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We show that if ËËÌ does not have small circuits, then there must exist a small number of formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if È AEÈ½℄ È AEÈ¾℄ , then the polynomial-time hierarchy collapses to Ë Ô ¾ ¦ Ô ¾ ¥ Ô ¾. Even showing that the hierarchy… (More)

We apply results on extracting randomness from independent sources to " extract " Kol-mogorov complexity. For any α, > 0, given a string x with K(x) > α|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y| = Ω(|x|), with K(y) > (1 −)|y|. This result holds for both unbounded and space-bounded Kolmogorov… (More)

We use hypotheses of structural complexity theory to separate various NP-completeness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is ¡ P T-complete but not ¡ P tt-complete. We provide fairly thorough analyses of the hypotheses that we introduce. AMS subject classifications.

We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: • The measure hypothesis: NP does not have p-measure 0. • The pseudo-NP hypothesis: there is an NP language that can be distinguished from any DTIME(2 n ǫ) language by an NP refuter. • The NP-machine hypothesis:… (More)

We prove that if there is a polynomial time algorithm which computes the permanent of a matrix of order n for any inverse polynomial fraction of all inputs, then there is a BPP algorithm computing the permanent for every matrix. It follows that this hypothesis implies P #P = BPP. Our algorithm works over any suuciently large nite eld (polynomially larger… (More)

We show the following results regarding complete sets. • NP-complete sets and PSPACE-complete sets are many-one autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are many-one autoreducible. • EXP-complete sets are many-one mitotic. • NEXP-complete sets are weakly many-one mitotic. • PSPACE-complete sets are weakly… (More)

Under the assumption that NP does not have p-measure 0, we investigate reductions to NP-complete sets and prove the following: 1. Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turing-complete for NP but not truth-table-complete. 2. Strong nondeterministic reductions are more powerful than deterministic… (More)

We clarify the role of Kolmogorov complexity in the area of randomness extraction. We show that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogorov extraction and randomness extraction.… (More)

We obtain the following new simultaneous time-space upper bounds for the directed reach-ability problem. (1) A polynomial-time, O(n 2/3 g 1/3)-space algorithm for directed graphs embedded on orientable surfaces of genus g. (2) A polynomial-time, O(n 2/3)-space algorithm for all H-minor-free graphs given the tree decomposition, and (3) for K 3,3-free and K… (More)

We study several properties of sets that are complete for NP. We prove that if L is an NP-complete set and S /spl nsupe/ L is a p-selective sparse set, then L -S is /spl les//sub m//sup p/-hard for NP. We demonstrate existence of a sparse set S /spl isin/ DTIME(2/sup 2n/) such that for every L /spl isin/ NP - P, L - S is not /spl les//sub m//sup p/-hard for… (More)