nonmonotonic logic for analogical reasoning · 2006-01-11 · nonmonotonic logic for analogical...

4
Nonmonotonic Logic for Analogical Reasoning Guiyou Qiu National Research Center for Intelligent Computing Systems P. O. Box 2704 Beijing 100080, P. R. China emaih [email protected] Abstract In this paper, we provide a nonmonotonic logic for analogical reasoning and a graphical represen- tation method to facilitate the calculation of analo- gies. 1. Introduction Analogical reasoning is one of major forms of humanreasoning, if not the most privileged form. Ascribing the capability of analogical reasoning to computers is one of the long-term goals of artifi- cial intelligence research. The method of reason- ing by analogy has been widely used in theorem proving, problem solving, machine learning, plan- ning, pattern recognition and someother fields, see Hall[6] for a survey. In contrast to other forms of reasoning(e.g, deductive reasoning and inductive reasoning), however, the logical aspect of reason- ing by analogy has been rarely studied, with a few exceptions such as Davies and Russell[2]. Among the reasons for the situation, there are mainly two reasons: the complexity of the concept of analogy and the defeasibility of analogical conclusions. The real role of analogical reasoning has not been recog- nized fully. We believe that logic can give us much help. In this paper, we will discuss some logical aspects of analogical reasoning from! the view of nonmonotonic reasoning. The plan of the paper is as follows. In section 2, we give a short review of previous work of logical approach to reasoning by analogy. Also, we dis- cuss analogical reasoning from a view of analogical reasoning as nonmonotonic reasoning. In section 3, we introduce a nonmonotonic ana- logical logic NAR. Some properties of our logic will be proved. In section 4, we propose a formalism of knowl- edge representation to facilitate the calculation of analogies. In section 5, we focus the integration of similarity-based and determination-based analogi- cal reasoning. Finally, in section 6, wegive someconclusions. 2. Logical analysis of analogy" a review Analogical reasoning may be defined as the pro- cess of inferring that a conclusion property Q holds of a particular situation or object T ( the target ) from the fact that T shared a property or set of properties P with another situation/object S (the source or base) which has property Q. The set of common properties P is the similarity between S and T, and the conclusion property Q is projected from S onto T. Generally, we can summarize the process schematically as follows: P(S) A Q(S) P(T) Q(T) Evidently, the above schema of argument is not deductive. Wecannot justify the plausibility of conclusion by the argument itself. The justifica- tion aspect of the logical problem of analogy leads Davies and Russell[2] to develop a theory of deter- mination. They define analogical reasoning within a deductive logical framework. If adding determi- nation rules to premises, the analogical conclusion can be followed soundly. To prevent the redun- dancy problem, determination rules rather than im- plicational rules are assumed. The problem of this theory is that in manycircumstances, it is not easy to get the determination rules if any. Moreover, the determination rules have exceptions. Loui[8] demonstrates the possibility of accounting for ana- logical reasoning in Kyburg’s theory of statistical inference. In this paper, we regard analogical reasoning as nonmonotonicreasoning. In this respect, the con- clusions of analogical reasoning have the defeasibil- ity. Also, we can explain the phenomena of mul- tiple conflict analogical conclusions. We can in- 161 From: AAAI Technical Report FS-94-02. Compilation copyright © 1994, AAAI (www.aaai.org). All rights reserved.

Upload: buicong

Post on 23-Jun-2018

213 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Nonmonotonic Logic for Analogical Reasoning · 2006-01-11 · Nonmonotonic Logic for Analogical Reasoning ... reasoning(e.g, deductive reasoning and inductive reasoning), ... [2]

Nonmonotonic Logic for Analogical Reasoning

Guiyou QiuNational Research Center for Intelligent Computing Systems

P. O. Box 2704Beijing 100080, P. R. Chinaemaih [email protected]

Abstract

In this paper, we provide a nonmonotonic logicfor analogical reasoning and a graphical represen-tation method to facilitate the calculation of analo-gies.

1. Introduction

Analogical reasoning is one of major forms ofhuman reasoning, if not the most privileged form.Ascribing the capability of analogical reasoning tocomputers is one of the long-term goals of artifi-cial intelligence research. The method of reason-ing by analogy has been widely used in theoremproving, problem solving, machine learning, plan-ning, pattern recognition and some other fields, seeHall[6] for a survey. In contrast to other forms ofreasoning(e.g, deductive reasoning and inductivereasoning), however, the logical aspect of reason-ing by analogy has been rarely studied, with a fewexceptions such as Davies and Russell[2]. Amongthe reasons for the situation, there are mainly tworeasons: the complexity of the concept of analogyand the defeasibility of analogical conclusions. Thereal role of analogical reasoning has not been recog-nized fully. We believe that logic can give us muchhelp. In this paper, we will discuss some logicalaspects of analogical reasoning from! the view ofnonmonotonic reasoning.

The plan of the paper is as follows. In section 2,we give a short review of previous work of logicalapproach to reasoning by analogy. Also, we dis-cuss analogical reasoning from a view of analogicalreasoning as nonmonotonic reasoning.

In section 3, we introduce a nonmonotonic ana-logical logic NAR. Some properties of our logic willbe proved.

In section 4, we propose a formalism of knowl-edge representation to facilitate the calculation ofanalogies.

In section 5, we focus the integration ofsimilarity-based and determination-based analogi-

cal reasoning.Finally, in section 6, we give some conclusions.

2. Logical analysis of analogy" a review

Analogical reasoning may be defined as the pro-cess of inferring that a conclusion property Q holdsof a particular situation or object T ( the target) from the fact that T shared a property or set ofproperties P with another situation/object S (thesource or base) which has property Q. The set ofcommon properties P is the similarity between Sand T, and the conclusion property Q is projectedfrom S onto T. Generally, we can summarize theprocess schematically as follows:

P(S) A Q(S)P(T)

Q(T)

Evidently, the above schema of argument is notdeductive. We cannot justify the plausibility ofconclusion by the argument itself. The justifica-tion aspect of the logical problem of analogy leadsDavies and Russell[2] to develop a theory of deter-mination. They define analogical reasoning withina deductive logical framework. If adding determi-nation rules to premises, the analogical conclusioncan be followed soundly. To prevent the redun-dancy problem, determination rules rather than im-plicational rules are assumed. The problem of thistheory is that in many circumstances, it is not easyto get the determination rules if any. Moreover,the determination rules have exceptions. Loui[8]demonstrates the possibility of accounting for ana-logical reasoning in Kyburg’s theory of statisticalinference.

In this paper, we regard analogical reasoning asnonmonotonic reasoning. In this respect, the con-clusions of analogical reasoning have the defeasibil-ity. Also, we can explain the phenomena of mul-tiple conflict analogical conclusions. We can in-

161

From: AAAI Technical Report FS-94-02. Compilation copyright © 1994, AAAI (www.aaai.org). All rights reserved.

Page 2: Nonmonotonic Logic for Analogical Reasoning · 2006-01-11 · Nonmonotonic Logic for Analogical Reasoning ... reasoning(e.g, deductive reasoning and inductive reasoning), ... [2]

fer Q(T)from F = {P(S), Q(S), P(T)}, but not obtain Q(T) from F = {P(S), Q(S), P(T)} {-~Q(T)}. The defeasibility of analogical conclu-sions is due to the incomplete information. If weadd some default rules, we can make incompleteinformation complete.

3. Nonmonotonic analogical logic system-NAR

3.1 An universal many-sorted first orderlanguage L

In this subsection, we define a many-sorted firstorder language. The definition of the language isbased on that in Thiele[13]. The purpose to usesuch a language is to facilitate the definitions ofsome concepts used in our system.

We are given the pairwise different symbolsc, v,g, q, and the set S~ with card S~ = R0. El-ements of S~ are called sorts and denoted by s(eventually with arbitrary indices ), the set S~ said to be the universal set of sorts.

The set of all natural numbers (including zero is denoted by N.

By the following definition, we form individualconstants, variables, functional symbols and predi-cate symbols.

Definition 3.1.11. for every s E Su and j E N,c] is an individual constant ( of the sort s 2. for every s E Su and j E N,

vf is an individual variable ( of the sort s 3. for every s E Su and j E N,g~ is a functional symbol ( of the type w 4. for every s E S~ and j E N,qjW is a predicate symbol ( of the type w We define terms and formulae (denoted by A

in a similar way to other many-sorted languages.

Definition 3.1.2(Exchange)Given a wffp, a, b be two terms or predicate sym-

bols. Exchange(p, a, b) is defined as the formulasobtained by exchanging a and b in all their occur-rences in P. Similarly, we define the exchange be-tween two terms or predicate symbols in a set ofwits.

Evidently, we have the following conclusions.

Exchange(Exchange(P, a, b), a, b)

Exchange(P, a, a) =

Definition 3.1.3 (Clusterabillty)

Given a consistent set of closed wffs F, a, b betwo terms or predicate symbols. We say a,b isclusterable in F denoted by Cr(a, b), if r(a, Exchange(F, a, b) isconsistent.

Theorem 3.1.4Given a consistent set of wiTs F, Cr is a reflexive

and symmetrical relation.

Definition 3.1.5 (Similarity)

Given a consistent set of closed wits F, as, bs betwo terms or tuples of terms (with the sort s ), say as, bs is similar in F( denoted Simr(as, bS)), if itexists predicate symbols p8 such that both PS(aS)and PS(bS) are in F.

Theorem 3.1.6 For two sets of first order wffsF1, F2, F1 C_ F2, if Simrl(a, b) then Simr2(a, b),where a, b are two terms or predicate symbols in L.

Definition 3.1.7 (Relevance)Given a consistent set of closed wffs F, a, b be two

terms or predicate symbols. We say a, b is analogi-cally relevant in F denoted by Rr(a, b), if Cr(a, b)and Simr(as, bS)).

Example 1 Given a consistent set of wffs S ={P(a),-~P(b),Q(a),P(c)}, we have Rs(a,c) butnot Rs(a, b).

3.2 A nonmonotonic analogical logic sys-tem NAP

Definition 3.2.1 (Analogical extensions)Let W be a set of closed wits. For any set of

closed wits S C L let F(S) be the smallest set sat-isfying the following three properties:

D1. W C_ r(s)D2. TH(F(S)) = r(s)03. If Cr(s)(a, b), Simr(s)(a, b) is true andCr(s)o{~p}(a,b) is false, then P E r(s) where

a, b are two terms or predicate symbols.A set of wits E C L is an extension for W iff

r(E) = Theorem 3.2.2Let S be a set of consistent wits. If

Cr(s)(a, b), Simr(s)(a, not Cr(s)u{.p} (a, b) tF(S) O {p} is consistent.

Proof. Suppose F(S) O {p} is inconsistent, have F(S) ~- -~p. Thus F(S) O -~p = F(S). This in contradiction with the premises.

3.3 Some properties of NAR

Theorem 3.3.1(Exchange is closed within theextensions)

162

Page 3: Nonmonotonic Logic for Analogical Reasoning · 2006-01-11 · Nonmonotonic Logic for Analogical Reasoning ... reasoning(e.g, deductive reasoning and inductive reasoning), ... [2]

Given a consistent set of first-order wffs S,a,b be two terms or predicate symbols. If

P E £(S), Simr(s)(a, andCr(s)(a, b) t rue, thenExchange(P, a, b) E F(S).

Proof. It is enough to show that

CF(s)u-.Exchange(p,a,b)(a, is fal se. Thelatt er istrue because F(S) U {--p} is inconsistent.

To well understand the behaviours of our logic,let us consider some examples.

Example 2. Given a consistent set of wffsS = {P(a),-,P(b),Q(a)}, we have the followingreasoning process:

(1) Cs(P, Q)(2) Sims (P, Q)(3) -’CSuQ(b)(P,Q)(4)Example 3. Given a consistent set of wffs S =

{P(a), P(b), Q(a)}, we have the following reasoningprocess:

(1) Cs(P, Q)(2) Sims(P, Q)(3) -~Csu-~Q(b)(P,Q)(4) Q(b)For some sets of wits, there may be multiple ex-

tensions.Example 4. Given a consistent set of wffs

W -- {P(a), R(a), R(c),-~P(b),-~S(b),-~S(c)}, wehave two extensions

Wt = Th(WU{’,S(a), P(c)}) and W2 Th(WU{--P(c), R(b))).

4. Graphical representation and computa-tion

To facilitate the calculation of analogical conclu-sions, a good formalism representing the informa-tion about analogy is important. Here, we presenta graphical knowledge representation method. Ourmethod is inspired by the work of Pearl[23]. Sim-ilar to belief network, it is difficult to detect theconditional independence relationship by numeri-cal computation. Here it is difficult to detect theclusterability relationship. Graphical representa-tion may reveal some important properties of clus-terability. We must develop concepts similar to thed-separation of belief network, etc. Further studyof this problem is beyond this paper.

We define types of predicate nodes and termnodes, thus represent a theory as a graph. Ac-cording to the types of different predicate symbols,

we can differentiate between the types of predicatenodes. The arcs link the predicate nodes and termnodes. We give two examples to explain the basicideas of our method.

Example 2.(cont.) Graphical representation:

@

Clustering:

P, Q is clusterable, a, b is not clusterable. Afterthe clustering, we have

CP’Q

From the new graph, we can obtain the conclu-sion -~Q(b).

Example 3. (cont.) Graphical representation:

Clustering:

P, Q is clusterable, a, b is also clusterable. Afterthe clustering, we have

163

Page 4: Nonmonotonic Logic for Analogical Reasoning · 2006-01-11 · Nonmonotonic Logic for Analogical Reasoning ... reasoning(e.g, deductive reasoning and inductive reasoning), ... [2]

Q,p )

£2.

.

,

From the new graph, we can obtain the conclu-sion Q(b).

5. Integration of similarity and determi- 5.

nation

In our logic NAR, the determination informa-tion is ignored. One reason is that there exist some 6.

situations where this information is not available.Another reason is for the simplicity. In this sec-tion, we discuss the integration of similarity anddetermination. 7.

As argued by many authors[2,3,11], the determi-nation is more important than the similarity infor-mation for establishing the validity of the analogi- 8.cal conclusion. We extend our logic to deal withthis problem. The basic idea is that one agentA makes similarity-based analogical inference, theother agent B makes determination based analogi-cal inference. If A get a conclusion p without con- 9.fliction with the conclusion made by B, we acceptP, otherwise accept the conclusion made by B.

6. Conclusions 10.

In this paper, we have discussed some logical as-pects of analogical reasoning. The major contribu- 11.tions of this paper are a nonmonotonic logic whichcan make some kind of analogical reasoning and 12.a formalism of knowledge representation for ana-logical reasoning. Our theory provides a suitableextendible framework to account for more logicalaspects of analogical reasoning. More specifically 13.speaking, both the nonmonotonic reasoning andthe graphical representation are easily extended.

Acknowledgement. This work is supported byChinese KR&R Project, National High-Tech Re-search and Development Program. Thanks to ShuoBai for his fruitful discussions. I would also like tothank John Case, Todd Davies, K. P. Jankte, YvesKodratoff and Stuart Russell for thief help.

References

1. Bai,S.[1994] A calculus for logical clustering,to appear.

Davies,T.R. and Russell, S.J. [1987] A logicalapproach to reasoning by analogy. IJCAI-87,

Davies,T.R.[1988] Determination, uniformityand relevance: normative criteria for gen-eralization and reasoning by analogy, inD.H.Helman(ed.) Analogical Reasoning, 227-250. Kluwer Academic Publishers.

Goebel.R.[1989] Asketch of analogy as reasoning with equalityhypotheses, in K.P.Jantke(ed.): it Analogicaland Inductive Inference, LNCS 397,243-253.

Greiner,R.[1988] Learning by understandinganalogies, in A. Prieditis(ed.) Analogica, 1-36.Pitman, London.

Hall,R.P.[1989] Computational approaches toanalogical reasoning: a comparative analysis.Artificial Intelligence 39, 39-120.

Kodratoff, Y.[1990] Combining similarity andcausality in creative analogy, ECAI-90, 398-403.

Loui,R.P.[1989] Analogical reasoning, defeasi-ble reasoning and the reference class. Proc.of Principles of Knowledge Representation andReasoning, 256-265.

Pearl,J.[1988] Probabilistic Reasoning in Intel-ligent Systems. Morgan Kaufmann Publishers,Inc. San Mateo, Califorlia.

Reiter,R.[1980] A logic for default reasoning,Artificial intelligence 13, 81-132.

Russell,S.J.[1986] Preliminary steps towardthe automation of induction, AAAI, 477-484.

Russell,S.J.[1988] Analogy by similarity, inD.H.Helman(ed.) Analogical Reasoning, 251-269. Kluwer Academic Publishers.

Thiele,H.[1987] A model theoretic oriented ap-proach to analogy, in K.P.Jantke(ed.): it Ana-logical and Inductive Inference, LNCS 265,196-208. Springer Verlag.

164