Complexity of Propositional Abduction for Restricted Sets of Boolean Functions

TL;DR

This paper classifies the computational complexity of propositional abduction problems when formulas are restricted to specific Boolean functions, revealing which cases are tractable or intractable and analyzing explanation counting complexity.

Contribution

It provides a complete complexity classification for propositional abduction with various Boolean function restrictions, including counting explanations.

Findings

Identifies NP-complete and polynomial cases for restricted Boolean functions

Provides a full complexity classification for explanation existence

Analyzes counting complexity of explanations

Abstract

Abduction is a fundamental and important form of non-monotonic reasoning. Given a knowledge base explaining how the world behaves it aims at finding an explanation for some observed manifestation. In this paper we focus on 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 SigmaP2-complete in general. We consider variants obtained by restricting the allowed connectives in the formulae to certain sets of Boolean functions. We give a complete classification of the complexity for all considerable sets of Boolean functions. In this way, we identify easier cases, namely NP-complete and polynomial cases; and we highlight sources of intractability. Further, we address the problem of counting the explanations and draw a complete picture for the counting complexity.