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P'rolog Technology for Default Reasoning Artificial Intelligence ELSEVIER Artificial Intelligence 106 (1998) l-75 P’rolog technology for default reasoning: proof theory and compilation techniques Torsten Schaub a**, Stefan Briining b$l a Institutfiir Informatik Universitiir Potsdam, Posrfnch 60 15 53, D-14415 Potsdam, Germany b TLC GmbH, HahnstraJe 43a, D-60528 Fran&curt, Germany Received 24 February 1997; received in revised form 3 June 1998 Abstract The aim of this work is to show how Prolog technology can he used for efficient implementation of query answering in default logics. The idea is to translate a default theory along with a query into a Prolog program and a Prolog query such that the original query is derivable from the default theory iff the Prolog query is derivable from the Prolog program. In order to comply with the goal-oriented proof search of this approach, we focus on default theories supporting local proof procedures, as exemplified by so-called semi-monotonic default theories. Although this does not capture general default theories under Reiter’s interpretation, it does so under alternative ones’. For providing theoretical underpinnings, we found the resulting compilation techniques upon a top-down proof procedure based on model elimination. We show how the notion of a model elimination proof can be refined to capture default proofs and how standard techniques for implementing and improving model elimination theorem provers (regularity, lemmas) can be adapted and extended to default reasoning. This integrated approach allows us to push the concepts needed for handling defaults from the underlying calculus onto the resulting compilation techniques. This meth’od for default theorem proving is complemented by a model-based approach to incremental consistency checking. We show that the crucial task of consistency checking can benefit from keeping models in order to restrict the attention to ultimately necessary consistency checks. This is supported by the concept of default lemmas that allow for an additional avoidance of redundancy. 0 1998 Elsevier Science B.V. All rights reserved. Keywords: Defmlt reasoning; Automated reasoning; Default Logic; Model elimination; P’lTP, Model-based consistency checking * Corresponding author. Email: [email protected]. ’ Email: [email protected]. 0004-3702/98/!6-see front matter 0 1998 Elsevier Science B.V. All rights reserved. PII: SOOO4-3702(98)00092-7 2 I: Schaub, S. Briining /Artificial Intelligence 106 (1998) 1-75 1. Introduction In many AI applications default reasoning plays an important role since many subtasks involve reasoning from incomplete information. This is why there is a great need for systematic methods that allow us to integrate default reasoning capabilities. In fact, the two last decades have provided us with a profound understanding of the underlying problems and have resulted in well-understood formal approaches to default reasoning. Therefore, we are now ready to build advanced default reasoning systems. For this undertaking, we have chosen Reiter’s default logic [71] as our point of departure. Default logic augments classical logic by dejbdt rules that differ from standard inference rules in sanctioning inferences that rely upon given as well as absent information. Knowledge is represented in default logics by default theories (D, W) consisting of a consistent set of formulas W, also called facts, and a set of default rules D. A default rule y has two types of antecedents: a prerequisite I_Iwhich is established if a! is derivable and a justijication ,6 which is established if /l is consistent. If both conditions hold, the consequent y is concluded by default. A set of such conclusions (sanctioned by default rules and classical logic) is called an extension of an initial set of facts: given a set of formulas W and a set of default rules D, any such extension E is a deductively closed set of formulas containing W such that, for any if a E E and -/I 4 E then y E E. (A formal introduction to default logic is given in Section 2.) In what follows, we are interested in implementing the basic approach to query answering in default logic that allows for determining whether a formula is in some extension of a given default theory. 2 Unlike other approaches that address this problem by encapsulating the underlying theorem prover as a separate module, we are proposing a rather different approach that integrates default reasoning into existing automated theorem provers. In order to comply with the methodology underlying query-oriented classical theorem provers, it is more or less indispensable to center the overall approach around local proof procedures (i.e., proof procedures that allow for deciding whether a set of default rules forms a default proof by looking at the constituent rules only). This is because such procedures permit validating a default inference step during the goal-directed proof search in a locally determinable way (see Section 2 for details). The methodology presented in this paper has its origins in an approach to default query answering proposed in [79]. This approach integrates the notion of a default proof into a calculus for classical logic, which renders it especially qualified for implementations by means of existing theorem provers. To be more precise, Schaub [79] furnishes a mating- based characterization of default proofs inside the framework provided by the connection method [ 151. This results in a connection calculus for query answering in so-called semi- monotonic default logics. (The advantage of these default systems is that they allow for the * Membership in all extensions is actually computable by appeal to a procedure testing membership in Verne extension (see [89]). 1: Schaub, S. Briining /Artij?cial Intelligence 106 (1998) I-75 3 aforementio’ned local proof procedures, as we detail in Section 2.) In fact, there are already numerous implementations of connection calculi. Most of them, like the high performance theorem prover SETHEO [47], are based on model elimination [53] which can be regarded as a member of the family of connection calculi. We draw on this relationship in this paper and show how an implementation technique for model elimination, namely Prolog Technology Theorem Proving (PTTP) [87,88], can be used for (default reasoning. Our overall contribution can thus be looked at from different perspectives: first, we provide implementation techniques for (semi-monotonic) default logics. Second, we extend an existing automated theorem prover by means for handling default information. In particular, we show how the notion of a model elimination proof can be refin’ed to capture (semi-monotonic) default proofs and how standard techniques for implementi:ng and improving model elimination theorem provers (regularity, lemmas) can be adapted and extended to default reasoning. And finally, one can view our contribution somehow as a logic programming system integrating disjunction, classical as well as default negation. To give a, more precise overview of our approach, we start by noting that default logic is among the consistency-based approaches to default reasoning. In these formalisms, a logical formalization of a consistency-driven procedure is added to a standard logic. As explained above, this is done in default logic by means of the justification of a default rule. In this way, default reasoning is mapped onto a deductive task and a consistency checking task. Of course, this carries over to the resulting proof procedures for query answering, too. As anticipated above, we address the deductive task of default query answering by appeal to techniques borrowed from Prolog Technology Theorem Proving. The idea is to translate a default theory along with a query into a Prolog program and a Prolog query such that the original query belongs to an extension of the default theory iff the Prolog query is derivable from the Prolog program. For providing theoretical underpinnings, we found the resulting compilation techniques upon a top-down proof procedure based on model-elimination. This proof procedure has its roots in the mating-based characterization of default query answering given in [79], which also integrates the notion of a default proof into a calculus for classical logic. This integrated approach allows us to push the concepts needed for handling defaults from the underlying calculus, over the corresponding proof procedure, into the resulting compilation techniques. As regards the task of consistency checking, the first interesting question is whether we can find a way of pruning “inconsistent subproofs” while reducing the computational efforts for consistency checking. For this, we observe that a formula is consistent (or satisfiable) iff it has a model. This leads us to the following incremental approach to consistency checking: we start with a model of the initial set of facts. Each time, we apply a default rule, we check whether the actual model satisfies the underlying default assumptions (which can be done in linear time in propositional logic). If this is the case, we continue proving. If not, we try to generate a new model of the initial set of facts satisfying the current as well as all default assumptions underlying the partial default proof at hand. If we succeed, we simply continue proving under the new model. Otherwise, we know that the considered default assumption cannot be assumed in a consistent way (since there is no model for the continuation of our current default proof with the default rule at 4 I: Schaub, S. Briining /Artificial Intelligence 106 (1998) 1-75 hand). In this way, we restrict the generation of new models to the ultimately necessary ones. The second interesting question is then whether a simultaneous treatment of theorem proving and satisfiability checking allows for proof procedures benefiting from structure and information sharing. This is important since the two tasks encompass genuine and even orthogonal sources of (putative) exponential complexity, as reflected by the fact that query answering in default logics is El-complete [42].
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