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Title: Search-To-Decision Reductions for Kolmogorov Complexity
{"Abstract":["A long-standing open problem dating back to the 1960s is whether there exists a search-to-decision reduction for the time-bounded Kolmogorov complexity problem - that is, the problem of determining whether the length of the shortest time-t program generating a given string x is at most s.\r\nIn this work, we consider the more "robust" version of the time-bounded Kolmogorov complexity problem, referred to as the GapMINKT problem, where given a size bound s and a running time bound t, the goal is to determine whether there exists a poly(t,|x|)-time program of length s+O(log |x|) that generates x. We present the first non-trivial search-to-decision reduction R for the GapMINKT problem; R has a running-time bound of 2^{ε n} for any ε > 0 and additionally only queries its oracle on "thresholds" s of size s+O(log |x|). As such, we get that any algorithm with running-time (resp. circuit size) 2^{α s} poly(|x|,t,s) for solving GapMINKT (given an instance (x,t,s), yields an algorithm for finding a witness with running-time (resp. circuit size) 2^{(α+ε) s} poly(|x|,t,s).\r\nOur second result is a polynomial-time search-to-decision reduction for the time-bounded Kolmogorov complexity problem in the average-case regime. Such a reduction was recently shown by Liu and Pass (FOCS'20), heavily relying on cryptographic techniques. Our reduction is more direct and additionally has the advantage of being length-preserving, and as such also applies in the exponential time/size regime.\r\nA central component in both of these results is the use of Kolmogorov and Levin’s Symmetry of Information Theorem."]}  more » « less
Award ID(s):
2149305
PAR ID:
10643224
Author(s) / Creator(s):
;
Editor(s):
Santhanam, Rahul
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
300
ISSN:
1868-8969
Page Range / eLocation ID:
34:1-34:20
Subject(s) / Keyword(s):
Kolmogorov complexity search to decision Theory of computation → Computational complexity and cryptography
Format(s):
Medium: X Size: 20 pages; 781484 bytes Other: application/pdf
Size(s):
20 pages 781484 bytes
Sponsoring Org:
National Science Foundation
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