### On Medium-Uniformity and Circuit Lower Bounds

Rahul Santhanam, Ryan Williams
2013 2013 IEEE Conference on Computational Complexity
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
more » ... 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 n ε inputs of length n, for all n.