1993-Abduction As Belief Revision: a Model of Preferred Explanations

From: AAAI-93 Proceedings. Copyright © 1993, AAAI (www.aaai.org). All rights reserved. A s Craig Boutilier and Verhica Becher Department of Computer Science University of British Columbia Vancouver, British Columbia CANADA, V6T 172 email: cebly,[email protected] Abstract “logic-based” frameworks also. Peirce (see Rescher (1978)) We propose a natural model of abduction based on the re- discusses the “plausibility” of explanations, as do Quine and vision of the epistemic state of an agent. We require that Ullian (1970). Consistency-based diagnosis (Reiter 1987; explanations be sufficient to induce belief in an observation de Kleer, Mackworth and Reiter 1990) uses abnormality in a manner that adequately accounts for factual and hypo- assumptions to capture the context dependence of explana- thetical observations. Our model will generate explanations tions; and preferred explanations are those that minimize that nonmonotonically predict an observation, thus general- abnormalities. Poole’s (1989) assumption-based framework izing most current accounts, which require some deductive captures some of these ideas by explicitly introducing a set relationship between explanation and observation. It also of default assumptions to account for the nonmonotonicity provides a natural preference ordering on explanations, de- of explanations. fined in terms of normality or plausibility. We reconstruct We propose a semantic framework for abduction that cap- the Theorist system in our framework, and show how it can tures the spirit of probabilistic proposals, but in a qualitative be extended to accommodate our predictive explanations and fashion, and in such a way that existing logic-based proposals semantic preferences on explanations. can be represented as well. Our account will take as central subjunctive conditionals of the form A + B, which can be interpreted as asserting that, if an agent were to believe I htroduction A it would also believe B. This is the cornerstone of our A number of different approaches to abduction have been notion of explanation: if believing A is sufficient to induce proposed in the AI literature that model the concept of ab- belief in B, then A explains B. This determines a strong, duction as some sort of deductive relation between an expla- predictive sense of explanation. Semantically, such condi- nation and the explanandum, the “observation” it purports tionals are interpreted relative to an ordering of plausibility to explain (e.g., Hemp&s (1966) deductive-nomological ex- or normality over worlds. Our conditional logic, described planations). Theories of this type are, unfortunately, bound in earlier work as a representation of belief revision and to the unrelenting nature of deductive inference. There are default reasoning (Boutilier 1991; 1992b; 1992c), has the two directions in which such theories must be generalized. desired nonmonotonicity and induces a natural preference Fist, we should not require that an explanation deductively ordering on sentences (hence explanations). In the next sec- entail its observation (even relative to some background the- tion we describe our conditional logics and the necessary ory). There are very few explanations that do not admit logical preliminaries. In Section 3, we discuss the concept exceptions. Second, while there may be many competing of explanation, its epistemic nature, and its definition in our explanations for a particular observation, certain of these framework. We also introduce the notion of preferred ex- may be relatively implausible. Thus we require some notion planations, showing how the same conditional information of preference to chose among these potential explanations. used to represent the defeasibility of explanations induces Both of these problems can be addressed using, for exam- a natural preference ordering. To demonstrate the expres- ple, probabilistic information (Hempel 1966; de KIeer and sive power of our model, in Section 4 we show how Poole’s Williams 1987; Poole 1991; Pearl 1988): we might simply Theorist framework (and Brewka’s (1989) extension) can require that an explanation render the observation sufficiently be captured in our logics. This reconstruction explains se- probably and that most likely explanations be preferred. Ex- mantically the non-predictive and paraconsistent nature of planations might thus nonmonotonic in the sense that Q may explanations in Theorist. It also illustrates the correct man- explain p, but cvAymay not (e.g., P(/?Icr) may be sufficiently ner in which to augment Theorist with a notion of predictive high while P (p 1cy A y ) may not). There have been proposals explanation and how one should capture semantic prefer- to address these issues in a more qualitative manner using ences on explanations. These two abilities have until now 642 Boutilier been unexplored in this canonical abductive framework. We conclude by describing directions for future research, and how consistency-based diagnosis also fits in our system. More 2 Conditionals and Belief Revision usible The problem of revising a knowledge base or belief set when new information is learned has been well-studied in AI. One of the most influential theories of belief revision is the AGA4 theory (Alchourr&r, Gardenfors and Makinson 1985; Gardenfors 1988). If we take an agent to have a (de- I ductively closed) belief set K, adding new information A to K is problematic if K l- -A. Intuitively, cermin beliefs in z K must be retracted before A can be accepted. The AGM theory provides a set of constraints on acceptable belief re- 0 vision functions *. Roughly, using Ki to denote the belief (a) <JT40-model (b) CO-model set resulting when K is revised by A, the theory maintains that the least “entrenched” beliefs in K should be given up Figure 1: CT40 and CO models and then A added to this contracted belief set. Semantically, this process can be captured by considering a plausibility ordering over possible worlds. As described (so that every logically possible state of affairs is possible). in (Boutilier 1992b; Boutilier 1992a), we can use a fam- The corresponding logics are denoted CO* and CT40*. Ax- ily of logics to capture the AGM theory of revision. The iomatizations for all logics may be found in (Boutilier 1992b; modal logic CO is based on a propositional language (over Boutilier 1992a). For a given model, we define the following augmented with two modal operators 0 and E. notions. We let llall denote the set of worlds satisfying for- Lcpt denotes the propositional sublanguage of this bimodal mula (Y(and also use this notion for sets of formulae K). We language Lg. The sentence Da is read as usual as “CXis true use min( cu) to denote the set of most pkausible a-worldd at all equally or more plausible worlds.” In contrast, &r is read “(Yis true at all less plausible worlds.” A CO-model is a triple M = (IV, 5, cp), where W is a set The revision of a belief set I< can bc represented using of worlds with valuation function ‘p and 5 is a plausibility CT40 or CO-models that reflect the degree of Iplausibility ordering over W. If w 5 v the w is at least as plausible accorded to worlds by an agent in such a belief state. To as v. We insist that 5 be transitive and connected (that capture revision of K, we insist that any such K-revision is, either w 5 v or v < w for all w, v). CO-structures model be such that ~~1~~~= min(T); that is, IlKI forms consist of a totally-ordered set of clusters of worlds, where the (unique) minimal cluster in the model. This reflects a cluster is simply a maximal set of worlds C E W such that the intuition that all and only K-worlds are most plausible w 5 v for each w, v E C (that is, no extension of C enjoys (Boutilier 1992b). The CT40-model in Figure l(a) is a K- this property). This is evident in Figure l(b), where each revision model for K = Cn( TA, B), while the CO-model in large circle represents a cluster of equally plausible worlds. Figure l(b) is suitable for K = Cn( ‘A). Satisfaction of a modal formula at w is given by: To revise K by A, we construct the revised set Ki by considering the set min(A) of most plausible A-worlds in 1. M l+,, Oa iff for each v such that v 5 w, M bV cr. AL In particular, we require that IIKi II = min(A); thus 2. M bw 60 iff for each v such that not v 5 w, M b,, cy. B E K; iff B is true at each of the most plausible A-worlds. We can define a conditional connective 3 such that A + B We define several new connectives as follows: Oa! =df is true in just such a case: lOl&; &! -df 76--a; &k =df Oa A fiCY;and &Y -df l&t. It is easy to verify that these connectives have the (A + B) -df B(A 1 O(A A q(A > B))) following truth conditions: Oa (&) is true at a world if Both models in Figure 1 satisfy A 3 B, since B lholds CYholds at some more plausible (less plausible) world; ECL at each world in the shaded regions, min(A), of the models. Using the Ramsey test for acceptance of conditionals (&) holds iff (;Yholds at all (some) worlds, whether more (Stalnaker 1968), we equate B E Ki with M + A j B. or less plausible. Indeed, for both models we have that Ki = Cn(A, B). If The modal logic CT40 is a weaker version of CO, where the model in question is a CO*-model then this characteriza- we weaken the condition of connectedness to be simple re- tion of revision is equivalent to the AGM model (Boutilicr flexivity.

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