Relations and equivalences between circuit lower bounds and Karp-Lipton theorems

Abstract

© Lijie Chen, Dylan M. McKay, Cody D. Murray, and R. Ryan Williams; licensed under Creative Commons License CC-BY 34th Computational Complexity Conference (CCC 2019). A frontier open problem in circuit complexity is to prove PNP 6⊂ SIZE[nk] for all k; this is a necessary intermediate step towards NP 6⊂ P/poly. Previously, for several classes containing PNP, including NPNP, ZPPNP, and S2P, such lower bounds have been proved via Karp-Lipton-style Theorems: to prove C 6⊂ SIZE[nk] for all k, we show that C ⊂ P/poly implies a “collapse” D = C for some larger class D, where we already know D 6⊂ SIZE[nk] for all k. It seems obvious that one could take a different approach to prove circuit lower bounds for PNP that does not require proving any Karp-Lipton-style theorems along the way. We show this intuition is wrong: (weak) Karp-Lipton-style theorems for PNP are equivalent to fixed-polynomial size circuit lower bounds for PNP. That is, PNP 6⊂ SIZE[nk] for all k if and only if (NP ⊂ P/poly implies PH ⊂ i.o.-PNP/n). Next, we present new consequences of the assumption NP ⊂ P/poly, towards proving similar results for NP circuit lower bounds. We show that under the assumption, fixed-polynomial circuit lower bounds for NP, nondeterministic polynomial-time derandomizations, and various fixed-polynomial time simulations of NP are all equivalent. Applying this equivalence, we show that circuit lower bounds for NP imply better Karp-Lipton collapses. That is, if NP 6⊂ SIZE[nk] for all k, then for all C ∈ {P, PP, PSPACE, EXP}, C ⊂ P/poly implies C ⊂ i.o.-NP/nε for all ε > 0. Note that unconditionally, the collapses are only to MA and not NP. We also explore consequences of circuit lower bounds for a sparse language in NP. Among other results, we show if a polynomially-sparse NP language does not have n1+ε-size circuits, then MA ⊂ i.o.-NP/O(log n), MA ⊂ i.o.-PNP[O(log n)], and NEXP 6⊂ SIZE[2o(m)]. Finally, we observe connections between these results and the “hardness magnification” phenomena described in recent works.