New Collapse Consequences of NP Having Small Circuits

Johannes Köbler, Osamu Watanabe
1998 SIAM journal on computing (Print)  
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
more » ... n 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.
doi:10.1137/s0097539795296206 fatcat:fgjq46fytfh2rfxp6tphe463wm