Autoepistemic logic

Autoepistemic logic is one of the principal modes of nonmonotonic reasoning. It unifies several other modes of nonmonotonic reasoning and has important applications in logic programming. In the paper, a theory of autoepistemic logic is developed. This paper starts with a brief survey of some of the previously known results. Then, the nature of nonmonotonicity is studied by investigating how membership of autoepistemic statements in autoepistemic theories depends on the underlying objective theory. A notion similar to set-theoretic forcing is introduced. Expansions of autoepistemic theories are also investigated. Expansions serve as sets of consequences of an autoepistemic theory and they can also be used to define semantics for logic programs with negation. Theories that have expansions are characterized, and a normal form that allows the description of all expansions of a theory is introduced. Our results imply algorithms to determine whether a theory has a unique expansion. Sufficient conditions (stratification) that imply existence of a unique expansion are discussed. The definition of stratified theories is extended and (under some additional assumptions) efficient algorithms for testing whether a theory is stratified are proposed. The theorem characterizing expansions is applied to two classes of theories, K₁-theories and ae-programs. In each case, simple hypergraph characterization of expansions of theories from each of these classes is given. Finally, connections with stable model semantics for logic programs with negation is discussed. In particular, it is proven that the problem of existence of stable models is NP-complete.