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• W4313227250 abstract "We consider the range avoidance problem (called Avoid): given the description of a circuit with more output gates than input gates, find a string that is not in the range of the circuit. This problem is complete for the class APEPP that corresponds to explicit constructions of objects whose existence follows from the probabilistic method (Korten, FOCS 2021). Motivated by applications in explicit constructions and complexity theory, we initiate the study of the range avoidance problem for weak circuit classes, and obtain the following results: 1)Generalising Williams’s connections between circuitanalysis algorithms and circuit lower bounds (J. ACM 2014), we present a framework for solving $mathscr{C}$-Avoid in FP <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>NP</sup> using circuit-analysis data structures for $mathscr{C}$, for “typical” multi-output circuit classes $mathscr{C}$. As an application, we present a non-trivial FP <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>NP</sup> range avoidance algorithm for De Morgan formulas./inlp>An important technical ingredient is a construction of rectangular PCPs of proximity, building on the rectangular PCPs by Bhangale, Harsha, Paradise, and Tal (FOCS 2020).2)Using the above framework, we show that circuit lower bounds for E <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>NP</sup> are equivalent to circuit-analysis algorithms with E <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>NP</sup> preprocessing. This is the first equivalence result regarding circuit lower bounds for E <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>NP</sup> . Our equivalences have the additional advantages that they work in both infinitely-often and almost-everywhere settings, and that they also hold for larger (e.g., subexponential) size bounds.3)Complementing the above results, we show that in some settings, solving $mathscr{C}$-Avoid would imply breakthrough lower bounds, even for very weak circuit classes $mathscr{C}$. In particular, an algorithm for AC <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>0</sup> -Avoid with polynomial stretch implies lower bounds against NC <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sup> , and an algorithm for $NC_{4}^{0}$-Avoid with very small stretch implies lower bounds against NC <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sup> and branching programs.4)We show that Avoid is in FNP if and only if there is a propositional proof system that breaks every non-uniform proof complexity generator. This result connects the study of range avoidance with fundamental questions in proof complexity." @default.
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• W4313227250 date "2022-10-01" @default.
• W4313227250 modified "2023-10-16" @default.
• W4313227250 title "On the Range Avoidance Problem for Circuits" @default.
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• W4313227250 doi "https://doi.org/10.1109/focs54457.2022.00067" @default.
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