On Uniformity and Circuit Lower Bounds

We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts: Lower bounds against medium-uniform circuits. Informally, a circuit class is “medium uniform” if it can be generated by an algorithmic process that is somewhat complex (stronger than LOGTIME) but not infeasible. Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against medium-uniform circuit classes, including:◦ For all k, P is not contained in P-uniform SIZE(nk). That is, for all k, there is a language Lk∈P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L_k \in {\textsf P}}$$\end{document} that does not have O(nk)-size circuits constructible in polynomial time. This improves Kannan’s lower bound from 1982 that NP is not in P-uniform SIZE(nk) for any fixed k.◦ For all k, NP is not in P||NP-uniformSIZE(nk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textsf P}^{\textsf NP}_{||}-{\textsf {uniform SIZE}}(n^k)}$$\end{document}.This also improves Kannan’s theorem, but in a different way: the uniformity condition on the circuits is stronger than that on the language itself.◦ For all k, LOGSPACE does not have LOGSPACE-uniform branching programs of size nk.Eliminating non-uniformity and (non-uniform) circuit lower bounds. We complement these results by showing how to convert any potential simulation of LOGTIME-uniformNC1 in ACC0/poly or TC0/poly into a medium-uniform simulation using small advice. This lemma can be used to simplify the proof that faster SAT algorithms imply NEXP circuit lower bounds and leads to the following new connection:◦ Consider the following task: given aTC0circuit C of nO(1)size, output yes when C is unsatisfiable, and output no when C has at least 2n-2satisfying assignments. (Behavior on other inputs can be arbitrary.) Clearly, this problem can be solved efficiently using randomness. If this problem can be solved deterministically in 2n-ω(logn) time, then NEXP⊄TC0/poly\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textsf{NEXP}} \not \subset {\textsf{TC}}^0/{\rm poly}}$$\end{document}.Another application is to derandomize randomized TC0 simulations of NC1 on almost all inputs: