the drum." . I · · · Tl :~

1: .,,. \ · \ ~)

• :.' {

:·

:(' ..

~- •

l :\

·.. 1J '. , ,

.' I •

t' \

c~~

r ~ •' •:,·.

'

, . , -,~ l

l

\

.\ • ,.\

::

~

l....,~(··: 1 ..~ii :i-

•

• ••

1

\

r,. , •· · · 1 ,

- ~; 1 • ;·•

I/:

1'

•!

\'\ 'fl, • ,•'

't i :\

..

'·

' \

'.

. :. ,_:

, ·.·.

t, J

:11•

.:• 1' J

(

... ·~

,·.,.· r

1

-t 1: ....

j_ t , . \

. ,_.,, ,

i ,•

l

, fl.. ~•~;·, l ~

,...• ,·

~• i.:.

'I

'

\: ~•. .1.,

l :. ,:_'. ,.,·~ ~·, ! \: r~

i

i~t

: .. ~

1 '•· '.

l

,

' ;~.

."1

' I.,, '

·, : i 1.:: . :_,

~ -~ "'! '..

1.' f;,:, \..l {~{·:• 1_ . '

' ?•

1

f•

\(

; . l•,\ ~

:,•·•

~

,, i · •. ,'

' :, :.1,

'''

;1_, :_ ~- ~,f!• :•',

···::

•,'? •·

:

:. \'.; -~ -.

. ~;~

' :.\1. ': :. ~-;i.·, . , ', ' !' ·' 1.. ,: . , 1:.:: . '

:\

... , . 'l

..

_.;_

,_~, ..;" ,.: ·:_ '1•;.,:·

'.

~ , ,; ,:

:.,·: . ::-:-~.,. . ~·-·

I,•

' '···

(

r;

.. ,..1;~

.,,.~~ t•

.

...

f• •, •.

:·.

••• • ': ~r•~\ •, . :;~

~•. I .;('

•·

~ ~

•~ ~l

-. 1'..-

\"t • f

;] . 1; ...

., ; ~•.

t_ ;· '-·

1_.\ •

- ~( t

,; .

\ •',J_· ·:1

1\

\.\I ; , .,_

, ~ :~~

.1:i

'.~.., . \ ,'

U: ·, .. !.r1--

-15-

__... ....

;I

;\

\ ,, '

·'

tt , ; •,,

.~\ ~.,,·:,· •..,i.·, (\'.I'.'

I--...,,,. '

',

' ' I.

,.

·i

·l . r

.

r'

I

.. 'd'

.•.,

..

,r '

·• t

q

: ·. 11·,\~

; L · t"

·~ ', I

I

'

I'

.._

',·,;

·.·

l r• ·

..

'•)

- 1-1 C

\• ; • 't: fEfift

t -, .-

; i

r · Db r :rt ·z

r r

. . ;'f t · 5 ·

·>" ) _ ..•·j _ ·T

-

21

d

ct dt

.___,

l

of the ...ntn1 of the conclualon• derived In the•• conatructlona. ·It I• thla very thin& which occur• In inductive conclusion•, where the ~mperfectlon of experience, which senerally ahowa up aa one of the baalc factor• In the relativity of human knowledse, Is manlfeated both In the Incompleteness of the analyala of the Information contained In . their empirical prealaea and in the claims of the conclu•lona In these Inference• to greater lnfol'lllatlon than was evinced In the premise•• And, althoush one of the cardlnai essisnmenta of Inductive logic 1• a• naach aa possible to eliminate the concluslonal indeterminacy which therefore occur•, or at least to appraise the desree of it, neverthel••• c~mplete achievement of the•e aims eludes ua and the stamp of lndeter■ lnacy always i1 borne by conclusions from Induction. .

The atten~lon of modem Anvesttsatora of Inductive logic la

••

being concentrated exactly on sizing up and developing methods to per■lt

I •

I

l

I,

iI

i

II

the fastest possible full and profound analysts of the Information contained In the premise• of Inductive Inference In order to make this conclusion a• determinate and separable from the premises as possible. Thus, the famous Inductive conclusion "All swans are white" had a determinate meaning only with respect to the limited experience on which It wa• made. From this aspect the discovery of the black swan tn Australia merely added to the preceding premises still another, which, to be •ure, •truck out that conclusion Itself, but only that one. Based on the old experience a certain conclusion had been made; based on the nev and fuller experience It proved to be groundless. In Itself this · fact, ao usual In human cognition, could hardly cause a sensation. It ls another thing If the conclusion "All swans are white" emerges as a the•I• entirely •eparate from Its premise• and moreover the separation of tht• conclu•ton from the premises ta achieved -by certain logical mean•• Then the discovery of the black swan subverts these means the••elve•, and we are now dealln& not so much with the appearance of a new private theory as with a "crlsla" In a definite logical conception. Viewed frOl9 IIIOdern inductive logic this situation ta explained by the lack tn traditional translational Induction of adequate means to analyze the Information contained In tt ■ premises -- lack of means to evaluate, and hence to a certain dearee to eliminate, the lndeter·•lnacy of •uch a conclusion. It may tn particular be demonstrated that tf the Inductive conclusion "All 'swan1 are white" la analyzed by several aean•· of tnfonnatlon analysts ela~rated tn IIIOdern Inductive logic, then even before any discovery of a black swan It would be clear that the lndeteralnacy of this conclusion la too great for It to be made. The method• of this sort of analysis •Y differ tn diverse theoretical models of Induction. Thus, If the Indeterminacy of an Induction conclu•lon I• thought to be Its lack of authenticity, and the latter la e•tlmated by a "probability" function Introduced by certain special rulea, which vary for the different systems of •nductlon, then the Te• sult will be that the proNblllty· of the conclusion "All swans are white,• even when 1,1ade on the basis of experience containing no contradictory example. Is extremely small. -17 -

I

It ...

~

•

,~-;~-:-:._i:..« ~ .:- ~ - -;:d ai

1

-·

. 1

117

IQ

.......

.-·

Iii

120·:i?< - -

a r

;n

·· ·>Sc ·, :

:

k-l

ris I r

;t i ·ftit·• ·

'

Here for exa•ple, ·· la how Reichenbach perform• thla analyala, 1 ualna hi• theory of different levels of fr•q~ency (1].· ·_He _. c·o natru~ta aeveral •atorey•" of Induction, that . la, regard•.. Induction not as ·o n1, one aequence of observation• of avana, but a• a nwnber of aequences : about different genera of vlldfowl. In the thus-obtal~ed table _the · horizontal line• are made up of obaervatlona on different . specimens of one and the aame genus of bird and characterize Induction by ■peclmen 1 . vhlle the vertical column• are now composed of observation ■ on specimen• of different genera and characterize Induction by genera. _· In thta caae Induction by apeclmen■ qy be evaluated by the first-order probability (wager) vhlch la described by the relative frequency of . specimen• of this genus which posse ■■ a certain distinctive feature (color In our case). Induction by gene·r a, however, may be evaluated by the second-order probability (wager) which la characterized by the relative frequency of the genera possessing some distinctive feature among all the genera Investigated (in our example color Is a constant characteristic within the genus). And then it becomes clear that color la not at all any constant characteristic within the biological genus of birds because the second-order probability here la extremely small. And al~hough the empirically established relative frequency fo~ Individual sequences wl th respect to specimens of the genus may Indeed · be unity (as was the case with the statement that all swans w.?re white which was made before the discovery of Australt.a), the probabll lty Itself · of the truth of this conclusion Is Insignificant • . Here the Inference of Induction through listing applied to one sequence Is corrected by another Induction which now regards every sequence ·as an e _lement.

.

,

Thia example also shows In particular the groundlessness of the . opposition between enwneratlve induction and ampllattve inference which was so Insistently emphasized In the traditional inductive logic of Bacon and Mil 1. Correctly conducted enumeratl ve Induct Ion .Is threat•. ened by a "crisis" In logical means of Inference to no greater ·d~gree than Is the logic of ampllatlve Inference, _which obviously contains logical means of analyzing Its premises and checking the validity of the conclusion. But aome logical means for this analysis are adequate In neither type of Induction,· and still other different sorts of extralogical. means are employed to effect .the most determinate and strictest possible Inference. In the Reichenbach analysis quoted this extra- · logical means Is a certain substantial knowledge of the biological classes of birds and their species which in the last analysis deter• mines the limitation which make• this principle of Induction effective. But If induction la Ineffectual . in Its purely logical principle . can inductive logic be regarded at all aa a logical theory proper? A negative opinion on t~I• acore. could be consistently Justified from the positions of extrem~ logical formallaaa and nominallsm. Meanwhile the effective Interrelation between logic and specific sciences In general and logical and extraloglcal means In particular la outlined for us on another plane .from the one on which It presented In the concepts Just

. -18 -

,.

I

__,.

t '

""'-..-,··•... ,

, · Ll.r · -, ·~rue ·1tt ~ .· . . et

---------------- _______..,__..._______

--•--·

~

if ~·

2

•

urt:

ttt ' •

kt

ti-

..·___:_

....... ......

...........

~

·:

--....

--'

·~-

~

,......,

I f

I· r

I I I

, I

l

.

. .ntt~ed~ 'Generally ·■ Ptaktna, any lo9lc ■y■t- (In contra■t . to a pro'p erl1 loilcal enunieratlon). t'ncludea not only ■yntaxta and deduction, but aho ' ■etMntlca· and prasmattc■.- The latter are aHoclated right with the feature■ of a particular application of logic. And lf · lt

prove ■ Inadequate to effect the construction of a logical syateni of · _purely logical mean■, then, aenerally ■peaklna, It la pemlaalble to : ,. _.. enltat even nonloalcal mean■• Thia, by the way, hold• true not only vlth re■pect to Inductive loglc 0 but also with respect to a number of deductive logical systems, for example. systems of metrical, In particular, probabilistic logics.

I

'

I

r-

r 1. t

I I

(

i

I

1. r

.I

!

·, . "J ' ....

) I.

I

l

I

I I. t

I

(

I

i

l I

The ·thua constructed syateru must be ·considered just as logic ayatema since other systems •• theories n • are built on top of them, aa on ,a logical basis. The la particularly dtsttnctly displayed in the case of constructing theories on the basis of formalized language~. In addition to the logical part a formalized language of theory also contains a certain specific part ~ n a apeclallzed language anaking It po•slble to describe precisely the concrete, substantial region whose description thia system la. As such a apectaltzed language a formalized language may contain the axioms of arithmetic, of classical or quantum mechanics, and ao on. If , however, the language of theory does not contain a formalized· logical part It la not a formalized language at all, but merely a specialized one.

,,

Thua, the language of quantum mechanics, although Including a solid mathematical apparatua, will not be a properly formalized language until · tts logical part la forullzed, that ts, those iogical means which are utilized In reasoning in thl• science. At the same time these logical meana themselves aomehow de~nd on the special language of the given science. To u■e A. Church'• expression, it may be said that the formalized language corresponds . to the logical form of the reasoning carried on ln the given ape·ctal language. The formal I zed language of theory therefore must contain all the logical mean• for effecting the necessary reaaonlng In the given special language. Then the logical part of the formalized language of theory may with complete Justification . be called the logic of this theory, and In caaea where the special languages of theory are aufficlently rich the logical part of the formalized language may organically include even extralogical meana. Thia ta precisely the situation In inductive logic, which deal• with a far richer world than does deductive logic and consequently muat also poasess richer means to describe It~

Any logic underatood to be an Interpreted system, that ts, a system with a cleart'y prescribed aemanttca, ha• cer~atn ontological premise• and la easentlally constructed a• the logical model of some ontological ayatem. Thus, Birkhoff and NeumaM note that classical logic la not suited to description of the mlcroworld since simultaneous obsenablllty of a nmber of characterletlcs of objects ts not ·' · , , aaUafled, and this makes the law of dhtrlbutlvtty unfulfillable In

-19 -

[.~

"

• ~ , Ill •

•~\~·\

I

1·

:1 ,J

r-u

I

1

5 1

-•

::

®

°t' ' t ·••-•.• 1»

111

ti:-: raw':it:rllft" I ti '"

:tt"thmwt't\1'.t · +or's

'·ivn-·:r#i-- - 1"

It• logic IIOdel [2]. !aH11tlally different ontological ayau111a, generally •peaking, alao require different logical models. Thls h , · · true both for deductive and for Inductive logics. Therefore on con- , dltlon of sufficiently apeclfic and delicate analysis we will have · 1 not one, but a 11Ultltude of systems of inductive loglc, each of which will be the apeclflc logic IIOdel of a certain sort of ontological aystem.

•

ln the actual construction of diffe~ent inductive logica th_la ls realized In the varlablllty of the extraloglcal postuales which are added to some original logic base, or even . In the variability of these bases themselves. We would note that with such an understanding of Inductive logic even the ao-called problem of Justlflcatlon of induction, as it has been traditionally formulated in philosophy since the tlme of H\llle, ls essentially eliminated, and moreover eliminated in a more natural way than by Reichenbach, who for this purpose actually resorts to the abstraction of potential realizability. lt must moreover by borne in mind that these terms them•elves •• "logical" and "extraloglcal" means e • are relative and correlative in nature. Means which are extraloglcal _in one context may make their appearance as logical means in another 9 and conversely. lf certain means in the construction of any theory belong to It specialized language, .. then they wt 11 be extraloglcal means wl th respect to thls theory. If, however, in the construction of the theory they belong to Its logical language, then they will now appear as logical means with respect to this -theory. Thus, certain substantial induct.ion premises are "extralogical" means with respect to the classic means of deductive logic. And these same means may be attributed the "status" of logical as soon. as we are dealing, for example, w_lth cases of inductive foundation of particular knowledge. The captious critic of the point of view expounded above should bear in mlnd that the term "extraloglcal means" which we previously used must be corrected ln precisely this correlative sense. 2,

•

Probabilistic Models of Induction

The founders of inductive loglc •· Francis Bacon, Herschel, and Hill, as well as some subsequent logicians contemporaries of ours•· John VeM, Grenevskly, and Reacher•· endeavor to construct induction as a rigorous conclusion li~e a _deductive one. And if they even accept an indeterminacy conclusion In it, they nevertheless do not apply the function of "probablllty" to the appralat1l of thh indeterminAC'I. Other. investigators, however-• Laplace, Keynes, Lindenbaum-Hoslasson, carnap, Kemeny, Reichenbach •• approach the solutiO!l of the problem of induction from the standpoint of probabilistic evaluation of this indeterminacy. Here probability la Interpreted a• a certain logic charactert~tic, however It ta measured •• It be the "degree of 11imllltude" ·(Laplace), "degree of confidence" (Keynes), "degree of confirmation" (Carnap)• or even "logical freq~ency• (Reichenbach).

-20 -

~;

I

I'/ i.

k,(d dt ,,.,_tt ,;ft t

I

\ts

c

he◄

e·

'

ttb )s· · H

nate ·· · a

z tr a: ·

:tr:-es 'ct

· r

ff

1t:1

· aw ·• ·

·tnsr

11 '

r • · •

b He

a

. . One of the flrat probablllatlc IIOdela ~f ·Induction la that of . . Laplace. He uaea the l~n1ua1e of the mathematical theory of probability to analyze enumerative Induction. Thia Induction• aa la well known. la defined aa follovas If a certain number n of cases of cl•••« be given which prove to be members of claH 13 .-and U moreover not one Ol. la known vh\c~ would not be •/.3• then the statements "the next case DC. vlll be • /J" ~:1d "all caaea Ol are both have a certain ·probability vhtch becomea lncreaalngly large•• the number of caaea examined la enlarged. Laplace propoaea thla sort of model for It: lf there are N + l ·identical uma each of vhlch contains N black and ·vhlte ball6 and all poaalble comblnatlona of black and white balls are found ln the urns. and If then~ balla· are selected from an urn taken at random and they prove to be whl_te. lt ts then asked ~ (1 ) what la the probability that th~ next ball taken from thla urn will be whlte and (2) what ls the probability that all the balls ln t he um wi l l be white.

r

I

l

ft t

·,•

·

11

u.

/3 "

!

t•

I

!

Laplace•• IIOdel has a rather dlatlnct .two-atage structure (3]. In the flrat atage Bayes• theorem 0 almpllfle4 by several supplementary aaaumptlon•• la used to determine the probability of the possible · cauae of a aerlea of events known to us (the drawing of the white ball ■). In the second stage, now vlth thia cause aa the starting point, a search la made for the probablilty of future events In the same aeries. In hla conatructlon Laplace employ• not only the prlnclplea and t~eorems of the theory of probabllltlea, but ·also assumptions from outside of probability theory. In determining the probability of a possible cause In his model Laplace introduces the assumption of the equtprobablllty of all the causes (urns). an assumption which cannot be derived from the data directly c•>nnected vlth our experience. hut la established from more extensive data, moat often from certain aprlorl consideratlona. Moreover In defining the probability of future events Laplace starts frOlll the assumption that their cause vlll be the same (that ls. will have the aa~ probability•• for th• past events). · Having the first Laplacian, assumptton ln mind Bertrand Russell properly calla It absurd [4]. It doe•• in fact. postulate an extremely powerful tdeallzatlon vhs'ch la ver.y far front the real conditions under which enumerattv• induction ta ·n~r used. On the other hand, thls sort of aaaumptton ta the alapleat ·for making calculations and tn general make• It possible to construct a 1raceful model of induction. lt ia Interesting to note that Boole and Edgeworth'• attempt to replace the Laplaclan assumption of the equality of the apriorl probabllities wlth ·· a aore complex one lead• to extreme compllcatlon of the model, right up to the practical lmpos■ lblll~y of making use of it.

' I· I

f t

I

r

I·. t l j

!•

! I

\

As for Laplace•• aecond assumption, in his model It obvious follova

•• +

..

•

frca the tact that all the experiments are conducted with• single urn, but It contain• ln hidden fora the profound principle which underlies all inductive conclusion• In general and which Hill formulates aa the "principle· of the unlforalty of the . structure of nature."

-21-

•'

\·

!

tr~ ..,.-~---------------......:.......;..---- - -- - - - -

>~• •

.:k...,JA L,.-t.

,.,

.~'\

·ror coaparlaon vlth Laplac••• IIOdel we ■hall adduce on• of , John ICeyne••• IIOdeh of· Induction (5). In. thla IIOdel Keyne■ : 1trlvea to deteralne th• probablllty Pn of the Inductive extension of I observed C■ Ns x1, x2, •••• x.. In the context of - ■ome system of knowledge!! -Pn • P(g/hAx1Ax2A ••• Ax..). Here ■GIN substantial aprlorl conald•

MaJUrd

eratlon Po• P(g/h) •• a characteristic of the inductive extension made under the condition■ of ·lcnov~edge system!! •• ta drawn upon tn addition to probablliatlc and logic conatderattona. The relationship derived by Keynea has the f ollowtng fora: p

h

'

-

P0 +

h-

'o.

P0 )•P(x1Axt'\ .... Ax11 !JJ\.h)

•

We aee that Pn • l If P0 • l, and thi1 occur■ In the case where inductive generallzatl~ I enauea from existing knowledge system 1! or If P(x1i\x2A ••• A>en/iAh) O, that ia 0 when in the context of the given knowledge 1! the cases x 1 , x 2 , •e ~, "n observed are incompatible and deny their Inductive exten■ton l• If Po• O, then Pn • 0 •· and th;l ■ occurs when inductive generalt'zatlon I la logically inconal ■tent with the extant aum of knovlege ,!!, and so on. Of the greatest Interest, however, la the set of Intermediate values of P~ In which must be COIII• puted the different relationships of the rate of approach to O or 1 by P(x1Ax2./\ ••• AXn/lAh) when Po la fixed. Here satisfactory result ■ may be obtained with very approximate calculations. Therefore the Keyneatan interpretation_of probability as a "degree of reasonable confidence" prove ■ to operative In these cases. At the same the -fundamental dlfflcultles _ln Keynes's model are associated with establishing the exact value of Po•

In comparison with Laplace•• model that of Keynes obviously still somehow minimizes the extralogical lnducttve postulate which, generally

apeaktng 1 la inherent In any form of Inductive conclusion. Instead of the set of aprtorl poaslbilltlea In the Laplactan model we are here dealing essentially with one such probability. It ts true that the matn difficulty ln both the Laplaclan and the Keynesian system la to ascertain the exact numerical value of th~. aprlort probability, but In the Keynesian model this difficulty is ,shtfted to a somewhat different plane •• tta solution Involves a lesser number of premises, la of less formal nature 1 and obviously presupposes a more ·substantial logtco-methodologlcal basis.

•

It ls our vlew that this minimization ts mainly achieved by meana of the new approach to the problea of induction. Induction, according to Keynes, ceases to be pre~lctlve ln the sense of forecasting new observable casea. It deals only vlth the relattonshtp of the hypothetical generallzatton to the observable data on which thls generalization ls built. It ls essentially with Keynes that the new (somewhat oneslded, we think) and presently predominating direction fn inductive logic beglna,

-22.

. ..... '

. and vhoN characterhtlc feature• are: (l) application .of the fol"llal. l&ed language of probabtll1tlc logic to analy•I• of Induction, and . ' · (2) deductlvl• ln the approach to the problem of Induction~- tran1fol"llllt.lon of thh problem into ·one of corroboration (verlflcatlon or -_~al ■lflcatlon) of hypoth••••• · In the tvo : tnductlon aodel ■ which we have cited (a■ , 1enerally · ■peaking, ln other al ■o) the . ta■k of determlntna the aprlorl poaslbilttle ■ flgurln1 ln the1e model ■-~ besides the action of logical principles-• l ■ -llkewlae perceived. It la thla very point which la the 1110st wlnerable

·:

ti

,.

and moot point ln lnductlve modeh(P Some believe that a certain extra• logical . postulate auat be utilized to determine the aprlorl probabllltlea ln Inductive logic 1110deh. For Laplace() for example, this postulate was the prtnclple of lnsufflelent reason. ¥eynes argued sharply with Laplace vho had attempted to apply the principle of inverse probabilities as the sole method of constructing probabilhtlc· models of Induction, and he dtd ■o just because thla route leads to the need to use precisely thl ■ extraloglcal postulate ln every specific Inductive conclusion.

('

!.

0

I l I

f

lCeynea•a conception la essentially that In every concrete sort of lnducttve Inference for finding aprlorf probabtl ltles one must be 1utded by practical considerations, that la, those making sense in regard to the case In hand,•• well as by considerations of analogy. The problem of the fwdamental extraloglcal postulate (or postulates) comes up In Keynes on another level•• that of the problem of basing or justifying Inductive logic. As such postulates Keynes puts forward the prln• . clple of restriction of independent ;diversity and • more pr~clae principle of Insufficient reason, which he calla the Indifference principle.

'

f.

[ I

I

I:

I ! [

f

The Keynesian prlnclp.l e of rutrlctlon of Independent diversity postulates that in the whole mul,tlfarlousnesa of the facts or properties under tnvestlgatlon a certain restricted set of elemental, that ls, Independent, constituents may be segregated and that from the combinations of these constituents all the properties of this multifariousness be cQmposed. The equlprobablllty of theJe constituents ls moreover postulated by the Indifference principle. These two postulate• thus underlie the fundamental possibility of computing the ultimate aprlori probabllltlea and thus the feaatblllty of utilizing the theory of logical probabllltle• In inductive logic.

1. I

I.

uy

1· -

!

t· I'

It 11 In this very approach that that which was new and Keynes•• contribution to the theory of Inductive logic made Itself felt. It ts our opinion that hitherto the crltlcl• reproaching Keynes for these I I.. principle• of hl ■ being completely inadequate for practical calculation I of aprlorl probablll~l•• have still underestimated this. Such reproach•• r . ,. are unjust because the aforementioned Keynesian postulate• play an eni' tirely different role In his theory ~f Induction•· the role, so to say, F I . .. I . . P.n .• of the axioms of edstence of ultimate aprlorl probabilities, and by no 111ean• that of prescriptive rules for their practical computation. 1· I

I

r-. I l l

. -2) -

It l• lntereatln& to. note that .t he gtat of the prtnctple of , f independent dtveratty ta aomehov ..t■plicttly asa\llled . tlon 0 reatr c f 1 · l al•o ln the conatructlon of Mi 11 '• induction aince, or examp e, in .·.· apeciflc case of lta application ve have (and can have) only a li■lted---.~ concomitant circU111stancea under investigation. Mi 11 •1 .. adoption of the concept of multiple causality has, in general terms, already created the prerequi ■ite for the opportunity to apply the probe. blllty concept in hla logic. The explicit formulation of .t he princi~l• . · of reatrlction of independent diversity, along with the introduction of the principle of insufficient reason formulated in some particular manner (neither formulation made in Mill) would lead to the construction of a probabillatic IIIOdel of Mill'• inductive logic, including hls methods. As ls known, however, the l4ea of a probabilistic approach to induction va• foreign to Mill himself. The principle of the uniformity of the structure of nature which he formulated •• rather phi losophlcal than logl~l ln its nature which made it possible. to speak in general of the existence of persistent (that ta, repetitive) causal connections•· has another significance and meaning in basing Mill's induction than the Keynesian principle of the restriction of independent diversity (6J.

•••rJ

•

The lCeyneslan approach to the construction of induction on the

bast ■ of probabilistic logic was essentially the first properly logical

and systematic approach vhlch exerted a perceptible effect on all succeeding probabilistic theories of induction. We find consistent fidelity · to lCeyne••• idea ■ and system in, for example, the works of LindenbaumHoaiasson [7). lCeynea•s conception ls developed on the baala of a more precise definition and more detailed elaboration of lts baste concepts and prlnctrles ln the Inductive logics of a. Carnap and J. lCemeny. Thus, Carnap. proceeding from the Keynesian principle of the · ~estrlctlon of independent diversity. introduces the new concept of description of the state" as fundaffle!ntal to his theory of induction [8]• . Thia enables Carnap to fonMllze inductive logfc more rigorously. It la true, because of the rather "rigid" formalization also of the language !n which may be described the world tn which the principles of hla tn• tuction prove to be effective. Thia la a::rather poor language •• tts Ype 1 • that of narrow calculation of one~place predicates with equtva1 0:nc:_-- and thus permitting induction to be , applled only to the world c ractertstlcs. not of relationships. In Carnap induction there ::itinuediretention of the significance of the lCeynesian principle e re•t r ction of independent diversity. which la manifested here in th requirements larlye important ' of f fl n it eneaa, as well as·- and this ls particucomprialng the b-- o the independence of the elemental propoalttons of induction per::~:g~c. Ca Under the condltlona of the formalization of propoattona consideratl map the requirement for thls Independence for application oft d tty impoverishes and limits the opportunities repeatedly pointed o:tu~ht:s'fac:-[9)~llel and Kemeny, for example, have

!;

•

·sttn arr wee·

/

r.

~

l f' i

t·

~ t

ff£?

·'n t't:?rr

1 · · tt

1 t

cc e's, ~, z ·

tNM ·

It •

j

l

L [··

'

t. ,. f .

!

f· .

;

1; !.

i. i-

i· I

l

I

1·

1.

The perceptible Influence of lOint & can be 0 :coepleted at the tiae T if the opponent U can compare his a-strategy to ' each a-strategy of the opponent Vin such a way that the trajectory z(t) \of the system (1) corresponding to these controls bin the set M no later · than during the tiae T . Throughout the entire paper we shall aasuae . that the righthand ■ide continuously differentiable with respect t,o z and discontinuous with respect to u and v . In addition, for any z and v the set f(z, U, v) is convex and (1) 1a

'" .

j.i/(:, u, 11) l ~ c(1

p

I..

l )

I

i

,~

+ 1.il 1).

\-

Proa the result■ of reference (2) it follows that under these conditiou the set of trajectories of ayetena (1) begjl.nning with the fixed t point s 0 1.a ccapact in the metric apace of the c011tinuous functions for 1 a fixed control v(t) and all possible allowable c1cmtrols u(t).

1

I

!

.

.

2. Definition 1. The operator Ta, a ~ 0 places in correspondence L. to each ■et X c E" the ilet Ta (X) of points z e E" such that for · each aeuurable control r,(l), u(t) e V, the measurable control u(l), t:{t) ~ U, I 1 1■ found for vhich the ■olution of ayatem (1) with the initial condition I .z(~) • a and u • u(t), v • v(t) fall■ in the ■et X no later than after

i I

the

.. 1

ts- a.

Pro,,arty 1.

T0 (X) c T 0 ,(X)

a)

ri1 •

b)

I, .

L ··· . I-~ "

c)

·.i

(,

•

I

O

'

..-.

I.' ,- • • ~} t_..1_( :

T.(X)cTa•(X') .

To(X)

r:c

X, Tu(X) ·=> X.

I

,

~• •

!-

I

I

; ,..

•

•

\ .°j

·r •• r' •1 (X)

C

T

••+•.: (X) • ·

•

• ' ' •

' ' '.'

"

\

I

\

'L

\.

when

'

I " 1.,·•-: ri ''··: Property ' 2 • \

vhen a'~ a;

,

I

·,

~ I

· Let ua liat the propertiea of the introduii:.ed operator vhich are l alaoet obvious results of its definition. ~! ~

i1

1

(:• ·

I

,1:,

- 2 -

X' ::>X;

froa ·

!iroperty 3'. ,', if X i■ cl.oaed. then T~ (X)' 1a ' cioa.a. and vhQ a ) 00. aeT.(X) it followa that· 1 eT.i~·cx). ,· ~,.: ,. , t, •: ,r.:, ' · ·

Property 4. If the cloaed Hte X,, t - 1, ·..• , each oiher, that t■, Xc+a then ·

ar• iabedde4 ~n

c: X,,

cio

I

'

'

• .

'

'

•

I'

.> •

·.~lr.cx,)ia ·T.(n·x,).1:1,

•

'

I

I.,

Property 5.

x

Por arbitrary aeta ~

.

1-1

..

Q

•

I

,. .,, _ , ,• (•

',

J' I

' - ; , •; ;

-'

·t .,·

. l , , ·•: ::-'f·,rt'> ... . .... ,.

n Tcr(Xa)=>To-(n Xa); ' ta

(• ! (/') l ' ' l.'1' 1

' '~ ..:.'f. l/ .. ,.. r: •,

.i

Cl .

.•, ,~

' J< •

iy "

.

,·i

· :, • 1

' !, . . •',,:'' •

\'

The arbitra_ry finite aeries •of rational number• 0, 1, ••• 0 !"•'.to= 0 will bJ called the rational eubdiv~•iQ.n

3. Definition 2. ff, '\'i

~ . 'ti+h i

=

~ lwl ~ 'fm,

Let ua set

We shall. atate ~hat the rational. ~\lb-

di~i■ion w' is finer than the subdivision

if

)fl>'I E; lwf

and all the number■

of the number■ 'tj

8

Cl)

-r,, -r; ~ ·1w'I, .,

defining the eubdivieion_(1) 6, == 'ti -

'f(-h

1

.- coincide with '

'

• . ~

.

'•

Let for the given w

'"v>'-< w),

(it i• denoted by

'

. , ,,

-

let~-

I =- 1, •·, • ,: n.. •"

ea.,

, .•

defi~

T.(X) == (Ta,"T'-1 ••• Ta, (X). · · ·· . Lau 1. If

w'

< w,

then

Tw,(X) c Tw(X) .~ :·

-

'

\

The proof of the leau follove fr• ."roperty 2 of th~ operato~T0 (X) and definition of the ratio

Definition 3.

T,(X)

It is obvioua that Lelllla 2.

-=z

i»' < ~:-

·.::

n T.. (X).

l•l>I

T,(X) c r,,(X'), ·

if t. •

-- t ,111.

and

...

x•

>

x.

T,..,,(i)-= r,,r,.(X).

Proof. we ahall denote by w the rational. •~~ivj,eione lw•I > t, 0 for which the rational aubdivieione w1 amw2 exiat, eatiafying the · ', condition• ·

r_

r, .

+tz,

f

l t-· !',

f. 3 -

I ••

' I I

I

i

I

I

j

I

!

i_ f

I I ·?"·~' .t i,:- ~

1,f ,

- , . •• , _

1

(i)

(i)

{ ~ " " • _~. • , ":'},

,~ ~ch that ... -r ~ ta, ~•· _.:, ~- ~ ·, 1•

••• ,

It i• obvioua that

T.}~

1■

Let ua coneider th: aubdiviaion

< ,-. ·- ~...,.... ex> -T; c: T.acx>. {0,-rc+a--;,. •••·, -r.---r},-~,-. ;,:{~c,-'l'i..... , -r,,T}. . - . ,.

; , Th ••• , T1, -r, -r,+t, (,.

_

the rational number 't Q

found

and

(3)

• f

r--· f · when

(I}, . .

.

.

(-i.,

Q -

Thua. for uch

aubdiviaion w a finer aubdiviaion w will be found. On the other band, · a · it .ta posaible to ~lace the subdivision w such that (2) will be fulfilled 0 111 correspondance to any two subdiviaiona w1 and

I

t:

'=· •

>·,.,

f I

I

f

Coneiderirtg

l

t

f,_,

I •

,..

\ ,'

only

l• I > ••• ••

. . ..

. " .. .

r .

,.

·, .

n r .. ,x>_:_ n

-- r,,,,.-==

•.

what_hu been said, we obtain · 1..al > &., I,

I

r .."__: n · n 1·...rw1 1, &•'I>••

f"

)

t

.

r •

•

•

,,.

:

••

:

I

r~:wlleni ·property _5 of r . aubdivision

L, _~vi•ione

f

•

operator T0 vae· used in the derivation.

to conetruct the sequence

r

ao that :.. ,.

1-

! ~ • other.

r,.cx>

t, lt I ·> ·,a, "Ji== 1, ••• ,

1a foraed u

Nov

f,,}',.(X)

Si11ce each

the intersection of

such that

■eta

41>1.+1


1, 'J'..-(X)) = I .., 0I >1, 1'.,( •-1 0 T... (X))-

n

r

!

1a defined by a aet of rational numbers; the number of aub«a>, 1,,)1·> t,,· ia denumerable. It turns out that it is poaaibl• ·

;·

I

1

.. I,.' I >le

= 1'l,+I, (X)• *""n1 r ... r... (X) => I,.., In>t,TI, T ..fl (X) .

(5)

I

f-

l

fr

1 •

·.. ,,.·

-i •

~a~i•OD ~f (4) and (5) coapletu the proof of the leaa.

. 4. Theor•. _Let the differential gae be described by ayat• (1), and let the closed tend.nal set M be given. Let l (:0 )

= U ;,:t, min t. (If)

- 4 -

t

~.e 1,(M), t _.,O,

If

then

+oo.

&(z1) -

Thell in order for it to b·e poa■ible to caaplete the

point

•o

a- froa the in the ti.lie t , it 1• necea■aiy and ■ufficient that t 0 be lua 0

tbm t(s0 ).

Let •

"i.a r,.(M). . th•

indicate the b~ic ideu of the ·proof. If ,. _. l(io), then · · If the control v(t) i• kDOVll i~ the ■epent (O,a], than froa ·

. relation■

"e

.

.

r,.(M) == r.r,.-.(M> c: r.r,._.(M)

it fo~~Olr■ that it i■ po■aible to ■elec~ the control u(t) ■uch that 1(~) e r,.~(M) for ■me 6, O< 6 ~ a. By. continuing_thi■ p,;ocea11 the . ·.•·, trajectory of ■yet• (1) for any a-■trategy . of the o~ponent Vis brought to the ■et H no later than ·in the time t since 'i~(i,f) s:i M. . · 0

If t

0

■uch that

< t(z ), then 0

lo

~e r.(M), l(l)l > ,._

e T,,(M)

and the subdivision

Cd exist■ ·

Therefore,

1oeT._T._1 ••• T11 (M), . 6. + 6--, + ... + 6, - 1(1)1 > to'

u■ing

l

•

. '

•

.

.

(

'

•

; . :" :: •·

the dt:f inltion of the . operator . . Ta , we · see that . for .V the · . ·.: . . 1 a-■trategy defined by the subdivision and the ·control of opponent U exiat■ ·:r.. ■uch that however the opponent U operate■, the trajectory of the system ' ; (1) vill not fall in M before the time t • · Row

0

Cybernetic■ Institute Ukrainian SSR Academy of Sciences

. Received 7 May 1968 · ·, . \

BIBLlOGl(A)'ttY . 1.

,. .

. · , ,. ,.

L. S. Pontryagin, DAM (Reports of the USSR. Academy of Scienc~1), Vol ·

175, No 4, 1967. · 2. G. S. Goodman, Mathematical Theory of Control, 1967, p. ·222.

·

I

. I

10,84S

cso,

1880.S - 5 -

..

·..

r

'

!·

! ·.··J c·...•,-,J·-;!, : r,l

!

I I

II

1

1 ·, 1_~;

/

,r

1 • ...

:~n .~_,) f1~ o u:;:1

,' .. .tn '~7 >~. .' 1. : .l r: tf

~ \ , 1;: \

'. .r .. ?,

.-

; , • ,. -, 1

..:.. -,-. - ( d.

j\,.

•

"'I

t "'\ 1

, f/

~

'"l 1

~il 1 (!.~

I

•

"i fl ljt;?f10J} ll'1 i

r-,

•, 1

• ~~, •~ , \ I

_I

l

•

~::a:: '-',~[ !!~; -·~. ... \ , 1c J J'.1d iJ'.~)(71., ir, J' (j :~ .:;n ,~·j } . . L ,J r ' . ,., \ '.) J•J ~ '!_ :;, .!:'J' , ' r!) J.~ , r..., c;r:, .~ ! .r-.~:,-1 t•'i lf~. \L)

•

1

·-

•J,:

{1\

1

~

.:i ,.i.

", .·

I·I .:-

I

'

_ ,

: 1- ;·: ;~;, ·it :,

1·

I

\

1

.. :, r

\

;.,.•,·i

! •·:· ~. 't°'·S•••

,J

$'.

.._ , ;,,.:~

[fr ,.:, ·1 ,-~(.l

1

' i '_ ; · ·.·

-·:· ' .r. (: h(

~~

, '. •.. (' '

·r· . -",,J J

! ·- ,.~

' --:·

,

• •.• ·t~

(

i°': ~ • (.

0

)· ,) :,. r.,·~\

#

~~-: ~ •.:.:

•, . I

,· I

.,

, ., .

. ,., .. .) ~1

!

.',

·.,'

.A PROP!Jtff OP

Y/ ,

"

,

--

'~--.

A ,,.

iii )

~•.,-, • ;l,~{l;

,.

t.

,.

,

-

, 1 ,.,.

--'.1_ -' - 'Lll ...

l l .f ,-.~;, -, ,,J

,:.·~

.c. .'

1 ,

,":. ,J ..,.

,,-.. ,~ ]

r1

•

0

!'

.

.J_,:"· ,., . . .. UDC "' 62~S04.533 .

- · ~-

~~ -1J;! ,"{!•~

C:•l

•; ,, "(

i:0("--

,. .,,

•

.. ; , ,

c , .I __!

mtIODi·c MOl'IOI( 1or .A··sIHGI&-CIRCUIT '.NONLiNEAll 1~ ! ~-

SYSTEM WITH AM INTEGRATING LINK

.

. ·:i ·

'

.

a.

1 -

-~t ..·

. ' , .., .

I

[Article by A. s. Alebeyev; Moa~,. :Doklady .Akf •'.iv-..ti Hauk SSSR • llueaiaa~ ·); Vol 184, No 2 1 1969, pp 307-308) . Let ue iDYe■ tigate the poHible periodic motion■ of a control · ■:,■-· t • the dyn-.ics of vhich i■ de■cribed by _the differential equations .

('

\

,·

?;,

t-= Az + lxp (t',)e ~ - ~ (g (t) + c1'z)6 (t - /t),

!

l

(1) ,·:

1-0 ' ·./ . • '.. -, -~ ••

I I

I .

Ai■

matrice■

o-

I

I .. 'I I

P

; ' .. ,·

a fixed IIOflaingular matrix (n x n); band care fixed colllllll -~ (n x 1); x(t) is the same column of desired. functions; ·· j, (t), rpb) and g(t) _are the desir~d and given acalar functiona and .the -given·ecalar function which is periodic with a period a -multiple of T, ---respectively~ Equation■ (1) are a special case · of equations (5) .. from . [1] • ,and · they represent the dynamics of a single-circuit i deal pulse -nonlinear control . aystem in the circuitry of which, in addit:ion to c0111on feedback, a sum- · · i _: Ming element, an ideal pulse element operating with a repetition period T, an 1deal integrating element, the time constant of which is ■elected as the time scale, a static nonlinear elenient with the nonlinear charac- ,.. -c:eriatic q,(11) and a linear element of order n are connected. · From a ,.,. system with different real pulse elements usually we can -convert to a ·. system with ideal elements [2-4]. · :, where

•

';_f J

· ··

··o ·· · ·

lf

·

The solution of system (1) for x(-0) • x and µ(-0) ·•~ .• can be · written (5,6] in the fora z (t) = el•':t~ +

I

fe-t (1-t)lxp (I'(,)) d,,

u

I· where ve ha•• denoted

gJ -

g(j-r-·O)

and

{/IT)

.

'

• '·:•.' ,;' ~

'

I' (t) .... t'! +· ~ (gl + cTzJ) 1 (t - /t), (2) 1-o '!

I

f

11 .: ;

z'•z(rr-0),J-0.1,

1 1.,'

•I I

1

i.~.

6.,;,

' I I

~

,j I~•

"'

I

f. -

-tiDI /J! • ii/JT- OJ IIDd _,,.ruDI -1o1oyaly Ill to tha pal.at · /. · trllUtforaatioa ~n th• phua correapondiq to the aotion of the . (1) ad realisable with each cycle of ._the pulse element of the •1•t•, ve . • . obtain fof j • 1, 2, • • • · (3) I ·::1+ E)A-•b,p("'>,

■pace

f,

tA' -

t

~ l

■yet•

,A•zH .~:.: 1(•~-~-~..~ ~~~(~~~-~1 == deLe-4•, (U)

I

!

i

Gor'kiy Phy•icotechnical Research Institute Gor'kiy State University illeni N. I. Lobachevskiy

1-

l

i .

r·

II .

t

Received 8 May 1968

· BIBLIOGRAPHY 1.

'r

:

which it ia euy to see by aubatracting the lower row multiplied from the left by the colmm (tA"-E)-4,-Sbcp'(11't from the upper n rows of (11) .•

II

Ii

,:

Alekaeyev, Isv. vyssh •. uchebn! zaved. 1 Radiofizika (Neva of the High of Learning, Radiophysics), Vol 9, No 6, 1966, page 1218. . Ya. z. T8ypkin, Teoriya impul'snykh sistem,(Pulse System Theory), 1 Moscow, 1958; Teo_riya lineynykh impul snykh •is!!! (The Theory of Linear Pulse Systems), _Moscov, 1963~ A.

s.

In■ titutions

.

2.

- 8 -

···---·-· .,.,,........ . . ..:..-..-~--~

..,._____._.....~-.__'"-"-'._......_-=--- --..::...:·~. : ~•;..,. ~

,,J• ' .. .,;,••·

r~ ~

J_l:

.

'1 I ,. I· I

I .i i;(_;': I• •1

..

·' r

;.,

11BL10GltAPHY

I•

(cont,)

:, ,'; if•• 1. t.,i,kiD·~Avt~tika i ·telemekh ·, (Autou.t ion' ud Tel. .chanica) ,· fl ,, ,· fol, 23, No, 12, 1962, page 1565; Vol 24, No 12, 1963, page 1601. . 'f•• z. T1ypkin, ~ (Report, of the USSR Acadeay of Sciences), Vol ), 145; 1962, page 52; Vol 152, 1963, page 302; Vol 155, 1964, page ~~.

.

Ay■eraan.

M A.

P, I, Gantmakher, PMM •(Applied Mathematica and ,. K;chaniCS) , No 20, 1956, page 639;· . · ■ateniaticbeskoy

1. Bellman~ 1. Glikeberg, 0, Gro1s, Nekotoryye voprosy ,6; control ceorii protseesov upravleniya (Some Problems in Mathematical Theory of Processes), Foreign Literature Publishing Bouse, 1962. I, Hill, Punktsional 'nyy analb i polugruppy (Functional Analyeia I aod Saigroupa) , Foreign Literature Publishing House, 1951, , 1

,I: J.··' . ·..• ,. : ' ::

- -'] "

,·., -

·:1 ;1,, :a

:

.

,·,

1

(,,,.: l

'

, • .· 11

~ ,I

\ ., .

l:1'

.;,!

.·· . : .., , ,

L,.,

I;.

,/

I

I

I , ..

,, .

,,

r,' .

. :•·

.

'

\·'

La1 ,1 I

/.

l ':

1 1 ;_;. ,(

. i :• ., •, _;,

,

r ,,

,

!j ,•, ..-,I , '• !\l,t •'~· ;J

•

• 1· ,

t l

I ~ • ....

t •

!

!· '

1·-1 , . H.J , -~

~

. ,

• ' ri ·

. . ' . •• • •

-

~

f-t,:i'.~l ~- . ,.,.,._, ... • •

J'

1 ··,

•.

i

f

•

1 ·1

,

,4,~ ••

. :, .

}

(·;t· \

,- ·

10,84S

·~--- --~---·

:! ~:; cso •

'\ (:', i(

•AGl\a !,,; 'J 1 1,r

• . ••',·

C

'

(.~

11 }-,, I/

1

I

' 1 •1 ··:

I.

I

'

'

I •. ' r, ' . ~

' ,· , .

I

,, ' '

"

•,

;;".j-.,..-.4~ -

,41~~,..._...,.,,J.,...,::,11-.........,.."~'; ----- ~ -

_._\,,_ __,_, _"'-,_.';.......

~ ,·· ,.

1

,:,,

I l

I· . i .

..

. · · . Thi■ can be don• by aa11y procedure•. Thia offera the po■■ibility , in each apecific cu• of ■electing the aoat .appropriate .representation. ·

t l

.

l

Let

A. Lat ua introduce the following notatioo.

L

k-

/a(u; ")-= /1(u; 11;) - 1/1/(u; 11),

v. -

u.-u.-u.

~ .

v.-

v.

[ Tb•·· ·.

I. cp(y; o)- cpg(y~ ;,)- 6Pn(Y~~)•sup (1/1/(u; o) +(Y, u))

r -. .

'/JEU

.

.

r•

vi.th the regiOQ _of definition · (re V) - (r e, .V11) - (ra; Va) - {(y; P) El e.E"+'/11e V, cp(y. ·v )'< oo); ' i • · .·., ~(u; 6) sa,i, (&&, 6) -~~ (0; 6)- h1f (1/ 11 /(u: o)-:--(6, 11)) 1 ( ...:~ . . : •··1 ,. . . 11&V .

!

•

J

t .

f \

•

r

1

, .-.

-,

l

~-

f with di• region of definition a E•+l i u e U; ,i>(u; 6)

' 1

:

_I ..

l", .-. -) -. •

!he functiona

I

,

(U1 ; 6a) ·

{(a; 6) a ··

~(y;o)

and

1'!(Ui 6)

are conjugate (■ee [SJ)• J

I

II. If the function

-

.

-.

~(y; 6)-g'i(r. 6)-g~ (y, 6)- inf sup('/1 /(u: 11)

r•

.

~

(Ua; Aa)

~. -. ·•

t.f -!

(U; A) -co}.

I,

I

('._ I

>

.eV KGU

+ (y, u)-(6, 11)J,

••.

\

f

f

haTiag finite value■ i11 y or the function

r.' r t.

[

1. J.; ,{ ·

l

s~ t'!, 6) - z; (y; 6)- .r.ip inf { /a/(u; u) + (y, u)-(6, 11,l,

•-.

1

-;.;/1'&V

J..

.

I

-r

X

a,

6. - .

I · :·.,:: ., I'

6) -

A ia aeaiconti11uoua frca belOII with r-p•ct to

al.ao finite 111 X t.. 1a ■-iconti11uoue froa above with respect to then g'(y; a) • g'(y·; «S) • g(y;«S) 1a • cloaed co11vu-co11cave functiOII in

I ••

t·.

•

•

r r [.

• If" (y;

,, .• • :i .. , .

r•

If 111 the definin,t relation■ for q,, ~

and g the signs 111 front of

the acalar products are changed to the oppo,dte signs, we obtain new functiODS with their . regi'on• of definition '4>(--"Y, u), (-r; V), ,., \-,· {U· -A) g(-y; -b), (-f; ~~).

-v,,

. .

B. Now let us formulate the followins optillization problems.

. ·. . 1 , '

:· . . : Problem I • .Fin,l

':~ ~~I (u; 11). ..

~

I

l. I•

I

- 11 -

I.9 '

~

1- .

~-

t,~

I

I

',t

Probleiit I'. Find

•

•

• ••' • ~ \

J

mnx Jnf /(u: u). .. IIEU ..l'GV

.

")I• I

. I,,.,

.

;1,

_.

J

_

,·1

• r

j

rJJI

,' (',

, , •

t,:-:-:.;.

.

;'

•

; •_,.,.., _. !)•: I;

, IJ ..:.

Probl• II. ,ind ,,: •.IGf••:'~!:!,-r; v/'P (y: 11) + q>(--: y; u)J: Problem .II'. Find

IIUlX ·

. . ["1(u;6)+"1(u;-6)1: '·:,

(11: l)(.:(U; A)n(U; -J.I

n .-_·:.!U

. t·

Prob la III ; Find

min.

J, •,

t

-.&rnc-r}IE4nf-4)

-

'

,

..

(

,.,•; j , . , ~ ..• •;

sup (G(y; 6),+ i (.:._ y;- 6)J. I

I•

•

,... .

,,..-r-•· ,. ;:

.,

\

r, •. ; ·•

Problem III,'. Find

max

· inr [g(y;6)+1(-r;~.6)J;

ae~nc-4, ,ernc-r,

,

·of

,

. . , -. ..

.

t ,: · ,,·

, 1.. ,..

'

.; ..•

,..

~

is

The p~oblm in which the optimal ·v alue the p~rpose· . 'func't ion achieved i• called resolvable, and in the .case where the opt~ value exiac. but is not achieved, weakly resolvable. ',

:

~

•

J

~

, .

The following atatements are tnie rith respect to weak resolvability of the foraulated problems. -.- :c., i -~·

Br

If the sets

n(-I'; V)

n

and (U; 4) (U~ -4) . -~·• -· ,. '· are not empty, all the formulated problems are weakly resolvable and their opttmal values are equal. · · · :; ·~ ·" ·· (r;~V)

B2 • In order that all the problems be weak,l y resolvable, . it is necessary and sufficient that the origin of the coordinates of the space k+Z .

E

belong _to the set

r

x

A.

.. 1;.::::

With respect to the resolvability of the probleaawe have: B • If problems II and II' are resolvable, all the problems are:· ruolvable and their optilllal values are equal.

s4 • Problems II and II' are resolvable when and only when· probleaa I and II' are resolvable where all the optimal values are equal. If the origin · of the coordinates of the states EJc+Z is an intern.I point of the set r x A, all the problems are resolvable and their optimal values are equal. i. B5 •

. c. Let us proceed to the problem of the ezistence ·of an equilibrim situation in the game G. · • . ,, Let us set

+

a =·(r;'il; 11,

-=s lf(y; 6) c(-y; -c5) . tlie '),game G.

r = rn

.

•

•

. .

", .. \

I'

•

where (-r), A~ AO(~). l(y; 6)-· Thus, the defined game l: is called reciprocal to·

'.

I I

~ --

•

..

·, , .,.'!·,I

. - 12 -

.

, ..... '

' ~

.~

...·

.,,,.,.-,.,-

. i: · 1

\ ~

-

-

$! 1.'

1I

r

.. i

I

II

r

Theor- 2.

,

}I.

n

The folloving conditiou ·are ·equivalntt

3) The expr•••ion■

of £-equUibriu. for AD7 £ > O.

: NiV fnf aup /(11;· 11) 11&U .

and

aup inf f(ra; ·.,)

that .ie, they uauae ·finite vaiuea.

ed•t

I

,

•

•

If," r I.

•

·

,

.

pte V

auch that· the mq,resaiou fnf /(u.; r,) ..v

, ::r""; ...i

.

-

u•e U"

· · · .4) Ve have

f

o.

. ...o ...~

I•

1

5) The aete

- · -f ~it~ · · - . , tr; V) 0 (-r; V) and

are not

{U; A) 0 (U; -A)

-

J

...,_.,.

/, · ·,·-: · · ·

f

Nt

r

X

/.

6) 'l'be origin cf the coordinate• of ·the epace Bk+t b.o . '

Tbeor- 3. 'l'he following atateaents exiat:

r

1) If the probl... II and II' are resolvable, the have a situation of equilibrium.

i

!

I

'

I .. r..

.,

to the

s-• C and G

8Xp'Nl8810D8

mi;i aup / (i:-

and

•~

•mnx int/ (u; u)

l'EVMfiU

[·' . ~lat ·.

ut:iU IIGV

lf

.-..

•·1

..

;-

.

'

. . The g•e~ G. and G have the aituation of equilibrita, and the ·prob1- of -tbeaatical progr...tng reciprocal to thea are resolvable · ~ the fo~lowing c..aeu :

i: ; :

t

I""

I ..

r ,:1 : : r

t t

.•

'

J ___

•

.

1. The

••t•

2~ 'l'h• aet

tr; V)O(-f';

V)

and

(0;-ll,

.er·. V) n c-r;_V) ccu. A) ncu. -4))

!J (U:

-~)

are bounded.

is bounded and baa a

-relatively internal p~~ot.

I

-

13_

I .

r.I

1·

~

3) '1'he 1 - G hu the eituation of equilibri1.a vhen and only vhn t~

f I.

,

~\

2) The problfllU II and II' are resolvable when and only when the bu the aituation of equilfbrima.

, ... C

I

belong■

'

I

I.

for

2) 'l'be , ... C .h u th• situation of £-quilibri1a for a fixed£>

.

f

n

'

1) The 1••_C hu th• eituation

{

I

••t• ...

If the tr; V) (-r; V)_ and (U; 4) (U; -4) ' . are aot empty, the l • N C and C hAYe eituatiou of £-equil:lbr1• AD7 £ > o. 'l'heona 1.·

/'

.a .

'r ~ ------------------------ - ---

I :•

3. The ternal

••t• ·

IJ'; I') 0 (-f; -V) and·

., • • L

I

have ta.;,

.(U;, A) 0 (Cl; -:--A)

point■•

4. The ••t•

U·and ··v

are bounded.

alt+t·

$. The origin of the coordinates of the , -~~ce ·b ~{~~- to the . ••tr x A and ia a relatively internal point of this set.

D. Let ua consider the following . problem of convex prograilin1° Let the conti~uous concave funct~o~• · p(u); _q, (u),. • •, q,(11)0 · fined in a clo■ed convex •et problea is •tated: Let ua find

u• e

ti~=

u(u e

{u e E• tu·-;;;. O}

E• I u ';;3: 0, g(u) -

.de-

be given. ·: Then the (g, (u),· ••• , _g1(u))

> OJ,

-· .·.

at vhich the scalar function p(u) reaches its greatest value. ,

The Lagrange function L(u; v) of thi■ problem has the fora L(u; r,) -=r p(u) + (v, q(u)} with the closed region of definition U x V where

V.. {v e E 1 Iv ';;3: O}.

By conatructioa of L(u; v) there is a closed concave-convex function and all the above-pre~ented results are applicable to it. However,' in . thi• cue another representation suggests itself, namely: J, (u; u)

= p(u),

l;(u;

v)

= (v,

q(u) ),

Ut == U1

..

U,

V1

-

V2

-

V.

::

Carrying out the conatructiona analogous to the constructiona of iteaa A and B (see also ( 41), we obtain two dual convex p:oblems of opti.ai- ·

zation and a dual game. q,, and y, (I= 1, 2):

Let us present the corresponding, relations· for·

q,, (y; u) =- sup (/1 (u; v) + (y, ~)J -== sup fp (u) + (y u)J ueu, ueu ' • (f,; Y) {(y; u) e E•+i Iv;:, 0, cp1 (y; u) < oo},

=

= sup (/1 (u; v)- (y, u)J == sup l(v, q(u))-(y u)J · w.eu, ..eu . • .• (-rz; V) = {(v: v) e E•+i Iv~ 0, q,z(-y; v) < ~} Y• (u; 6) = p(u), (U; Aa) == {(u; 6) e El+l I~-;;;, 0, u > O},

,)

fl'• (-y; v)

,Pa(u; 6) • 0, (U; -A1) = {(u; 6) eE"-N/ g(,i)

'

, 1·

'

+6·> O' 11 ,.. J O,\,. . ...

l

- 14 -

•~

~

·. ~

:

lo • '

:

Probl• II , _,,, ,..

I•

I.. Pi~

't.J'.

.,,:I ' ...) .

-~

,lti(r. •)+~--r. t)). . .

.

. hi •>&lil(~,·

V)

.

Probl• 1111 Vill be called . ' . . .. '. th• buia of th~ Lagrange function reciprocal to the initial. p~oblea oa 11• ill ~he duality occurring when•uaThe duality of probleni1 Il'a ·and exaple, ( 1) & Chapter 7) • ing conjugate function, (aee, for . Theorea 4 • The Lagrange f1mct1 L( · · ) · 1ituaticm of equilibrium) when and ~ u; v has a saddle point (the •ex prograning and the problem r:t.c~ Y when the initial problem of confunction are resolvable ·and· have equ~rocal.. it on the bads of the La~range · a1 opt.uua1 value, (compare (3, 6)).

!~

The full proof of the presented reaulta appears in [7]. '/

'

~ ,

I'

Received 8 May 1968

BIBLIOGRAPHY •

#

I•

' ~

!(atematiche■ldye metody v teorU igr 1 progr8Jlllllirovan11 i ekoncmilc.e, (Mathematical Methods in Gaml! TheoT) ,Programming . and Economics), Moecow, 19.6 4. . L. v. lCantorovich, G. P. 'Ak~lov, Punktdona1 1 nyy analb v normir~v•nn,kh proe~nstvakh (Functional 'Analysis in Normalized Spaces), Moscow, 1959. H. W. Kuhn, A. W. Tacker, Proc. II Berkeley Symposium on Mathematical Statietice and Probability, 1951, ~age 481. V. N. Lebedev, N. T. Tynyanakiy, DAN (Repoi;:ta of the USSR I cademy of Science■), Vol 154, No 6, 1967. w. Fenshel, Canad. J. Matl!.!_ No 1, 1949, page 73. t. J. ArrrN,A• Gurv1te, lCh. Udzava, Isaledovsniye po lineynom~li rogrammirovaniyu (Investigation with Respect to Linear ~d Pr~amming), Foreign Literature Publishing House, 1962. lf Ty kiy Oenovy teorii dvoyatvennosti zadach nelineyno_8! • T• nyans ni ; i differentsial 'nyye igty (Fundamentals of Duality programmirov•~l i Nonlinear Programing and Differential Games),,

· 1. : S. Karlin, 2. 3.

: ' 4.

5.

6.

1. 1·

\:

::~raJ

Theory o Prob ems n Moscow, 1968.

i i

10,845 CSO: 18~S

- 15 -

. :j:

' -.

...-

·•

' .

,. :

.. :-. (

•. l

: .. 1

'

.

\ ! , j' '

.

!

-

,.,

.

- _; I'•

. . ·. . DBACK . RELIABILITY OP LOGICAL CillCUITS WITH PEE .

.

'.

· . ;·, UDc .

,

• ·• • •

.._

l

I

_.•

.J .. ;

•

••

,.'

,,,,

i6~

•,• :

J. I

,._ b A VI.. Giotgadze· Tbilisi, Soobsh'c heniya Akadeai i Nauk [Ar t i c.- 1 • "'1 • ' 2 1968 pp 315-318) Gruzinsltoy SSll, Russian, Vol 52, No , • . Let · 118 consider the logical circuit L having • binary inputs. on, output and a feedback circuit connecting the output L to one of it■ element• (1). It 1■ convenient to represent , the functioning · of L with the help •f the transition chart of the Muhr automaton in the following vay. Let us introduce tw0 states corresponding to values of .1 and O in the output channel of L. t.et a be a letter of the input alphabet, 1 • . i . • 1, 2, ••• , 2•; P(ai) be the probability of occurrence of the letter . 1 The probabilities P(ai) are independent of time e,. en in the aggregate,

a,

9

The transition matrix of an ideally operating circuit L when the letter ai . i■ input to it willbe ~•noted by A1 and we shall compile the generalized matrix A• ·.., ~At · .P ('-',)._ The transition matrix of the circuit L permit•

°,

ting f ailurea will be denoted by Ai and we mall introduce A _ j Finally, let B be the matrix of the circuit Lj obtained when

~

~

A p

' (Cl',), ' ' the element

aj fails, j • 1, 2, ••·• N; N is the number of elements of L. The representation of Lin the fora of illustrated by an example (Figure l,a, l,b). an automaton f.a

Figural • 16

6

~

- ~

z,o

ex

·x

1,

. . . s -~, ·

r

so·: ~

-- --d

.

'~·tt· t a

!. ·, i

11 · : 1 a;-q-.

---.

I

II I

In Pi~re 1.a we have a 101.tical circuit with a feedback circuit

,/

aod in Figure l,b, its automaton representation.

1 / /

i · ,rariable■ x

For values of the input

~

0 _a nd x 2 • 0 (that is, the letter ,a 1 ) and with y • 1 at 1 the output (the autoaaton ia in the atate ·y • 1) the output value in the

·1

I / atxed cycle will be O (the aut0111aton is converted into state y • O).

f

The

letters a 2 , a 3 and a 4 are tested analogously.

I

We

■hall

coneider the average number of

cycle■

of proper operation

!I T .. the desired reliability chararacteristic of the circuit L under the

aHuaption that from the beginning of ope1:ation to the time of occurrence an incorrect resu~t at the system output one failure can occur. No Uait 1■ imposed on the multiplicity of failures.

!. of

r

I

Let us write the condition of logical equivalence of operations of th• ideal automaton Land the automaton L' subject to failures:

,,: V UV-• Here 7 and 7 1 are ihe output

th• automaton with

II

'

-' I

'11, /·

function■ failure■ re■ pectively,

of the ideal aut01U1ton and

Let u■ compile the matrix B describing the Markov chain. the set of atates (s 1 ~ s 2 , • s• e sk) of whiat is the cartesian product of the ■et■ of ■tates of the automata Land L'. For exm,mple (1,a, l,b) thi■ aet ta (0, O), (0, 1), (1, o~. (1 , 1). The element P(s1sj) of the matrix s 1 t■ the probability of simultB.lleous conversion of the automaton L from state 1 to state j and the automaton L' from state i to state k. State 1 of aut0111aton Land Z of automaton L', the pair (1, Z) _• Si, fora the atate s 1 • ~; pair (j, It) • •j forms sj; 1, j, k, ,z • 1,. 2, ••• , n; •i•j • 1, 2, ••• , 11 •

°

Let u■ convert the aatrix B1 into B1 in the following way: let u■ introduce the additional state •o into the set (a , s , •••• •1t>• In 2 1 addition, ve a hall i•olate the set of state■ . c- (s1l , s1a, •••, s,41 such that , all , 1 EC are formed by pairs of the type · s1 - (/, {). The probabilities P(a aj) for EC and • 1 • 1, 2, ••• , k are the probabilities

6;

1

of tranaition of the automaton• Land L' to identical etates. 1

/.

r

For the investigated example P(O, 0) and 'P(l,1). In the matrix 10 all the probabilities p (s,s,), s1 ~ C; all the probabilitiea p (s,s,), s, Ec, 1 s,- 1. _2•••• , k:: P (s.,s,), s,-i. 2, ••• , k will be set equal to zero." The elements

P(si•o>• a •

aeut

1

• 1, 2, ••. , It will be aet equal ·co

P - 1. - 17 •

1- -EP(s,s1); •J E: C

the ele-

""

1·

i

0 Thue, the etate •o le the ab■orbing atat• for the network de■crl~ld by 1 1 • It le eaay to Hi that p (s s) ls the probability of trari-, . ' I .

f

'J C

·.· .

■ition

of the autoaata Land L' to identical atate■ _frca the ■ tatee frgmins,.~1- The analogous •• of the t-th degree element• o.f the matrix B1 , ia the ,robability that int c,clea the automata-Land L' will Mke the: tran■ ition froa the atatea foraing a

atatea.

L

Thu■, the aatrix

B~

penitting failures.

'I

i to one and the eae aequence . ·of characterizes the reliabilic, ·of thecircuit · 0

.

..

II I

I

.

It is posaible to obtain the matrix 1 analoRously for th• reli2 ability characterietic of an automaton with a failur• in one of the eleaents. Let a failure occur in the element _aj in the t-th operating c,cla of the automaton L. Then the automaton L degenerates into the automaton Lj which differs from tin that the function .!(aj) of the element xj in ; the fonaula■ deacribing the operation of th• automaton with• failure 11 replaced by w• (a,) + w(a,). Perfol"llling the■e operations on the matrices of the automata Land

LJ.

we obtain the matrix B~.

Let ua assume that it ia given that tj(t) is the probability of . occurrence of a failure in the element aj in the t-th cycle. Let ua de- . note the probability that the autolll!lta Land L' will pass through the same sequence of stataduring the course oft cycles by f ~ and by f 2t 1 let us denote the probability that the autOl'll&la Land Lj will pass through the same sequence of states. In addition, let the probabtlity that the automata L and U will be in the state■ i and j during the cycl;e' t be Q' [s, (i, /)) under the condition that the initial state of the auto-, aaton Lie given.

.

..

Then the probability of proper operation of an automaton subject to breakdownsand failuain aj during the course of r cycles be written in the fora

r-1

R (r) .;.

L

k.

.

L, f{ •r.-

1, ct>

l=l •1•1

1 (t) •Q' (s1) ·

and the average naber of cyclee of proper operation by "' T= L/R(r),

,.1

- 18 - .I

I·

I

I·

I

I

·O O . It 1• poaeibl• to calculate the aatricea B and 112 and al■o 1

t.. j

\

-,

(.

i.;

o~

I'•

! 1_Qt.( ';j) . vith

the help

I ~, -L',


j j - : if p u the nmlber of the initial_ ~ent of the operation and Pj is the 1 r ·· uuaber of the final event of this operation. Thia numbering vill be " carried out by the Ford algorithm (2). New event numbers will be written, · ·re■pectively, in columns P1 and Pj of Table 2.

! I

II ' t

;'' '

l

'

'

(

•

~ f •'

' :. • 't

•

Table 2

"··,• J •

J -

P.

~

1

K

!. \.

. I

l

,,,

s

t,, t' r:"• t"J J' -front (1) • and let u■· denote this ■ua by Rl"l. .

.,

•

The fros on which the relation

I.

I •. 1•

ia fuUillecl will be called the norul front, . and in the opposite caee:, .' the critical front. Our problem is to convert. the. critical front into the nonaal front if thi■ is poaeible. ·i• ••. . ·, . I

1 Step. Let us find the first elementary front: . let us find the· elements of the colUllll\8 'ti" and -:j -:, .,,;; min lti) and -c1 ... min l-ci,, -:jl >-c.. vhf.ch gives the fint elementary front {To, ~ 1 ). The operations perfo~, on thia front will be operations the dates of occurrence of'. the initial · events of which are 'tO• Let us sum the intensities of these operationa~ and if the relation(*) occurs, then, the elementary front will be no~l and the elements of the columns 't1", 't/ and µij will be cartied over without change to the columna 'ti*,. 'tj* and µij*. If the relation(*) is: violated, that is, thr. elementary front is critical. then we find the distribution of resources for which the relation(*) will be satisfied. i Let ua number the operations of this front as follows: the first number■ will be assigned to the operations which have operAtiOQS following directly in the process on this front (this can be defined as follows: the elements of the column Pj of the selected operation will be compared with, the element of column Pi; if any element of col\lllln P1 ' is equal to any · element of column Pj, the operation corresponding to the latter element will ·be a■ signed the first number). The next numbers will be aasignedl to the operations in increasing order:

• •

Prom the first elementary front, the operatioM which have large will be carried over U>the second front located directly to the right ao that the relation(*) will be satisfied. It is ·clear that the intial and final dates of the transferred operation will be

number■

respectively. - 24 -

' -·

L-

:'ttr'

>tn

Wt

3d

tt t • a: L ••

r t b17Sh rt

'IMhl\

It r 2·1 -

.

-·

•.:

a

• t;;t:

' ·~- -

1211Ctt JJ eht'xQ' to

· · - --

- -- -

1 ..:..,_

- - - · re+

~

I

•·-th Step • _ • .... t ua f incl the . el....ta 9C1Ual to . . Cf•-th el...ntary front: ·

give the ·4'-th

occur for the

.

,,_, • min I~ ..i'\ • " · •11

•n

cl

el• entu, front · f'f.,_1, t ).

~

,:• • min \-r,, ·

let ue find the which

> •11 -r..-s,

• -"'

· .' The following relatione vill ~ted operations of thie front:

:•t

If ltbG Si~• f fr• :lt h, the • ~ iii Ciriti~ lc th1: opHaU.ou rill be trauferred ing the operatiou ••eribec2 rul«h The clifferemc• U.ee only in' number• ginn1ng 1n the : th• firet number~ ar~ assign.ad t0 th• operations betiona 111 anal preceding fronte and the a1mbering ~f tite remaining operaogm.ae to the firet fr~t . · Thie ie th• ••• for all r~iui~

f!'0ntz 0

The operationa in agricultural production &iH perfonaed on one or •neral fronta. The nmbera of these fronts will bG introduced i~to the Cf-th co1'811l of Table 2, and thia will completely fill out the given table. The above-■tated problea ie completely aolved, the reeourcee are diatributed, and the calendar date■ of the beginning and end of the operations are calculated.

However, ve 11Uat coneider that after operatinn of the algoritha critical fronts can adll reaain. In thia cue it ta neceHary to reexamine the plan: either the plan cannot be fulfilled under the given conditions and it auat be corrected or it le neceHary to in~o·duce addi-• tional quantitiea of reeource■• 'l'heae prob'lellltl are aolved directly by aan.agement (the board of director• and adJlini■ tration) depending on the existing situation. Ac.deay of Sciencea of th• Georgian SSR Cybernetic■ lnatitute, Tbilisi BIBLIOGRAPHY

1.

o. ~.

Aburdshaniya, B. G. Shukakidse, Soobahcheniya AN GSSR (BulletiD of th• Academy of Scieacea of the Georgian SSR), Vol. XLVIII, No. 3, 1967. . ' 2. s. I. zukhovit■kiy, l. A·. lladchik, Mateaaticheakiye ■etody aete\lDg~ planirovaniya (Kathwtic•~ PERT Methods), Moacov, Nauka Publiehing . Bouse, 1965. · 3. D. 1. Golenko, Trudy ~n-~:b ~tem:tiki SO AN ~SS~:...(Worke of the Matheaatice lnatitute oft e · er an epartment o t e USSR Academy of $cieace•>• No. II, 1964. - END -

10,845 CSO: 1880-S - 25 ...

.. I