Towards automatic autoepistemic reasoning

Abstract

Nonmonotonic reasoning is one of most important and active areas of research in knowledge representation and reasoning. Autoepistemic logic introduced by Moore offers a very promising approach to formalize nonmonotonic reasoning. The key concept in autoepistemic reasoning is a stable expansion of a set of premises. Stable expansions are defined non-constructively using a fixed point equation which is rather hard to reason about. In this paper first-order autoepistemic logic is studied where quantifying into a modal context is not allowed. A simple new syntactic characterization of stable expansions is developed. For each stable expansion a unique set of sentences, a kernel, is constructed which completely determines the expansion. For finite sets of premises the kernel is also finite and thus stable expansions can be given a finite representation even though they are infinite sets of sentences. Using this new characterization conditions ensuring that a set of premises has at least one stable expansion and exactly one stable expansion can be stated and an upper bound on the number of stable expansions of a finite set of premises can be given. Furthermore, preliminary results on decidability and complexity of autoepistemic reasoning are obtained. E.g. it is shown that autoepistemic reasoning is decidable iff the underlying monotonic consequence relation is decidable.

The financial support of the Foundation of Technology is gratefully acknowledged.