Relating Default Logic and Circumscription

David W. Etherington1

Artificial Intelligence Principles Research Department AT&T Bell Laboratories 600 Mountain Avenue Murray Hill, NJ 07974 ether^allegra(a btl.csnet

Abstract approaches are different on both the syntactic and Default logic and the various forms of cir• semantic levels. cumscription were developed to deal with similar Default logic and circumscription were developed problems. In this paper, we consider what is known at approximately the same time as attempts to formal• about the relationships between the two approaches ize nonmonotonic inference, which was recognized as and present several new results extending this an important facet of intelligent behaviour. In the knowledge. We show that there are interesting cases intervening years, there has been considerable activity in which the two formalisms do not correspond, as in this area. Both formalisms have been extensively well as cases where default logic subsumes cir• studied, and several new paradigms have been pro• cumscription. We also consider positive and negative posed. These two, however, have had remarkable results on translating between defaults and cir• staying power, and applications and refinements con• cumscription, and develop a context in which they can tinue to surface. Since both attempt to capture similar be evaluated. phenomena, the natural question is whether either subsumes the other. Is there a direct mapping by which default 1. Introduction theories subsume circumscription, or vice versa"! Circumscription [McCarthy 1980, 1986; Lifschitz There is, as yet, no definitive answer to this question. 1984] and default logic [Reiter 1980] are both formal• In what follows, we draw together results that begin to isms for reasoning in the absence of complete informa• provide answers. Because there are many criteria that tion. Often, the available knowledge about the state can be used to determine "subsumption", a plurality of of the world (problem domain, task) is incomplete, in "answers" is perhaps the best that can be provided. the sense that required details are unknown. Certain As we proceed we outline the assumptions that under• plausible assumptions can sometimes be made - to fill lie the partial answers that can be provided to date. in some of these missing details - that may further the We generally refer to Lifschitz' [1984] generaliza• goals of the reasoning system. These assumptions may tion of circumscription, which we simply call "cir• or may not prove correct when further information cumscription". This version allows the denotations of becomes available, for example through further obser• terms, as well as of predicates, to be varied. Where vation. The availability of additional information may thus lead to the retraction of certain conclusions. This appropriate in what follows, we draw attention to the property is called nonmonotonicity. effect of the presence or absence of variable terms. In what follows, we present very brief sketches of Both default logic and circumscription can be semantic underpinnings for the two formalisms. These used to enforce "policies" (defaults, preferences, ...) sketches are then contrasted and compared to help that (ideally) lead to highly plausible conjectures in determine some of the relationships between the tech• the absence of definite information. The actual world niques. is assumed to be among some restricted subset of those worlds that meet the criteria of what is known. This selection of "preferred" worlds may be externally jus• 2. Circumscription tified by convention, probabilities, etc, but these exter• nal justifications do not play a direct role in the One form of preference is the "Closed-World theory. Beyond these intuitive similarities, the two Assumption", or CWA. This is the assumption that all the facts about the world have been stated [Reiter 1978]. Those facts that do not follow from the given 1 Parts of this work were done at the University of facts can thus be assumed to be false. The assumption British Columbia, and supported in part by an I. W. of complete knowledge about the world can be prob• Killam Predoctoral Scholarship and NSERC grant A7642. lematic if applied indiscriminately, but it is sometimes

Etherington 489 reasonable to assume complete information, especially consistent to believe certain "gating facts" or "justifi- about particular predicates. If such an assumption is cations^ (£). A default theory consists of a set of warranted, the world can be characterized by its being defaults and a set of first-order axioms. Ideally, the among those models having the smallest possible defaults induce (possibly several) stable, maximal, spe• extension for the predicates in question, given the con• cializations - called extensions- of the theory given by straints of what is known. Syntactically, this amounts the axioms. to assuming the negation of every atomic fact (over In spite of the apparent simplicity and naturalness the predicates for which complete information is of defaults, the semantics of default logic is more assumed) not entailed by the original theory. complex than that of circumscription. There are two In general, this idea can be extended to predicates reasons for this. First, defaults are by nature global: about which only incomplete information is available, each refers to everything that is believed or not so long as it is reasonable to assume that they cover as believed. This means that the semantics must be based few individuals as necessary. In such cases, the on sets of models (which we call world-descriptions), models satisfying this "generalized" CWA [Minker rather than individual models, to allow the concepts of 1982] need not agree on the extensions of the predi• belief and unbelief to be represented. Second, since cates concerned, only on the minimality (with respect the justification (p) and consequent (y) of a default to the original theory) of those extensions. need not coincide, defaults can predicate conjectures on the continuing lack of belief in certain propositions The idea of minimization can be generalized from without asserting or otherwise enforcing this lack of subset minimality of predicates' extensions to minimal• belief. Thus, although local comparisons between ity according some arbitrary pre-order on models. world-descriptions can be used to partially represent Circumscription is a syntactic device that characterizes the preference relation, information not inherent in what is true in a theory's minimal models.2 The the world-descriptions must be considered in the final theory is augmented by a second-order axiom that can determination of preference. The semantics must be obtained deterministically, given a particular somehow make this information available. choice of an ordering on models. This axiom represents the assumption that the world is among the Given these caveats, we can define a semantics theory's minimal models. The conjectures of interest for default logic analogous to the minimal model then follow from this extended theory by normal semantics of circumscription. The defaults define a deduction. preference ordering that, in turn, determines a set of A typical application would be to axiomatize the minimal world-descriptions. These must be further "normal" state of the world and then circumscriptively restricted to exclude those that violate the lack-of- minimize the set of abnormalities. conjecturing that belief constraints that implicitly gave them rise. Each things are as normal as possible [McCarthy 1986]. This of the remaining world-descriptions consists of all the models of some extension of the default theory, and approach allows circumscription to be used for some 3 forms of default reasoning, although the full details each extension can be so characterized. remain to be worked out. 4. Default Logic as Subsuming 3. Default Logic Circumscription Default logic sanctions conjectures based on infer• Since circumscription's proof-theory avoids the ence rules, called defaults. These have the form consistency checking (in general not semidecidable) associated with default logic proofs, it would be nice if P , and can be read "If a is believed, and p is not y circumscription subsumed default logic. This would disbelieved, then y can be assumed". Because of P's allow us to deal only with circumscriptive theories. role, defaults allow conjectures based on both what is Unfortunately, this is not the case. known and what is not known. Intuitively, the defaults provide criteria for preferring world- Theorem 1 descriptions over one another. Roughly speaking, each default. " , sanctions a conjecture (y) that Default logic can make conjectures that cannot specializes a world-description provided certain be obtained by generalized circumscription "prerequisites" (a) are believed, so long as it remains without variable terms. |

2 For simplicity's sake, we ignore the issue of incom• pleteness. See [Perlis & Minker 1986] or [Etherington 3 This semantics is developed in detail elsewhere [Eth• 1987a] for a discussion. erington 1987 a,b].

490 KNOWLEDGE REPRESENTATION Etherlngton 491 The reasons that only prerequisite-free defaults (provability) vs local (consistency) comparisons in the have modular translations become clear if we contrast model-theory (proof-theory), and statements about the semantics we have sketched for default logic and "unnamed" individuals. In all but the last of these the minimal-model semantics of circumscription. The categories, default logic is stronger. This suggests that former is based on orderings over sets of models, the the search for a direct embedding of circumscription latter on orderings over individual models. Sets of in default logic might be more successful than the con• models are necessary to capture the proof-theoretic verse attempt. The answer to this is, "Yes, and no.". notion of provability. Since the prerequisites of a One facet of generalized circumscription is com• default must be provable for the default to have any pletely absent from default logic. Circumscription effect, default logics semantics must be based on sets may hold some predicates constant, while others are of models. Circumscription's submodel relation, how• allowed to vary. In default logic, there is no way to ever, only considers pairs of models. Prerequisite-free restrict the repercussions of the defaults to some par• defaults fit nicely into circumscription precisely ticular set of predicates (and/or individuals). Because because they are prerequisite-free. Such defaults predicates can be richly interconnected, the effects of impose no (global) provability requirements, only con• a default inference may be arbitrarily far-reaching. sistency requirements. Consistency can be determined This seems to preclude a completely-general, direct by the existence of a single model, and so can be embedding. locally determined. This model-theoretic perspective In restricted cases, better results obtain. We can yields a much simpler, more direct, proof that cir• rule out unnamed individuals by requiring the domain cumscription can provide a modular translation of to be closed, and decide the equality theory to prevent default theories - even in the propositional case - only defaults from affecting equality. If we then consider for prerequisite free defaults. circumscription, with all predicates required to be It is arguable that the requirement that a theory variable, we get the correspondence outlined in and its "translation" have identical sets of theorems is Theorem 2. too strong. For example, we have noted that default logic is a "brave" reasoner while circumscription is "cautious". It seems reasonable to expect that a cir• cumscriptive translation of default theories would reflect this cautious nature, perhaps returning those facts true in all extensions. Conversely, circumscrip• tive conjectures apply to all individuals, whereas those resulting from open defaults apply only to individuals having names in the language. It might, therefore, be reasonable to expect circumscriptive versions of default theories with open defaults to prove certain stronger conjectures (at least for theories without domain closure axioms). Perhaps such side-effects of translation are sufficiently benign that they can be accepted. These considerations suggest that Imielinski's results might be taken as a "worst case" scenario, leav• ing open the possibility of acceptable translation schemes for defaults with prerequisites, given a weaker notion of "acceptable". The question remains open, and we do not further consider this possibility here.

6. Translations from Circumscription to Default Logic The dual of the question we have been examining is whether default logic can be used to circumscribe (in any but the trivial sense mentioned at the begin• ning of this paper). The previous section outlined some of the different capabilities of the two formal• isms: brave vs cautious, effects on equality, global

492 KNOWLEDGE REPRESENTATION Corollary 4

For this restricted class of theories, the default theory. has the advantage that different extensions (assuming there are more than one) allow the reasoner to explore the various alternative minimizations. The brave/cautious distinction can be seen to be exactly the difference between considering only one of or all of the default theory's extensions. In a sense, captures the brave circumscription of P in T with every predi• cate variable. Notice that Theorem 2 requires that T have a domain-closure axiom and decide the equality of each pair of terms mentioned therein. If the theory does not decide the equality of these terms, then the default theory becomes stronger than the circumscrip• tive theory, in the sense that Corollary 3 continues to hold but Theorem 2 and Corollary 4 do not. Because of the limitations of open defaults concerning unnamed individuals, none of the results generalize to theories with no domain-closure axiom, and we have Theorem 5.

Theorem 5

Even more pessimistic is Theorem 6, which states that fixed predicates preclude the straightforward transla• tion of circumscription to default logic we have con• sidered, even for closed-domain, unique-name theories.

Theorem 6

We experimented with an extended version of default logic that allowed for the specification of "fixed" predicates. Although we were able to show that the results in [Reiter 1980, chapters 2 and 3] hold

Etherlngton 493 Theorem 8 that should be carefully considered in determining adequacy conditions for translations. If there are no variable predicates, then ECWA(T) augments T with every instance of the circumscription schema. References Etherington, D.W. [1987a], Reasoning from Incomplete We have seen that - if such a thing exists - any Information, Pitman Research Notes in Artificial general translation from circumscription to default Intelligence, Pitman Publishing Limited, London. logic, short of adding defaults for each instance of the Etherington, D.W. [1987b], "A semantics for default circumscription schema, requires more power than the logic", Proc. Tenth International Joint Conference on closed-world default provides. The existence of an Artificial Intelligence, Milan, Italy. appropriate translation remains open. Gelfond, M., and Przymusinska, H. [1985], Negation as Failure: Careful Closure Procedure. University of Texas at El Paso, unpublished draft. 7. Autoepistemic Logic Gelfond, M., Przymusinska. H., and Przymusinski. T. In light of recent developments, we should men• [1986], "The extended closed-world assumption and tion Moore's [1985] autoepistemic logic in this context. its relation to parallel circumscription", Proc. ACM Autoepistemic logic is a modal language allowing non• SIGACTSIGMOD Symp. on Principles of Database monotonic reasoning, based on agents' introspection Systems. 133-139. on their knowledge. Default reasoning is achieved by Grosof, B. [1984]. Default Reasoning as Circumscrip• explicitly predicating conjectures on the absence of tion, Technical Report, Stanford University. Stan• knowledge. Any fact not a consequence of what is ford. CA. known is explicitly marked as unknown. Imielinski. T. [1985]. "Results on translating defaults Konolige [1987] has shown that autoepistemic to circumscription", Proc. Ninth International Joint logic and default logic - though superficially dissimi• Conference on Artificial Intelligence, Los Angeles. lar - are formally equivalent. From this it is possible CA, Aug. 18-23, 114-120. to infer that the statements we have made - and the Konolige, K. [1987], "On the relation between default theorems we have proven - on the relationship theories and autoepistemic logic", Proc. Tenth between default logic and circumscription apply International Joint Conference on Artificial Intelli• equally to that between autoepistemic logic and gence, Milan, Italy. circumscription. Lifschitz, V. [1984], Some Results on Circumscription, Technical Report STAN-CS-84-1019, Stanford University, Stanford, CA. 8. Conclusions McCarthy, J. [1980], "Circumscription - a form of We have considered the relationship between non-monotonic reasoning", Artificial Intelligence default logic and circumscription. We showed that the 13, North-Holland, 27-39. closed-world default sometimes coincides with cir• McCarthy. J. [1986], "Applications of circumscription cumscription; that, in a particularly useless way, to formalizing commonsense knowledge". Artificial default logic subsumes circumscription; and that Intelligence 28, 89-116. default logic is capable of affecting the equality theory McDermott, D. [1982], "Non-monotonic logic II", J. while generalized circumscription (without variable ACM 29(1). terms) is not. Minker, J. [1982], "On indefinite databases and the closed-world assumption", Proc. Sixth Conf. on For particular classes of theories, we have pro• Automated Deduction, New York, 7-9 June, vided a translation from circumscription to default Springer-Verlag, NY. logic. This translation applies only to theories with Moore, R.C. [1987], "Semantical considerations on domain-closure axioms, and then only when the cir• nonmonotonic logic". Artificial Intelligence 25, cumscriptive theory specifies all predicates as variable. We showed that the introduction of fixed predicates North-Holland, 75-94. and applications to open domains each provide cir• Pedis, D. & Minker, J. [1986], "Completeness results cumscription with capabilities not available using sim• for circumscription". Artificial Intelligence 28, 29- ple closed-world default theories. 42. Reiter, R. [1978], "On closed-world data bases", in Finally, we used semantic comparisons to Gallaire, H. and Minker. J. (eds), Logic and Data highlight some of the essential differences between the two approaches. This allowed us to suggest that some Bases, Plenum Press, 55-76. of the work on translations between the two formal• Reiter, R. [1980], "A logic for default reasoning", isms may not have noticed the essential characteristics Artificial Intelligence 13. 81-132.

494 KNOWLEDGE REPRESENTATION