Revision by Conditional Beliefs

Both the dynamics of belief change and the process of reasoning by default can be based on the conditional beZief set of an agent, represented as a set of “if-then” rules. In this paper we address the open problem of formalizing the dynamics of revising this conditional belief set by new if-then rules, be they interpreted as new default rules or new revision policies. We start by providing a purely semantic characterization, based on the semantics of conditional rules, which induces logical constraints on any such revision process. We then introduce logical (syntax-independent) and syntax-dependent techniques, and provide a precise characterization of the set of conditionals that hold after the revision. In addition to formalizing the dynamics of revising a default knowledge base, this work also provides some of the necessary formal tools for establishing the truth of nested conditionals, and attacking the problem of learning new defaults. Consider a child using a single default “typically birds fly”, to predict the behavior of bids. Upon learning of the class of penguins and their exceptional nature she considers revising her current information about birds to include the information that penguins are birds yet “typically penguins do not fly”. This process is different from that usually modeled in approaches to nonmonotonic reasoning and belief revision, where upon discovering that Tweety is a (nonflying) penguin she simply retracts her previous belief that ‘Iweety does fly. Instead, the example above addresses the issue of revising the set of conditional beliefs, namely, the default rules that guide the revision of our factual beliefs. In this p are concerned with the dynamics of such conditional beliefs. Our objective is to characterize how the conditional information in a knowledge base evolves due to the incorporation of the new conditionals, which rules should be given up in case of inconsistency, and what principles guide this process.’ Qne well-known theory addressing the dynamics of factual beliefs is that proposed by Alchourron, Gardenfors and Makinson (1985; 1988). The AGM theory t&es epistemic ‘We will not address the important question of why and when an agent decides to revise its conditional beliefs or defaults. Moishs Goldsddt Rockwell International 444 High Street Palo Alto, CA 94301 U.S.A. [email protected] states to be deductively closed sets of (believed) sentences and characterizes how a rational agent should change its set K of beliefs. This is achieved with postulates constraining revision functions *, where Ki represents the belief set that results when K is revised by A. Unfortunately, the AGM theory does not provide a calculus with which one can realize the revision process or even specify the content of an epistemic state (Bout&r 199% Doyle 1991; Nebell991). Recent work (Boutilier 199% Goldszmidt AGM revision can be captured by assumi has a knowledge base (KB) containing subjunctive conditionals of the form A + B (where A and B are objective formulae). These conditionals define the agent’s belief set and guide the revision process via the Ramsey test (Stalnaker 1968): A ---+ B is accepted iff revision by A results in a belief in B. Such conditionals may be given a probabilistic interpretation (Goldszmidt 1992): each A + B is associated with a conditional probability statement arbitrarily close to one. They may also be interpreted a statements in a suitable modal logic (Boutilier 1992a). The corresponding logics (and indeed semantics) are identical (Boutilier 1992a), and furthermore there is a strong relation between these conditionals and conditional default rules (Boutilier 1992c; Goldszmidt and Pearl 1992a). The AGM theory has two crucial limitations. First, the conditionals (or revision policies) associated with K, that determine the form of K& provide no guidance for determining the conditionals accepted in Ki itself. The theory only determines the new factual beliefs held after revision. Even if conditionals are contained in K, the AGM theory cannot suggest which conditionals should be retained or retracted in the construction of Ki. Subsequent revisions of Ki can thus be almost arbitrary. Second, the theory provides no mechanism for revising a belief set with new conditionals. Thus, the ision policies of an agent cannot, in general, be changed.2 is paper provides a solution to this second problem, and extends our recent work on a solution to the first problem (Bout&r 1993; Goldszmidt and Pearl 1992b). 2Surprisingly, these two issues have remained largely unexplored, due largely to the G%rdenfors (1988) triviality result, which points to difficulties with the interpretation of conditional belief sets. But these can be easily circumvented (Boutilier 1992~). Representation and Reasoning 649 From: AAAI-93 Proceedings. Copyright © 1993, AAAI (www.aaai.org). All rights reserved.