Fixpoint 3-valued semantics for autoepistemic logic

TL;DR

This paper introduces a constructive three-valued fixpoint semantics for autoepistemic logic, providing a unified framework that simplifies computation and relates to existing semantics like Moore's stable expansions and well-founded semantics.

Contribution

It presents a novel 3-valued semantics for autoepistemic logic based on a derivation operator's least fixpoint, simplifying previous approaches and connecting to established semantics.

Findings

The semantics characterizes a unique, possibly three-valued belief set.

Complete fixpoints correspond to Moore's stable expansions.

The proposed semantics is computationally simpler than Moore's, assuming the polynomial hierarchy does not collapse.

Abstract

The paper presents a constructive fixpoint semantics for autoepistemic logic (AEL). This fixpoint characterizes a unique but possibly three-valued belief set of an autoepistemic theory. It may be three-valued in the sense that for a subclass of formulas F, the fixpoint may not specify whether F is believed or not. The paper presents a constructive 3-valued semantics for autoepistemic logic (AEL). We introduce a derivation operator and define the semantics as its least fixpoint. The semantics is 3-valued in the sense that, for some formulas, the least fixpoint does not specify whether they are believed or not. We show that complete fixpoints of the derivation operator correspond to Moore's stable expansions. In the case of modal representations of logic programs our least fixpoint semantics expresses well-founded semantics or 3-valued Fitting-Kunen semantics (depending on the embedding used). We show that, computationally, our semantics is simpler than the semantics proposed by Moore (assuming that the polynomial hierarchy does not collapse).