The Use Of Informal Calculus in Problems Of Artificial Intellect

THE USE OP' IDOIUIAL CALCULUS I1f PROBLEIIS OP' AR!IP'ICIAL IITBLLECT

by A ··· Tiro Research Institute of Systems of Planning and l&nagement in Electrical Industry 20 Pirita Road, 200106, Tallinn

u.s.s.R.

ror taking decisions in complicated situations, especially with the help of computers, various heuristic methods are often used. However, their use is indi­ vidual in character and requires special apparatuses for every new problem.

The present report outlines prin­ ciples of a possible general apparatus of informal calculus and observes its application to the problem of taking

intellectual decisions.

By human intellect we mean a complex of mental abilities that ensures man's behaviour in complicated situations. Accordingly, by one of the principal intellectual functions of some artifi­ cial apparatus, first and foremost a computer, we may mean its ability to come to independent reasonable deci­ sions. To reflect these sensible deci­ sions or operations in intellectual problems the present paper dwells upon the use of operations of the informal calculus apparatus suggested by the author (1).

The realization of these problems will be carried out by an abstract com­ puter, referred here to as a system com­ puter. It can also be a man with the necessary training, or, for some types of problems, a computer supplemented with the model operations of informal calculus.

The ess~ntial credit of these operationa is their non-determination, i.e. ambiguity and informality of their re-. . sults. Consequently, the realization of various intellectual problems may pro­ ceed in different ways. Thus we have a calculus on band (2, p.203), i.e. we have permission to carry out some operations.

ror example, if we add 1 and 8 with the aim of getting an arithmetic sum, we shall get number 9. However, if we add 1 and 8 with the intention of getting a new two-figure number, we shall get 1& rurther, if we add 1 and 8 with the aim of getting a new symmetrical figure, we shall get $ , but if we simply aim at any new figure, we shall get figures 18 or $ or 81 , i.e. three different pos­ sible results. Obviously, it is the realization of some general summation operation that below will be referred to as the system summation operation.

It ~ould also be noted that tor the man informal calculus operations are strictly enough determinable by intu­ ition as they, first and foremost, for­ malize common arithmetic operations with objects and notions. However, the use of such operations in intellectual problems does not serve the purpose of copying the structure of man's thinking.

P'or realizing of informal calculus operations it is necessary to give rules governing the representation of real system on the level of a computer which In this case, some real system (A) is represented by some set /A/, oalled here the system representing structure, of the symbols of the chosen alphabet

for describing

in the system [A) , the context (1) of which is meaDingful for its goal K. In other words, (1) and (2) together pro­ vide for an unambiguous representation of the system A in the structural form of /A/, retaining the goal K. The goal K makes it possible to eliminate from the observation a great number of elements aDd practically all the links of the real system, aDd it also dictates a strict succesion in (1).

Let us call such a complex of ale­ mots and links of the system [ A J , ar­ ranged into a succession according to the goal K a sequence of numerals of the system [A] with respect to the goal K, and the very successive elements and links of this system in consideration of their interaction - numerals of the sys­ tem (A) with respect to the goal K.

It is possible to eliminate the un­ necessary elements and links from the real system and to arrange the remaining ones into a sequence only in case we have sufficient general knowledge in the field of science to which the given sys­ tem and its respective goal K belong.

lor the initial operation of infor­ mal calculus the taking of a decision when two objects or notions tA J and [B] coincide according to the goal K has been chosen. Here, this operation is de­ signated as the system coincidence oper~ tion and is marked by the following symbol,

Due to its contents this operation is on the level of elementary human intellectual decisions. The process of learning occupies an important place in the realization of this problem.

lor the realization of the system coincidence operation one should estab­ lish for the representing structures /A/ and /B/ of real systems or their oor - responding pair-elements

a sign of physical, technological or meaningful coincidence with respect to the given goal K,

In the latter case, the presence of coincidence or lack of coincidence is given separately for the possible pairs ai :bi of the structures /A/ and /B/·

Here are some examples of e7stem coincidence:

1. I< DBSK-LAJIP') I (· ('POBIU.!r!OB OP LAIIP-PI~~D�S ') )/ < ST.ABD.ABD LAIIl')/ .. YES.
2. In a shop of discrete production it is necessary to control the normal state of production with respect to the realization of the plan by the end of the month. The state of production is characterised by the system (A] repre­ senting a number of actually completed single operations a 3

At the same time, we have a lay-out plan

• [B J which requests an amount bj of these completed operations j by the given date:

It is generally known that a fore­ man, despite the condition a 3 ~ b 3 in separate element-pairs, can, neverthe­ less, determine that ·the state of pro­ duction is normal·, i.e.

/A/ ( · (lfORIIAL S!ATB OP PBo­

DUCTIOlf) ) /B/ -+ IES

by subconsciously taking into consider­ ation a wide range of factors.

The indicated problems may be re­ lated to the intellectual ones as the direct apparent information in this case is incomplete. If the problems are real­ ized in a system computer, the partici­ pating ob3ects or notions, which will be chosen by the goal K and transferred by it into sequences, are called in the ap­ paratus described here - the sequences

of numerals.

As an example, let us follow the formation of the sequence of numerals of the system [ C] with the respective goal K1:

K 1 - "formation of pieces of

fUrniture for sitting·.

It is known that there are exc ep­ tionally many types of such pieces (i.e. systems), like chairs, armchairs, seats

of various styles.

The formation of the sequence of numerals should follow the complex of knowledge pertaining to ( C ) and K 1 · However, in the present report examples have been simplified and this complex of knowledge has been replaced by the following two rules which take into ac­ count only functional properties of the system I C 1 elements, though their tech­ nological, ornamental and other possible properties should also be considered.

1. An element of the system with the highest functional meaning should stand in the sequence of numerals before an element with a lower functional meaning.
2. The sequence of numerals must be

preserved in correspondence with the given goal K if elements of lower func­ tional meaning are excluded.

In this case, the sequence of numer­ als for the system [ C J with respect to the goal K 1 acquires the following structure,

/C/ · /<SEAT>, (LEGS> , <BACK) ··· /(3) fhe circumstance, that the following

structures

/H/ · / (LEGS), (SEA!">,<BACK) · · ·I (4)

/M/ · I (BACK') ,<LEGS) , (SEA!) ··· ; (5) are incorrect sequences of numerals of the system (C) and do not satisfy the goal K, can easily be proved if we apply to them the above-stated rules.

On the other hand, we can see that when we check the ccnfomity of the highest functional elements of (4) and (5) following the goal K with those of (3) as a whole, we accomplish the system

• coincidence operation,

lfO - stands for the lack of system

coincidence.

The following operation, that in system sense is opposite to that of sys­ tem coincidence operation, should be of great importance in intellectual prob­ lems. ·It is "the operation of searching for a coincidental system",

• [A] is the initial system,
• K the goal in the search for a coincidental system for [A) ,
• /B/ the result of the operation, i.e. ( B J is the system that has system coincidence with /A/ with respect to the goal K.

This operation will be realized by searching for the coincidental system from the whole capacity of the memory of the system computer which can be: an abstract system computer, man with ad­ equate training, and for some sets of problems, a usual computer with a model that, in addition, includes the model of the system coincidence operation.

~he system summation operation

represents the joining of two systems (A) and [B) , and their resultant system [D) will have new qualities determined by the goal x. Here IAaEsT·Bzms~/ are the unused elements of the systems [A) and [B].

The application of the described operation allows to lay down the reali­ zation of a highly intellectual problem, as for example, the formation of a new system of general!zed complex notions from two systems of elementary notions. In the development of science we find well-known examples of this operation, like the formation of mathematical ecc - nomy from the concepts o~ mathematics and eccnollll", or physical geography from physics and geography.

This requires the joining of the structures representing initial systems [Al and [B1 of elementary notions ai and bJ:

into a new system (D] of complex notions d 0 , the representing structure of which

will be

If the problem is presented in a generalized form, the contents of the goal of the system summation operation will be the following,

K - "formation of complex notions".

Let us consider the required algo­

rithm of the system summation operation in an example.

1. Using the system coincidence op­ eration with respect to the goal X we check the general principal potentiality of each element ai of the system ( A ] to form complex notions with the elements of the system [ B J :

As a result we get initial or origi­ nal variants of complex notions, for example,

2. In order to draw from these initial complex notions a final comprehen­ sive system of complex notions, we must introduce a basio sy.atem [ E] ,

As a rule, [ E] represents a system of generalized notions for the presup­ posed elements of the system ( D] · The system [El may also be taken from the branch of science analogical to that of the system [ B ].

In this way, the application of ( E] must guarantee the meaningfulness of the next step of the decision.

). Elements of the final system-va­ riant

come forth with the use of the operation of searching for a coincidental system for each element of the system E sep­ arately. The search here includes the structure of initial variants of complex

notions /Ai:B 3 /:

In this case, the goal x 1 stands for, K1 - "to secure the greatest meani~ulness".

The operation of searching for a co­ incidental system as stated before, is realized by the use of system coincidenoe operation in the structure of co111111on algori thlls. Consequently, the system summation operation, according to the described algorithm, can be realized on condition of the realization of the system coincidence operation.

:ror ~DBDY problems of taking intel­ lectual decisions a model of operation ot system coinoidenoe can be implaented tor a USQ&l computer.

ODe of these models has been suc­ cessfully tested by the author for the realization of the problem "deteraination of the normal state of production", which has been described in the beginning of the report.

Reference

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2. MapKOB A.A. TeopHH aAropBTYOB. TpyltV MaT8M8'l'B~8CKOro HHCTBTy'l'a AH CCCP BY. B.A.CT8MOB8, XLII , Hslt · .AH CCCP, !954.