Abstract
When we represent a decision problem, like CIRCUIT-SAT, as a language over the binary alphabet, we usually do not specify how to encode instances by binary strings. This relies on the empirical observation that the truth of a statement of the form “CIRCUIT-SAT belongs to a complexity class C” does not depend on the encoding, provided both the encoding and the class C are “natural”. In this sense most of the Complexity theory is “encoding invariant”.
The notion of a polynomial time computable distribution from Average Case Complexity is one of the exceptions from this rule. It might happen that a distribution over some objects, like circuits, is polynomial time computable in one encoding and is not polynomial time computable in the other encoding. In this paper we suggest an encoding invariant generalization of a notion of a polynomial time computable distribution. The completeness proofs of known distributional problems, like Bounded Halting, are simpler for the new class than for polynomial time computable distributions.
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Notes
And they are not provided one-way permutations exist, see Theorem 4.
To find the largest standard segment inside a given segment, we can use binary search.
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Acknowledgements
The author is grateful to Alexander Shen for bringing his attention to the fact that PComp might be not encoding invariant. The author is grateful to Dmitry Itsykson and to anonymous referees for a number of helpful suggestions.
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This is the full version of the paper “An Encoding Invariant Version of Polynomial Time Computable Distributions” appeared in Proceedings of the 5th International Computer Science Symposium in Russia, CSR 2010, Kazan, Russia, June 16–20, 2010. Lecture Notes in Computer Science, vol. 6072, pp. 371–383.
The work was in part supported by a RFBR grant 12-01-00864.
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Vereshchagin, N. Encoding Invariance in Average Case Complexity. Theory Comput Syst 54, 305–317 (2014). https://doi.org/10.1007/s00224-013-9517-5
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DOI: https://doi.org/10.1007/s00224-013-9517-5