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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 AEȽ℄ È AEȾ℄ , 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 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 $$(2^{{n}^\epsilon})$$ language by an NP refuter. The NP-machine(More)
We report progress on the NL versus UL problem. We show that counting the number of s-t paths in graphs where the number of s-v paths for any v is bounded by a polynomial can be done in FUL: the unambiguous log-space function class. Several new upper bounds follow from this including $${{{ReachFewL} \subseteq {UL}}}$$ and $${{{LFew} \subseteq(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)
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)