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

Authors Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, Yuichi Yoshida

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Author Details

Mahdi Cheraghchi
  • Department of EECS, University of Michigan, Ann Arbor, MI, USA
Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Dimitrios Myrisiotis
  • Department of Computing, Imperial College London, London, UK
Yuichi Yoshida
  • National Institute of Informatics, Tokyo, Japan

Funding

  • Cheraghchi, Mahdi: M. Cheraghchi’s research is supported in part by the NSF award CCF-2006455.

Acknowledgements

We would like to express our gratitude to Emanuele Viola and Osamu Watanabe for bringing to our attention the works by Kalyanasundaram and Schnitger [Bala Kalyanasundaram and Georg Schnitger, 1992] and Watanabe [Osamu Watanabe, 1983], respectively, and for helpful discussions. In particular, we thank Emanuele Viola for explaining to us his works [Elad Haramaty et al., 2018; Emanuele Viola, 2019]. We thank Rahul Santhanam for pointing out that {N}ečiporuk’s method can be applied to not only MKtP but also MKTP. We thank Chin Ho Lee for answering our questions regarding his work [Chin Ho Lee, 2019]. We thank Paul Beame for bringing his work [Paul Beame et al., 2016] to our attention. We thank Valentine Kabanets, Zhenjian Lu, Igor C. Oliveira, and Ninad Rajgopal for illuminating discussions. Finally, we would like to thank the anonymous reviewers for their constructive feedback.

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Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, and Yuichi Yoshida. One-Tape Turing Machine and Branching Program Lower Bounds for MCSP. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 23:1-23:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.23

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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Minimum Circuit Size Problem
  • Kolmogorov Complexity
  • One-Tape Turing Machines
  • Branching Programs
  • Lower Bounds
  • Pseudorandom Generators
  • Hitting Set Generators

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