NP trees and Carnap's modal logic

Abstract

In this paper we consider problems and complexity classes definable by interdependent queries to an oracle in NP. How the queries depend on each other is specified by a directed graph G. We first study the class of problems where G is a general dag and show that this class coincides with Δ p 2 . We then consider the class where G is a tree. Our main result states that this class is identical to P NP [O(log n)], the class of problems solvable in polynomial time with a logarithmic number of queries to an oracle in NP. This result has interesting applications in the fields of modal logic and artificial intelligence. In particular, we show that the following problems are all P NP [O(log n)] complete: validity-checking of formulas in Carnap's modal logic, checking whether a formula is almost surely valid over finite structures in modal logics K, T, and S4 (a problem recently considered by Halpern and Kapron [1992]), and checking whether a formula belongs to the stable set of beliefs generated by a propositional theory. We generalize the case of dags to the case where G is a general (possibly cyclic) directed graph of NP-oracle queries and show that this class corresponds to Π p 2 . We show that such graphs are easily expressible in autoepistemic logic. Finally, we generalize our complexity results to higher classes of the polynomial-time hierarchy.