On the Parameterized Complexity of Default Logic and Autoepistemic Logic

TL;DR

This paper explores the parameterized complexity of default and autoepistemic logic, providing efficient algorithms for some problems and complexity barriers for others, highlighting the nuanced computational landscape.

Contribution

It applies Courcelle's Theorem and logspace algorithms to logic problems, establishing both fixed-parameter algorithms and complexity lower bounds.

Findings

Efficient fixed-parameter algorithms for certain logic problems.

Complexity barriers for other problems unless P=NP.

Identification of simple structures resistant to fixed-parameter algorithms.

Abstract

We investigate the application of Courcelle's Theorem and the logspace version of Elberfeld etal. in the context of the implication problem for propositional sets of formulae, the extension existence problem for default logic, as well as the expansion existence problem for autoepistemic logic and obtain fixed-parameter time and space efficient algorithms for these problems. On the other hand, we exhibit, for each of the above problems, families of instances of a very simple structure that, for a wide range of different parameterizations, do not have efficient fixed-parameter algorithms (even in the sense of the large class XPnu), unless P=NP.