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- W2343516500 abstract "We study connections between the Natural Proofs of Razborov and Rudich, derandomization, and the problem of proving “weak” circuit lower bounds such as ${sf NEXP} notsubset {sf TC}^0$, 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 the following: (i) Constructivity is unavoidable, even for ${sf NEXP}$ lower bounds. Informally, we prove for all “typical” nonuniform circuit classes ${cal C}$, ${sf NEXP} notsubset {cal C}$ if and only if there is a polynomial-time algorithm distinguishing some function from all functions computable by ${cal C}$-circuits. Hence ${sf NEXP} notsubset {cal C}$ is equivalent to exhibiting a constructive property useful against ${cal C}$. (ii) 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. In general, we develop and apply several new connections between the existence of certain algorithms for analyzing truth tables, and the nonexistence of small circuits for problems in large classes such as ${sf NEXP}$." @default.
- W2343516500 created "2016-06-24" @default.
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- W2343516500 date "2016-01-01" @default.
- W2343516500 modified "2023-10-18" @default.
- W2343516500 title "Natural Proofs versus Derandomization" @default.
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- W2343516500 doi "https://doi.org/10.1137/130938219" @default.
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