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 NC
¹
in ACC
⁰
/poly or TC
⁰
/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 TC
⁰
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 deterministically in 2
^{n-ω(log n)}
time, then NEXP ⊄ TC
⁰
/poly. The lemma can also be used to derandomize randomized TC
⁰
simulations of NC
¹
on almost all inputs: ; Suppose NC
¹
⊆ BPTC
⁰
. Then for every ε > 0 and every language L in NC
¹
, there is a (uniform) TC
⁰
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.