The Complexity of Reasoning for Fragments of Default Logic

TL;DR

This paper provides a comprehensive complexity classification for propositional default logic when restricting propositional connectives, revealing a detailed hexachotomy for extension existence and similar classifications for reasoning problems.

Contribution

It systematically classifies the complexity of default logic decision problems across all Boolean function restrictions using Post's lattice, extending prior partial results.

Findings

Complexity varies across six classes for extension existence.

Credulous and skeptical reasoning have similar complexity classifications.

Identifies trivial cases and detailed complexity boundaries.

Abstract

Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified the complexity of the extension existence problem for propositional default logic as $\SigmaPtwo$-complete, and the complexity of the credulous and skeptical reasoning problem as SigmaP2-complete, resp. PiP2-complete. Additionally, he investigated restrictions on the default rules, i.e., semi-normal default rules. Selman made in 1992 a similar approach with disjunction-free and unary default rules. In this paper we systematically restrict the set of allowed propositional connectives. We give a complete complexity classification for all sets of Boolean functions in the meaning of Post's lattice for all three common decision problems for propositional default logic. We show that the complexity is a hexachotomy (SigmaP2-, DeltaP2-, NP-, P-, NL-complete, trivial) for the extension existence problem, while for the credulous and skeptical reasoning problem we obtain similar classifications without trivial cases.