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One-Tape Turing Machine and Branching Program Lower Bounds for MCSP

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Abstract

For a size parameter s: ℕ → ℕ, the Minimum Circuit Size Problem (denoted by MCSP[s(n)]) is the problem of deciding whether the minimum circuit size of a given function f : {0,1}ⁿ → {0,1} (represented by a string of length N : = 2ⁿ) is at most a threshold s(n). A recent line of work exhibited "hardness magnification" phenomena for MCSP: A very weak lower bound for MCSP implies a breakthrough result in complexity theory. For example, McKay, Murray, and Williams (STOC 2019) implicitly showed that, for some constant μ₁ > 0, if MCSP[2^{μ₁⋅ n}] cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time N^{1.01}, then P≠NP. In this paper, we present the following new lower bounds against one-tape Turing machines and branching programs: 1) A randomized two-sided error one-tape Turing machine (with an additional one-way read-only input tape) cannot compute MCSP[2^{μ₂⋅n}] in time N^{1.99}, for some constant μ₂ > μ₁. 2) A non-deterministic (or parity) branching program of size o(N^{1.5}/log N) cannot compute MKTP, which is a time-bounded Kolmogorov complexity analogue of MCSP. This is shown by directly applying the Nečiporuk method to MKTP, which previously appeared to be difficult. 3) The size of any non-deterministic, co-non-deterministic, or parity branching program computing MCSP is at least N^{1.5-o(1)}. These results are the first non-trivial lower bounds for MCSP and MKTP against one-tape Turing machines and non-deterministic branching programs, and essentially match the best-known lower bounds for any explicit functions against these computational models. The first result is based on recent constructions of pseudorandom generators for read-once oblivious branching programs (ROBPs) and combinatorial rectangles (Forbes and Kelley, FOCS 2018; Viola 2019). En route, we obtain several related results: 1) There exists a (local) hitting set generator with seed length Õ(√N) secure against read-once polynomial-size non-deterministic branching programs on N-bit inputs. 2) Any read-once co-non-deterministic branching program computing MCSP must have size at least 2^Ω̃(N).

BibTeX - Entry

@InProceedings{cheraghchi_et_al:LIPIcs.STACS.2021.23,
  author =	{Cheraghchi, Mahdi and Hirahara, Shuichi and Myrisiotis, Dimitrios and Yoshida, Yuichi},
  title =	{{One-Tape Turing Machine and Branching Program Lower Bounds for MCSP}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{23:1--23:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13668},
  URN =		{urn:nbn:de:0030-drops-136681},
  doi =		{10.4230/LIPIcs.STACS.2021.23},
  annote =	{Keywords: Minimum Circuit Size Problem, Kolmogorov Complexity, One-Tape Turing Machines, Branching Programs, Lower Bounds, Pseudorandom Generators, Hitting Set Generators}
}

Keywords: Minimum Circuit Size Problem, Kolmogorov Complexity, One-Tape Turing Machines, Branching Programs, Lower Bounds, Pseudorandom Generators, Hitting Set Generators
Seminar: 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)
Issue date: 2021
Date of publication: 10.03.2021


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