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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 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)

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)

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 ⊇ L is a p-selective sparse set, then L − S is ≤ p m-hard for NP. We demonstrate existence of a sparse set S ∈ DTIME(2 2 n) such that for every L ∈ NP − P, L − S is not ≤ p m-hard for NP. Moreover, we prove for every L ∈ NP − P, that there exists a… (More)