On Medium-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: 1. 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(n^k). That is, for all k there is a language L_k ∈ P that does not have O(n^k)-size circuits constructible in polynomial time. This improves Kannan´s lower bound from 1982 that NP is not in P-uniform SIZE(n^k) for any fixed k. ; For all k, NP is not in P_||^NP-uniform SIZE(n^k). 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 n^k. 2. Eliminating Non-Uniformity and (Non-Uniform) Circuit Lower Bounds. We complement these results by showing how to convert any potential simulation of LOGTIME-uniform NC¹ 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 a TC⁰ circuit C of n^O(1) size, output yes when C is unsatisfiable, and output no when C has at least 2^n-2 satisfying assignments. (Behavior on other inputs can be arbitrary.) Clearly, this problem can be solved efficiently using randomness. If this problem can be solved deterministical- y in 2^{n-ω(log n)} time, then NEXP ⊄ TC⁰/poly. The lemma can also be used to derandomize randomized TC⁰ simulations of NC¹ on almost all inputs: ; Suppose NC¹ ⊆ BPTC⁰. Then for every ε > 0 and every language L in NC¹, there is a (uniform) TC⁰ circuit family of polynomial size recognizing a language L´ such that L and L´ differ on at most 2^nϵ inputs of length n, for all n.