91 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Unprovability of strong complexity lower bounds in bounded arithmetic

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    While there has been progress in establishing the unprovability of complexity statements in lower fragments of bounded arithmetic, understanding the limits of Jeˇr ́abek’s theory APC1 [Jeˇr07a] and of higher levels of Buss’s hierarchy Si 2 [Bus86] has been a more elusive task. Even in the more restricted setting of Cook’s theory PV [Coo75], known results often rely on a less natural formalization that encodes a complexity statement using a collection of sentences instead of a single sentence. This is done to reduce the quantifier complexity of the resulting sentences so that standard witnessing results can be invoked. In this work, we establish unprovability results for stronger theories and for sentences of higher quantifier complexity. In particular, we unconditionally show that APC1 cannot prove strong complexity lower bounds separating the third level of the polynomial hierarchy. In more detail, we consider non-uniform average-case separations, and establish that APC1 cannot prove a sentence stating that ∀n ≥ n0 ∃ fn ∈ Π3-SIZE[nd] that is (1/n)-far from every Σ3-SIZE[2nδ] circuit. This is a consequence of a much more general result showing that, for every i ≥ 1, strong separations for Πi-SIZE[poly(n)] versus Σi-SIZE[2nΩ(1)] cannot be proved in the theory Ti PV consisting of all true ∀Σb i−1- sentences in the language of Cook’s theory PV. Our argument employs a convenient game-theoretic witnessing result that can be applied to sentences of arbitrary quantifier complexity. We combine it with extensions of a technique introduced by Kraj ́ıˇcek [Kra11] that was recently employed by Pich and Santhanam [PS21] to establish the unprovability of lower bounds in PV (i.e., the case i = 1 above, but under a weaker formalization) and in a fragment of APC1

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Approximate degree in classical and quantum computing

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    In this book, the authors survey what is known about a particularly natural notion of approximation by polynomials, capturing pointwise approximation over the real numbers.FG-2022-18482 - Alfred P. Sloan Foundation; CNS-2046425 - National Science Foundation; CCF-1947889 - National Science FoundationAccepted manuscrip

    Computer Aided Verification

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    This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum
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