Tractable Reasoning in Knowledge Representation Systems

Abstract

This document addresses some problems raised by the well-known intractability of deductive reasoning in even moderately expressive knowledge representation systems. Starting from boolean constraint propagation (BCP), a previously known linear-time incomplete reasoner for clausal propositional theories, we develop fact propagation (FP) to deal with non-clausal theories, after motivating the need for such an extension. FP is specified using a confluent rewriting systems, for which we present an algorithm that has quadratic-time complexity in general, but is still linear-time for clausal theories. FP is the only known tractable extension of BCP to non-clausal theories; we prove that it performs strictly more inferences than CNF-BCP, a previously-proposed extension of BCP to non-clausal theories. We generalize a refutation reasoner based on FP to a family of sound and tractable reasoners that are "increasingly complete" for propositional theories. These can be used for anytime reasoning, i.e. they provide partial answers even if they are stopped prematurely, and the "completeness" of the answer improves with the time used in computing it. A fixpoint construction based on FP gives an alternate characterization of the reasoners in this family, and is used to define a transformation of arbitrary theories into logically-equivalent "vivid" theories -- ones for which our FP algorithm is complete. Our final contribution is to the description of tractable classes of reasoning problems. Based on FP, we develop a new property, called bounded intricacy, which is shared by a variety of tractable classes that were previously presented, for example, in the areas of propositional satisfiability, constraint satisfaction, and OR-databases. Although proving bounded intricacy for these classes requires domain-specific techniques (which are based on the original tractability proofs), bounded intricacy is one more tool available for showing that a family of problems arising in some application is tractable. As we demonstrate in the case of constraint satisfaction and disjunctive logic programs, bounded intricacy can also be used to uncover new tractable classes