Towards Efficient Default Reasoning

A decision method for Reiter's default logic is developed. It can determine whether a default theory has an extension, whether a formula is in some extension of a default theory and whether a formula is in every extension of a default the­ ory. The method handles full propositional de­ fault logic. It can be implemented to work in polynomial space and by using only a theorem prover for the underlying propositional logic as a subroutine. The method divides default rea­ soning into two major subtasks: the search task of examining every alternative for extensions, which is solved by backtracking search, and the classical reasoning task, which can be imple­ mented by a theorem prover for the underly­ ing classical logic. Special emphasis is given to the search problem. The decision method em­ ploys a new compact representation of exten­ sions which reduces the search space. Efficient techniques for pruning the search space further are developed. 1 Introduction In this paper we develop a theorem-proving method for default logic of Reiter [1980]. Default logic is one of the most well-known nonmonotonic logics [Marek and Truszczyriski, 1993]. There is a body of results indicat­ ing that default logic captures a large number of different forms of nonmonotonic reasoning. Default logic is closely related to logic programs and deductive databases [Gel-fond and Lifschitz, 1990). Connections have been es­ tablished between default logic and autoepistemic logic and McDermott and Doyle style nonmonotonic modal In default logic knowledge is represented by a default theory, which consists of ordinary first-order formulae and nonmonotonic inference rules, default rules. Possi­ ble sets of conclusions from a default theory are defined in terms of extensions of the default theory. In this pa­ per we consider three basic reasoning tasks: extension existence (whether a default theory has an extension), brave reasoning (whether a formula is in some extension of a default theory) and cautious reasoning (whether a formula belongs to every extension of a default theory). Computational properties of nonmonotonic reason­ ing have received considerable attention. This research provides a valuable basis for developing nonmonotonic theorem-proving techniques. The complexity of proposi­ tional default reasoning has been located at the second level of the polynomial time hierarchy [Garey and John­ son, 1979]: extension existence and brave reasoning are Ep/2-complete and cautious reasoning Ilp/2-complete [Got-tlob, 1992]. This means that full propositional default logic can be implemented in polynomial space and that it …