New collapse consequences of NP having small circuits

Abstract

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 of Karp, Lipton, and Sipser [KL80] stating a collapse of PH to its second level Σ P 2 under the same assumption.As a further consequence, we derive new collapse consequences under the assumption that complexity classes like UP, FewP, and C = P have polynomialsize 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.

ORACLES IN $\Sigma^p_2$ ARE SUFFFICIENT FOR EXACT LEARNING

“…We use the following refinement of this result. Lemma 3 [13] Let X be a collection of at most 2 n different subsets of{0, l} m ? each of which has cardinality at most 2 k~l .…”

Computing with New Resources

“…This is a paper on some question that I have been considering from time to time since I published a paper [KW98]. It is about some important topic in computational complexity theory, which is also important in general, I believe, to understand the nature of computation.…”

“…Some years later, Köbler-Watanabe pointed out [KW98] that the learning technique of Bshouty et al [Betal96] can be used to improve the Kannan's result, showing such a hard problem in ZPP NP , which has been improved further to S 2 by [Cai01]. These techniques are relativizable 2 .…”