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On Medium-Uniformity and Circuit Lower Bounds

Rahul Santhanam, Ryan Williams

2013
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2013 IEEE Conference on Computational Complexity
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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
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... 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.

doi:10.1109/ccc.2013.40
dblp:conf/coco/SanthanamW13
fatcat:mm4ls445rjd37mkuqv5sg2lqiy