Circuit lower bounds for nondeterministic quasi-polytime: an easy witness lemma for NP and NQP
Author(s)
Murray, Cody (Cody Daniel); Williams, R Ryan
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We prove that if every problem in NP has nk-size circuits for a fixed constant k, then for every NP-verifier and every yes-instance x of length n for that verifier, the verifier’s search space has an nO(k3)size witness circuit: a witness for x that can be encoded with a circuit of only nO(k3) size. An analogous statement is proved for nondeterministic quasi-polynomial time, i.e., NQP = NTIME nlogO(1) n . This significantly extends the Easy Witness Lemma of Impagliazzo, Kabanets, and Wigderson [JCSS’02] which only held for larger nondeterministic classes such as NEXP. As a consequence, the connections between circuit-analysis algorithms and circuit lower bounds can be considerably sharpened: algorithms for approximately counting satisfying assignments to given circuits which improve over exhaustive search can imply circuit lower bounds for functions in NQP, or even NP. To illustrate, applying known algorithms for satisfiability of ACC ◦ THR circuits [R. Williams, STOC 2014] we conclude that for every fixed k, NQP does not have nlogk n-size ACC ◦ THR circuits.
Date issued
2018-06Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence LaboratoryJournal
Proceedings of the Annual ACM Symposium on Theory of Computing
Publisher
Association for Computing Machinery (ACM)
Citation
Murray, Cody D. and R. Ryan Williams. “Circuit lower bounds for nondeterministic quasi-polytime: an easy witness lemma for NP and NQP.” Paper in the Proceedings of the Annual ACM Symposium on Theory of Computing, June-2018, STOC 2018: 50th Annual ACM SIGACT Symposium on Theory of Computing, Los Angeles, CA,, June 25-29 2018, Association for Computing Machinery (ACM): 890–901 © 2018 The Author(s)
Version: Author's final manuscript
ISBN
9781450355599
ISSN
0277-0261