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.

Learning Reductions to Sparse Sets

“…This was later improved to a collapse of PH = ZPP NP [24,14], and finally PH = S show a similar result for disjunctive truth-table reductions. The best known consequence of this is a collapse of PH to P NP [11].…”

Different Approaches to Proof Systems

“…The classical Karp-Lipton Theorem states that NP ⊆ P/poly implies a collapse of the polynomial hierarchy PH to its second level [KL80]. Subsequently, these collapse consequences have been improved by Köbler and Watanabe [KW98] to ZPP NP and by Sengupta and Cai to S p 2 (cf. [Cai07]).…”

Hardness Hypotheses, Derandomization, and Circuit Complexity

“…(5) By the results of Kannan [28], Bshouty et al [10], and Kobler and Watanabe [30], ZPP NP ⊆ io-SIZE(n k ) for any k > 0. Since the NP-machine hypothesis implies BPP NP = P NP , P NP ⊆ io-SIZE(n k ) for every k > 0.…”