Natural Proofs Versus Derandomization
  We study connections between Natural Proofs, derandomization, and the problem of proving "weak" circuit lower bounds such as ${\sf NEXP} \not\subset {\sf TC^0}$. 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:   $\bullet$ Constructivity is unavoidable, even for $\sf NEXP$ lower bounds. Informally, we prove for all "typical" non-uniform circuit classes ${\cal C}$, ${\sf NEXP} \not\subset {\cal C}$ if and only if there is a polynomial-time algorithm distinguishing some function from all functions computable by ${\cal C}$-circuits. Hence ${\sf NEXP} \not\subset {\cal C}$ is equivalent to exhibiting a constructive property useful against ${\cal C}$.   $\bullet$ There are no $\sf P$-natural properties useful against ${\cal C}$ if and only if randomized exponential time can be "derandomized" using truth tables of circuits from ${\cal C}$ as random seeds. Therefore the task of proving there are no $\sf P$-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, including improved ${\sf ACC}^0$ lower bounds and new unconditional derandomizations. 
Ryan Williams
cs.CC