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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 time f such that f (x, y) ∈ L ⇐⇒ x ∈ L and y ∈ L. L has an OR function if there is a polynomial time g such that g(x, y) ∈ L ⇐⇒ x ∈ L or y ∈ L. While all NP complete sets have these functions, Graph… (More)

The research presented in this paper is motivated by the following new results on the complexity of the unique satisfiability problem, USAT.

We show that if the Boolean hierarchy collapses to level k, then the polynomial hierarchy collapses to BH 3 (k), where BH 3 (k) is the k th level of the Boolean hierarchy over P 2. This is an improvement over the known results 3], which show that the polynomial hierarchy would collapse to P NP NP O(log n)]. This result is signiicant in two ways. First, the… (More)

The research presented in this paper is motivated by the some new results on the complexity of the unique satissability problem, USAT. These results, which are shown for the rst time in this paper, are: if USAT P m USAT, then D P = coD P and PH collapses. if USAT 2 coD P , then PH collapses. if USAT has OR ! , then PH collapses. The proofs of these results… (More)

We study FP NP k , the class of functions that can be computed with nonadaptive queries to an NP oracle. We show that optimization problems stemming from the known NP complete sets, where the optimum is taken over a polynomially bounded range, are hard for FP NP k. This is related to (and, in some sense, extends) work of Chen and Toda CT91]. In addition, it… (More)

1 About Relativization In this column we explore what relativization says about space bounded computations and what recent results about space bounded computations say about relativization. There is a strong belief in computational complexity circles that problems which can be relativized in two contradictory ways are very hard to solve. We believe that… (More)