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SIGACT News complexity theory column 32
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Competing provers yield improved Karp-Lipton collapse results
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A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
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Competing provers yield improved Karp-Lipton collapse results
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On proving circuit lower bounds against the polynomial-time hierarchy: positive and negative results
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A tight Karp-Lipton collapse result in bounded arithmetic
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The shrinking property for NP and coNP
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We show that if a self-reducible set has polynomial-size circuits, then it is low for the probabilistic class ZPP (NP). As a consequence we get a deeper collapse of the polynomial-time hierarchy PH to ZPP(NP) under the assumption that NP has polynomial-size circuits. This improves on the well-known result in Karp and Lipton [ Proceedings of the 12th ACM Symposium on Theory of Computing, ACM Press, New York, 1980, pp. 302--309] stating a collapse of PH to its second level $\Sigmap_2$ under the same assumption. Furthermore, we derive new collapse consequences under the assumption that complexity classes like UP, FewP, and C=P have polynomial-size circuits. Finally, we investigate the circuit-size complexity of several language classes. In particular, we show that for every fixed polynomial s, there is a set in ZPP(NP) which does not have O(s(n))-size circuits.