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标题：On the (Non) NP-Hardness of Computing Circuit Complexity
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作者：Cody D. Murray ; R. Ryan Williams
期刊名称：Theory of Computing
印刷版ISSN：1557-2862
电子版ISSN：1557-2862
出版年度：2017
卷号：13
出版社：University of Chicago
摘要：he Minimum Circuit Size Problem (MCSPMCSP) is: given the truth table of a Boolean function ff and a size parameter kk, is the circuit complexity of ff at most kk? This is the definitive problem of circuit synthesis, and it has been studied since the 1950s. Unlike many problems of its kind, MCSPMCSP is not known to be NPNP-hard, yet an efficient algorithm for this problem also seems very unlikely: for example, MCSP∈PMCSP∈P would imply there are no pseudorandom functions.Although most NPNP-complete problems are complete under strong “local” reduction notions such as polylogarithmic-time projections, we show that MCSPMCSP is provably not NPNP-hard under O(n1/2−ε)O(n1/2−ε)-time projections, for every ε>0ε>0, and is not NPNP-hard under randomized O(n1/5−ε)O(n1/5−ε)-time projections, for every ε>0ε>0. We prove that the NPNP-hardness of MCSPMCSP under (logtime-uniform) AC0AC0 reductions would imply extremely strong lower bounds: NP⊄P/polyNP⊄P/poly and E⊄i.o.-SIZE(2δn)E⊄i.o.-SIZE(2δn) for some δ>0δ>0 (hence P=BPPP=BPP also follows). We show that even the NPNP-hardness of MCSPMCSP under general polynomial-time reductions would separate complexity classes: EXP≠NP∩P/polyEXP≠NP∩P/poly, which implies EXP≠ZPPEXP≠ZPP. These results help explain why it has been so difficult to prove that MCSPMCSP is NPNP-hard.
关键词：circuit lower bounds; reductions; NP-completeness; projections; minimum circuit size problem