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Title: On Black-Box Meta Complexity and Function Inversion
{"Abstract":["The relationships between various meta-complexity problems are not well understood in the worst-case regime, including whether the search version is harder than the decision version, whether the hardness scales with the "threshold", and how the hardness of different meta-complexity problems relate to one another, and to the task of function inversion.\r\nIn this work, we present resolutions to some of these questions with respect to the black-box analog of these problems. In more detail, let MK^t_M P[s] denote the language consisting of strings x with K_{M}^t(x) < s(|x|), where K_M^t(x) denotes the t-bounded Kolmogorov complexity of x with M as the underlying (Universal) Turing machine, and let search-MK^t_M P[s] denote the search version of the same problem.\r\nWe show that if for every Universal Turing machine U there exists a 2^{α n}poly(n)-size U-oracle aided circuit deciding MK^t_U P[n-O(1)], then for every function s, and every not necessarily universal Turing machine M, there exists a 2^{α s(n)}poly(n)-size M-oracle aided circuit solving search-MK^t_M P[s(n)]; this in turn yields circuits of roughly the same size for both the Minimum Circuit Size Problem (MCSP), and the function inversion problem, as they can be thought of as instantiating MK^t_M P with particular choices of (a non-universal) TMs M (the circuit emulator for the case of MCSP, and the function evaluation in the case of function inversion).\r\nAs a corollary of independent interest, we get that the complexity of black-box function inversion is (roughly) the same as the complexity of black-box deciding MK^t_U P[n-O(1)] for any universal TM U; that is, also in the worst-case regime, black-box function inversion is "equivalent" to black-box deciding MK^t_U P."]}  more » « less
Award ID(s):
2149305
PAR ID:
10643223
Author(s) / Creator(s):
;
Editor(s):
Kumar, Amit; Ron-Zewi, Noga
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
317
ISSN:
1868-8969
Page Range / eLocation ID:
66:1-66:12
Subject(s) / Keyword(s):
Meta Complexity Kolmogorov complexity function inversion Theory of computation → Computational complexity and cryptography
Format(s):
Medium: X Size: 12 pages; 738545 bytes Other: application/pdf
Size(s):
12 pages 738545 bytes
Sponsoring Org:
National Science Foundation
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