Abstract.Chang and Kadin have shown that if the difference hierarchy over NP collapses to level k, then the polynomial hierarchy (PH) is equal to the kth level of the difference hierarchy over E~. We simplify their poof and obtain a slightly stronger conclusion: if the difference hierarchy over NP collapses to level k, then PH collapses to (P~P_ l)_tt) NP, the class of sets recognized in polynomial time with k-1 nonadaptive queries to a set in NP NP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any class C that has

Abstract. We show that if the Boolean hierarchy collapses to level k, then the polynomial hierarchy collapses to BH3(k), where BH3(k) is the kth level of the Boolean hierarchy over E2 P. This is an improvement over the known results, which show that the polynomial hierarchy would collapse to P NPNP [O(Ign)]. This result is significant in two ways. First, the theorem says that a deeper collapse of the Boolean hierarchy implies a deeper collapse of the polynomial hierarchy. Also, this result points to some previously unexplored connections between the Boolean and query hierarchies of A and A3P. Namely,

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This paper is a study of the existence of polynomial time Boolean connective functions for languages. A language L has an AND function if there is a polynomial timefsuch that f(x,y) E L r x E L and y E L. L has an OR function if there is a polynomial time g such that g(x,y) E L r x G L or y E L. While all NP complete sets have these functions, Graph Isomorphism, which is probably not complete, is also shown to have both AND and OR functions. The results in this paper characterize the complete sets for the classes D P and pSAT[O(log n)] in terms of AND and OR, and relate these functions to the structure of the Boolean hierarchy and the query hierarchies. Also, this paper shows that the complete sets for the levels of the Boolean hierarchy above the second level cannot have AND or OR unless the polynomial hierarchy collapses. Finally, most of the structural properties of the Boolean hierarchy and query hierarchies are shown to depend only on the existence of AND and OR functions for the NP complete sets.

This paper explores the bounded query complexity of approximating the size of the maximum clique in a graph (Clique Size) and the number of simultaneously satisfiable clauses in a 3CNF formula (MaxSat). The results in the paper show that for certain approximation factors, approximating Clique Size and MaxSat are complete for corresponding bounded query classes under metric reductions. The completeness result is important because it shows that queries and approximation are interchangeable: NP queries can be used to solve NP-approximation problems and solutions to NP-approximation problems answer queries to NP oracles. Completeness also shows the existence of approximation preserving reductions from many NP-approximation problems to approximating Clique Size and MaxSat (e.g., from approximating Chromatic Number to approximating Clique Size). Since query complexity is a quantitative complexity measure, these results also provide a framework for comparing the complexities of approximating Clique Size and approximating MaxSat. In addition, this paper examines the query complexity of the minimization version of the satisfiability problem, MinUnsat, and shows that the complexity of approximating MinUnsat is very similar to the complexity of approximating Clique Size. Since MaxSat and MinUnsat share the same solution space, the``approximability'' of MaxSat is not due to the intrinsic complexity of satisfiability, but is an artifact of viewing the approximation version of satisfiability as a maximization problem.