Circumscription and the closed world assumption with its variants are well-known nonmonotonic techniques for reasoning with incomplete knowledge. Their complexity in the propositional case has been studied in detail for fragments of propositional logic. One open problem is whether the deduction problem for arbitrary propositional theories under the extended closed world assumption or under circumscription is $\Pi^P_2$-complete, i.e., complete for a class of the second level of the polynomial hierarchy. We answer this question by proving these problems $\Pi^P_2$-complete, and we show how this result applies to other variants of closed world reasoning.

This paper introduces a unified solution to the problem of extending stratified DATALOG to express DB-complexity classes ranging from P to D^P. The solution...

Abstract. Rice’s Theorem says that every nontrivial semantic property of programs is undecidable. In this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UP-hard with respect to polynomial-time Turing reductions. For generators [31] we show a perfect analogue of Rice’s Theorem. Mathematics Subject Classification: 03D15, 68Q15. Keywords: Rice’s Theorem, Counting problems, Promise classes, UP-hard, NP-hard, generators.

A knowledge base is redundant if it contains parts that can be inferred from the rest of it. We study the problem of checking whether a CNF formula (a set of clauses) is redundant, that is, it contains clauses that can be derived from the other ones. Any CNF formula can be made irredundant by deleting some of its clauses: what results is an irredundant equivalent subset (I.E.S.) We study the complexity of some related problems: veri cation, checking existence of a I.E.S. with a given size, checking necessary and possible presence of clauses in I.E.S.'s, and uniqueness.