the logic of nonmonotonicity

ARTIFICIAL INTELLIGENCE 365

RESEARCH NOTE

The Logic of Nonmonotonicity

John Bell Computer Sctence Department, Umversity of Essex, Wivenhoe Park, Colchester, UK, C04 3SQ

ABSTRACT

Shoham [14] presents a general semanttc framework m whtch many nonmonotomc logws (the so-called "preference logtcs") can be represented We present another means of gtvmg general semanncs for preference logws using models o f the condmonal logic C This has the advantage that we also provide an axtomanzatton for preference logics Insofar as all nonmonotomc logtcs are preference logics, C is the logw of nonmonotontclty

1. Introduction

Traditional (monotonic) logics cannot and were not designed to represent common sense reasoning. This is because such reasoning involves drawing conclusions that are defeas~ble from incomplete information. Common sense reasoning is central to AI and many researchers have suggested ways of adding nonmonotomc extensions to traditional logics in order to represent ~t. Typical examples are Reiter's default logic [13], McCarthy's predicate circumscription [10], and Moore's autoepistemic logic [11]. These and similar formalisms are superficially very different but recent work has suggested underlying similarities among them. For example, Konolige [8] shows that default logic and a slightly restricted form of autoepistemlc logic are formally equivalent, and Etherington [6] explores the connections between default logic and circumscription. Shoham [14] suggests a general semantic framework in which many nonmonotonlc logics can be defined. Shoham calls such logics preference logics as each such logic is defined by means of a preference criterion on its interpretations. There is a striking analogy between this approach and conditional logic. Shoham notes that this was pointed out to him by van Benthem. The connection is explored in this note. We present the conditional logic C and

Artlfictal lntelhgence 41 (1989/90) 365-374 0004-3702/90/$3 50 © 1990, Elsevier Science Pubhshers B V (North-Holland)

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use C-models to give general semantics for preference logics ~ This has the advantage of providing an axiomatizat ion for preference logics. Insofar as all n o n m o n o t o n i c logics are preference logics, C is the logic of nonmonoton ic i ty .

2. The Semantic Construction of Nonmonotonic Logics

We begin with Shoham ' s definition of preference logics. Let L be a s tandard logic (proposi t ional calculus, predicate calculus, or a modal logic) and let < be a strict partial o rder on the class I of all in terpretat ions of L (where " in terpreta t ion" is used to refer to a t ruth value assignment in the case of proposi t ional calculus, an interpreta t ion m the case o f predicate calculus, and a Kripke- model in the case of a modal logic2). Intuit ively, z < t ' means that the interpreta t ion t is preferred over the interpreta t ion i ' 3 Then L and < define a new logic L< called a preference logic. The syntax of L< Is identical to that of L As for the semantics, the not ions of satisfaction, validity, and en tadmen t are defined using those of L.

Definition 2.1. Let L be a s tandard logic, and let A and B be two sentences in L. The fact that an in terpreta t ion t satisfies A is deno ted t ~ A In this case we say that i is a model of A or an A-mode l . A is vahd lff A is satisfied by all interpretat ions. A entails B (written A ~ B) if all models of A are also models of B.

Definition 2.2. A n Interpreta t ion t preferenUally satisfies A m the f rame for L . (written i ~ < A) iff t ~ A and there is no o ther interpretat ion t ' such that t ' < z and i ' ~ A. In this case we say that t is a preferred model of A.

Clearly, if t ~ < A then i ~ A.

Definition 2.3. A preferentially entatls B (written A ~ < B) iff any interpretation t such that i ~ < A is such that t ~ B; that is, if the models of B (preferred or otherwise) are a superset of the prefer red models of A.

Mono tomc i ty can then be defined as follows.

Definition 2.4. L< is monotonic iff for all A , B, and C in L, if A ~ < C then A & B ~ < C also.

ivan Benthem [15] investigates the connection between C and clrcumscnpnon The work presented here represents an independent, and more general, development of the analogy

2 This ~s a s~mphficat~on of Shoham's treatment As we wall only be interested m sentences which are vahd in Knpke-models, we need not consider individual worlds m such models

Shoham writes this the other way around The reason for writing it as we have will become clear in the sequel

THE LOGIC OF NONMONOTONICITY 367

So, as the special case where < is empty, L is monotonic. The Deduction Theorem holds for any preferential logic L<; that is, for any

sentences A, B, and C in L<,

if A&B ~< C then A ~ < B D C .

However , the converse holds only if L< is monotonic. (If L ts nonmonotonic we have A ~ C and A&B ~t C for some A, B, and C. But then we have A ~ B D C a n d A & B ~ t C . )

Shoham defines the preference relation (the minimality criterion) for three nonmonotonic logics (see Section 6).

3. A Restriction

Definition 2.2 only makes sense if < is well-founded; that is, if there is no infinite sequence i, i ', i " , . . , such that i > t ' > i" > • • •. For example, if < has no minimal elements then no valid sentence A of L is preferentially satisfiable; no interpretation i of L is such that i ~ < A as there is always an interpretation i ' such that i ' < i and i ' ~ A. Consequently we will subsequently require that < be well-founded if L< is to be a preference logic. As we shall see, this places no real restriction on the preference criteria that we are likely to use.

4. The Conditional Logic C

The language of C, L c, has as its vocabulary a denumerable set P of atomic sentences; the truth-functional connectives --1, &, v , D , and ~-; and the intensional conditional operators ~ and 9 . Lc is the smallest superset of the atomic sentences that is closed under compounding by means of the connectives and the conditional opera tor with the aid of suitable punctuation. 4

4.1. Semantics

Definition 4.1. A model for C (a C-model) is a triple M = ( I , ~<, V) w h e r e / i s a nonempty set (of possible worlds), ~< is a well-founded partial ordering on I, and V is a valuation function on I; V: P x I--->{true, false}. Let (I,,<~,) be the subframe of ( I , ~ < ) such that I, = {j: i<~]) and ~<, is the restriction of ~< to / s .

Truth of a sentence A at a world i in a model M (written M,t ~ A) is then defined as follows.

4 So as to avoid excessive use of parentheses we define the following order of precedence on the connectwes and operator: --7, & and v , D and ~-, ~ and

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Definit ion 4.2. Let M = (I , 4 , V ~ be a C-model and let A and B be sentences of L c. Then

M , i ~ p iff V ( p , t ) = t r u e , M , l ~ - a A lff M , t ~ A , M , I ~ A & B iff M , t ~ A and M , i ~ B , M , t ~ A=:)B fff f (A , OC_l[BII M

Where ]lm]l M = {t: M,t ~ A} and f (m, i) is the set of <,-closest A-worlds to i in (1,, 4 ) ; that 1S,

f (A, t) = ( j : i < , l & l e IIAII M

& ~(3k ) (k <, j & k E IlZll M & k ~ j ) > .

The remaining truth-functional connectives are introduced by the standard definitions. The ~ o p e r a t o r is defined as follows:

df A ~ B = --a ( A ==~ ~ B ) .

The semantic clause for ~ would be:

M , t ~ A ~ B lff f(A,i)nlIBllM~

4.2. An ax iomat izat ion

The conditional logic C is the smallest subset of L~ that is closed under the following axiom schemas and inference rules.

Axiom schemas

(PC) Truth-functional tautologtes.

(1D) A ~ A .

(CS) A&B D (A ~ B)

(MP) ( A ~ B) D (A D B).

(CC) (A ~ B)&(A ==)> C) ~ (A ~ B&C) . (AD) (A ==> C)&(B ~ C) D (A v B ~ C).

(ASC) ( A ~ B ) & ( A ~ C ) D ( A & B ~ C ) .

Inference rules

(Mp) From A and A 3 B infer B .

(RCM) From B D C m f e r ( A ~ B ) D ( A ~ C ) .


Derived rules

(RI) From A 3 B infer A f f B .

(RCEA) From A -= B infer (A ~ C) = (B ~ C) .

(RCK) F r o m B l & - - - & B n D C i n f e r ( A ~ B 1 ) & - - - & ( A ~ B n ) D ( A ~ C ) for n~>0.

Theorems

(CF~) (A (CI) (A

( t l ) ( A ~

(t2) (A

(t3) (A

(t4) (A

The naming counterfactual

(AS) ( A ~

B)&(B~ A) D ( ( A ~ C) D ( B ~ C)). B D C) D ( ( A ~ B) D ( A ~ C)). e D C) D ( ( A ~ B) D ( A ~ C)). e ) D ( ( A ~ C) =- ( A ~ B ~ C)).

B) ~ ((A ~ C) -~ (A&B ~ C)) . B)D ( ( B ~ C) =- (A v B ~ C)).

conventions are those of [4] and [16]. C is equivalent to Lewis' logic VC [9] without the axiom:

B )&(A ~ C) D (A&B ~ C)

Theorem. C is complete and decidable.

Proofs of this is are given in [1, 3, 16].

5. C-models for Preference Logics

Let L be a standard logic, and let ~< be a well-founded partial order on the class of all interpretations of L. Intuitively, i ~ t ' means that interpretation i is preferred over or equal to interpretation i'. Then L and ~< define a new logic L<_ called a preference logic. The syntax of L< extends that of L as follows: if A and B are sentences of L then A f f B is a sentence of L<. The semantics for L~ are given by a unique C-model.

Definition 5.1. M = (I , 4 , V) is the C-model for L< iff I is the set of all interpretations of L, 5 ~< is the preference relation on I, and V : L × I---~ {true, false) is such that V(A, t) = true iff t ~ A.

There is only one such model for each L~ as V is unique, and changing L or ~< changes L<.

5 Strictly speaking 1 is a class, but for present purposes we can ignore this

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Definition 5.2. If A and B are sentences of L and M = ( I , ~<, V) is the C-model for L~< then

M,I ~ A lff V(A, t) = true, M , t ~ A ~ B iff f ( A , i ) c _ l l B I I M

Where f is the function induced by ~< and V.

We can now define preferential satisfiability and preferential entai lment for sentences of L in the model M for L<. (i is a mimmal element of M if i is in M and there is no other i ' in M such that t ' ~< t.)

Definition 5.3. An interpretation t in M preferentially satisfies a sentence A (written M,i ~<_ A) lff i ~ f ( A , i') for some minimal e lement t ' in M. In this case we say that t is a preferred model of A.

So A is preferentially satisfiable in M iff M,t ~ A ~ A for some minimal e lement z. If M,i ~ A ~ B for some minimal element l in M we say that " A preferentially satisfies B (at i ) ," or even that " B is believable (at t) given A . "

Definition 5.4. A sentence A preferenttally entails a sentence B in M (written A ~ < B) iff M,i ~ A ~ B for all minimal elements t in M, or equivalently, iff f (A, i)C_ IIBll M for all minimal elements t m M.

6. Preference Criteria for C-models

Adapting the preference criteria given by Shoham is easy. In the case of predicate circumscription the preferred interpretations are

those where the extension of a relation R, the circumscribed predicate, is as small as possible. So t ~ < t ' iff

(1) I and l ' have the same domain and agree on the interpretation of variables, function symbols, and relations other than R,

(2) t ' ~ R(Y) if t ~ R(Y) for all tuples ~.

Bossu and Siegel [2] present a logic which amounts to circumscribing all relations simultaneously. So ~ ~< i ' tff

(1) t and i ' share the same domain and agree on the interpretation of variables and function symbols,

(2) t ' ~ R(Y) if t ~ R(Y) for every relation R and tuple Y.

The logic of Halpern and Moses [7] is designed to characterize what an agent knows if it "knows only A . " KA is to be read " A is known." Interpretat ions


are S5-models. Preferred interpretations are those where fewest base sentences (sentences containing no modal operators) are known. So t ~ < i' iff

i' ~ KA if i ~ KA for all base sentences A.

Halpern and Moses argue that the agent can "know only A" only if there is a single preferred model of KA, and what it knows is all that is known in this model.

We can also give a preference criterion for the shghtly restricted form of Moore 's autoepistemic logic which konolige [8] proves equivalent to default logic. Autoepistemic logic aims to characterize the beliefs of an ideally rational agent reflecting upon its own beliefs. BA is to be read "A is believed." Given the premises A the agent's beliefs can be characterized in terms of the stable extensions of A that are strongly grounded in A (cf. [8, 11]). Each of these sets has what Moore [12] terms a possible-worlds model which consists of a pair (M, v ) in which M is an S5-model and v is an assignment of truth values which is consistent with that for one of the worlds m M. M represents the agent's beliefs and v represents what is actually the case. Each (M, v)-pair can be transformed into a KD45-model (nomenclature from [5]) as follows. If M = ( I, R, V) then let M ' = ( I ' , R', V ' ) ; where t 1s a new world, I ' = I U {t}, R' extends R by having iR'i' for some i' in I, and the assignment by V' of truth values to the atomic sentences at l is consistent with v (see Fig. 1). It is easily seen that R' is lrreflexive, serial, transitive, and euclidean. So M' is a KD45-model. Clearly also, M ~ BA iff M' ~ BA. So M and M' are equivalent.

We can now define the preference criterion on the KD45-models that correspond to strongly grounded stable extensions. Preferred interpretations are those where fewest base sentences are believed. So i ~< t' iff

i' ~ BA if i ~ BA for all base sentences A.

O

K , q

,s

<i %

Fig 1 The KD45-model, M', corresponding to (M, v)

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Proposition 6.1. The preferred models of A comctde with the stable extenstons of A that are strongly grounded m A.

Proofl Suppose that t is a preferred model of A and that its corresponding possible-worlds model is (M, v) . Then B ( i ) = B ( M ) ; that is, the set of sentences believed in t is the same as the set of sentences beheved in M As M is an S5-model there is a stable set T such that T = B(M) [12, Theorem on p 349]. So T = B(t). As t is a preferred model of A the number of base sentences in B(1) is minimal subject to A E B(i). So the number of base sentences in T is minimal subject to A E T. So T is minimal for A [8, Definition 2 3]. So T is a stable expansion of A that is strongly grounded in A [8, Proposition 2.10] Conversely, suppose that T is a stable expansion of A that is strongly grounded in A. Then T is minimal for A and T = B(M) for some S5-model M. Consider any possible-worlds model (M, v) and corresponding KD45-model g As T is minimal for A, t is a preferred model of A. []

So the preferred models of A coincide with what the Ideally rational agent ought to believe given that it believes only A

Fur thermore , the ordering has a least element; the model in which only the theorems of KD45 are believed. Suppose that t is this least element, then we can define two different notions of autoeplstemlc consequence corresponding to the two different notions of autoeplstemlc theoremhood suggested by Moore. If the autoeplstemlc theorems of A are the intersection of the stable extensions of A that are strongly grounded in A, then the autoeplstemic consequences of A are the set:

{B:M, t~ A ~ B }

And if the autoepistemlc theorems of A are the union of the stable extensions of A that are strongly grounded in A, then the autoeplstemlc consequences of A are the set:

{B:M, i~ A ~ B }

7. Relating the Two Approaches

A formula A of L is preferentially satisfied by an interpretation t in the frame for L< if it is preferentially satisfied in the C-model M for L~; that is,

t ~ < A lfM, t ~ < A ,

and consequently

A ~ < B I f A ~ < B .


Fig 2 Example of the mequahty of ~< and ~

The converse is not generally true as we might have the situation pictured in Fig. 2. Suppose this is a portion of the C-model for L~ and that I and t' are minimal elements, then 1' ~ ~ A as j ' E f(A, i'). However , viewed as a portion of the frame for L<, with i and i' as minimal elements, 1' ~ < A as ] < ] ' and j~A .

If the preference criterion is such that no interpretation i is related to two minimal elements then the two notions of preferential satisfaction (and consequently those of preferential entailment) coincide.

This is the case with the nonmonotonic logics discussed. In the case of predicate circumscription the ~<-related worlds form a collection of disjoint lattices; in the bottom element of each lattice the extension of R is empty, and m the top element of each lattice its extension ~s the whole domain. With Bossu and Siegel's logic we again get a collection of disjoint lattices; the bottom element of each lattice has the extension of every relation empty, and the top element of each lattice has a product of the domain as the extension of each relation. For the logic of minimal knowledge we get a lower semilattice; in the bottom element only the S5-tautolog~es are known, and in the maximal elements either A or --qA is known for every sentence A The case for autoepistemlc logic is similar.

Note that each of the above orderings is well-founded because the preference criterion concerned is defined usmg set inclusion on countable sets. All preference criteria are defined using a partial order that restricts C . So, as we are unlikely to define preference criteria on uncountable sets, the requirement that the ordering ~s well-founded imposes no real restriction

8. The Logic of Nonmonotonicity?

We have seen that each preference logic L< has a unique C-model, and that the conditional operator ~ encodes the ~< relation in the sense that

A ~ < B iff M , i ~ A ~ B for all minimal e l e m e n t s i .

So theorems and rules of C in which the conditional operator is not nested express properties of ~ ~<. As these properties apply to ~ ~< they apply to ~ <

374 J BELL

also. Fu r the rmore ~ < has no proper t ies which a ren ' t so expressed. (For a n o n - t h e o r e m (non-rule) we can always give a coun te rmode l with a least

e l emen t (by defini t ion of inval idi ty) , and we can always construct a preference logic which replicates this model (by supplying an appropr ia te preference cri ter ion).) Similarly appropr ia te axioms and theorems con ta in ing the

opera to r express proper t ies of preferent ia l satisfaction. Consequen t ly , C is the logic of preference logtcs; the different no t ions of n o n m o n o t o m c inference character ized by these logics all share the c o m m o n g round of proper t ies s tated

by the appropr ia te theorems and rules of C. It may be that all n o n m o n o t o n i c logics can be character ized as preference logtcs. To the extent that this is the case, C is the logic of nonmono ton i c i t y .

ACKNOWLEDGEMENT

I would hke to thank Johan van Benthem, Yoav Shoham, Ray Turner, Wayne Wobcke and everyone else who read and commented on earlier versions of this note for their helpful suggestmns This research was supported by the Umted Kingdom Science and Engineering Research Councd

REFERENCES

1 J Bell, PredJctwe condmonals, nonmonotonlclty and reasoning about the future, Ph D Thesis, University of Essex, Colchester (1988)

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3 J P Burgess, Quick completeness proofs for some logics of conditionals, Notre Dame J Formal Logtc 22 (I) (1981) 76-84

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(1987) 388-393 15 J F A K van Benthem, Semantic parallels m natural language and computation, m M

Garndo, ed , Logw Colloqutum, Granada 1987 (North-Holland, Amsterdam, 1988) 16 F J M M Veltman, Logics for condltmnals, Ph.D. Thesis, University of Amsterdam (1985)

Received June 1988; revised verston recetved January 1989

the logic of nonmonotonicity

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