Complexity of Default Logic on Generalized Conjunctive Queries

Reiter’s default logic formalizes nonmonotonic reasoning using default assumptions. The semantics of a given instance of default logic is based on a fixpoint equation defining an extension. Three different reasoning problems arise in the context of default logic, namely the existence of an extension, the presence of a given formula in an extension, and the occurrence of a formula in all extensions. Since the end of 1980s, several complexity results have been published concerning these default reasoning problems for different syntactic classes of formulas. We derive in this paper a complete classification of default logic reasoning problems by means of universal algebra tools using Post’s clone lattice. In particular we prove a trichotomy theorem for the existence of an extension, classifying this problem to be either polynomial, NP-complete, or Σ2P-complete, depending on the set of underlying Boolean connectives. We also prove similar trichotomy theorems for the two other algorithmic problems in connection with default logic reasoning.