NP trees and Carnap's modal logic

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 /spl Delta//sub 2//sup P/. We then consider the class where G is a tree. Our main result states that this class is identical to P/sup NP/ [O(log n)], the class of problems solvable in polynomial time with a logarithmic number of queries to an oracle in NP. Using this result we show that the following problems are all P/sup NP/[O(logn)] 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, and checking whether a formula belongs to the stable set of beliefs generated by a propositional theory.<
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