@MISC{Williams_naturalproofs, author = {Ryan Williams}, title = {Natural Proofs Versus Derandomization}, year = {} }

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Abstract

We study connections between Natural Proofs, derandomization, and the problem of proving “weak” circuit lower bounds such as NEXP ⊂ TC 0, which are still wide open. Natural Proofs have three properties: they are constructive (an efficient algorithm A is embedded in them), have largeness (A accepts a large fraction of strings), and are useful (A rejects all strings which are truth tables of small circuits). Strong circuit lower bounds that are “naturalizing ” would contradict present cryptographic understanding, yet the vast majority of known circuit lower bound proofs are naturalizing. So it is imperative to understand how to pursue un-Natural Proofs. Some heuristic arguments say constructivity should be circumventable. Largeness is inherent in many proof techniques, and it is probably our presently weak techniques that yield constructivity. We prove: • Constructivity is unavoidable, even for NEXP lower bounds. Informally, we prove for all “typical” non-uniform circuit classes C, NEXP ⊂ C if and only if there is a polynomial-time algorithm distinguishing some function from all functions computable by C-circuits. Hence NEXP ⊂ C is equivalent to exhibiting a constructive property useful against C. • There are no P-natural properties useful against C if and only if randomized exponential time can be “derandomized ” using truth tables of circuits from C as random seeds. Therefore the task of proving there are no P-natural properties is inherently a derandomization problem, weaker than but implied by the existence of strong pseudorandom functions. These characterizations are applied to yield several new results. The two main applications are that NEXP ∩ coNEXP does not have n log n size ACC circuits, and a mild derandomization result for RP. 1