Autoepistemic logic

Abstract

Autoepistemic logic is one of the principa~modes of nonmonotonic reasoning. It umties several other modes of nonmonotonic reasoning and has important applications in logic programming. In the paper, a theory of autoepisternic 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 N 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 stratitied 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 se-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.

Greedy algorithms in Datalog

“…This contrasts with the general intractability of finding stable models for general programs: in fact, we know that checking if a Datalog program with negation has a stable model is NP-complete (Marek and Truszczynski, 1991).…”

A Top-Down Query Answering Procedure for Normal Logic Programs Under the Any-World Assumption

“…Such statements were the main motivation for non-monotonic logics like Default Logic [22], Autoepistemic Logic [8,18,20,21] and Circumscription [19]. We can formulate such a statement in a natural way, using abnormality theories, as A ← ϕ ∧ ¬Ab and Ab ← ¬A, where Ab stands for abnormality, and then consider the hypothesis H(Ab) = f, i.e., by default there are no abnormal objects.…”

Operational Semantics for DyLPs

“…to find a stable model of a given program) is known to be NP-Complete, while the inference problem (i.e. to determine if a given proposition is true in all the stable models of a program) is co-NP-Complete [17]. Hence, from the fact that DyLPs extends the class of normal LPs and from theorems 1 and 2 it immediately follows that such problems are still NP-Complete and co-NP-Complete also under the refined semantics for DyLPs.…”