Complexity Classifications for Propositional Abduction in Post's Framework

TL;DR

This paper provides a detailed complexity classification of propositional abduction problems, analyzing how different Boolean connectives and problem variants affect computational difficulty.

Contribution

It offers a comprehensive classification of the complexity of propositional abduction across various Boolean function sets and problem variants, including counting explanations.

Findings

Identifies NP-complete, coNP-complete, and polynomial cases for abduction complexity.

Provides a complete complexity classification for counting explanations.

Highlights sources of intractability in propositional abduction.

Abstract

In this paper we investigate the complexity of abduction, a fundamental and important form of non-monotonic reasoning. Given a knowledge base explaining the world's behavior it aims at finding an explanation for some observed manifestation. In this paper we consider propositional abduction, where the knowledge base and the manifestation are represented by propositional formulae. The problem of deciding whether there exists an explanation has been shown to be \SigPtwo-complete in general. We focus on formulae in which the allowed connectives are taken from certain sets of Boolean functions. We consider different variants of the abduction problem in restricting both the manifestations and the hypotheses. For all these variants we obtain a complexity classification for all possible sets of Boolean functions. In this way, we identify easier cases, namely \NP-complete, \coNP-complete and polynomial cases. Thus, we get a detailed picture of the complexity of the propositional abduction problem, hence highlighting sources of intractability. Further, we address the problem of counting the explanations and draw a complete picture for the counting complexity.