The Complexity of Circumscriptive Inference in Post’s Lattice

Abstract

Circumscription is one of the most important formalisms for reasoning with incomplete information. It is equivalent to reasoning under the extended closed world assumption, which allows to conclude that the facts derivable from a given knowledge base are all facts that satisfy a given property. In this paper, we study the computational complexity of several formalizations of inference in propositional circumscription for the case that the knowledge base is described by a propositional theory using only a restricted set of Boolean functions. To systematically cover all possible sets of Boolean functions, we use Post’s lattice. With its help, we determine the complexity of circumscriptive inference for all but two possible classes of Boolean functions. Each of these problems is shown to be either $\protect \ensuremath {\mathrm {\Pi ^{\mathrm{p}}_{2}}}$ -complete, coNP-complete, or solvable in logspace. In particular, we show that in the general case, unless P=NP, only literal theories admit polynomial-time algorithms, while for some restricted variants the tractability border is the same as for classical propositional inference.