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

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Abstract

For a size parameter \(s:\mathbb {N}\to \mathbb {N}\), 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 \(\text {MCSP}[2^{\mu _{1}\cdot n}]\) cannot be computed by a one-tape Turing machine (with an additional one-way read-only input tape) running in time N1.01, 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 \(\text {MCSP}[2^{\mu _{2}\cdot n}]\) in time N1.99, 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 \(N^{1.5-o\left (1\right )}\). 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, Electron. Colloq. Comput. Complexity (ECCC) 26, 51, 2019). En route, we obtain several related results: (1) There exists a (local) hitting set generator with seed length \(\widetilde {O}(\sqrt {N})\) 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 \(2^{\widetilde {\Omega }(N)}\).

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Notes

  1. Observe that ∃μ, (P(μ) ⇒ Q) is logically equivalent to ∃μ, (¬P(μ) ∨ Q), which is equivalent to ¬(∀μ,P(μ)) ∨ Q.

  2. It is worthy of note that Theorem 4 mildly improves the lower bounds of [12, 36] to \({\Omega }(N^{2} / \log ^{2} N)\) by directly applying Nečiporuk’s method, which matches the state-of-the-art lower bound for any explicit function up to a constant factor.

  3. We emphasize that the notion of PRGs secure against these three computational models is different. See Definitions 14, 17, and Lemma 19.

  4. It should be noted that before Haramaty, Lee, and Viola [17] and Viola [42], the problem of designing PRGs of polynomial stretch that fool RTMs was wide open despite intense research efforts.

  5. We note that our definition of PRG is different from that of [42] in that a random tape is not regarded as an input tape.

  6. Let U be an efficient universal Turing machine. For a string x ∈{0, 1}∗, the resource-unbounded Kolmogorov complexity of x is defined as \(\mathrm {K}(x):=\min \limits \{|d|\mid U^d(i) \text {outputs} x_i \text {for every} i\in [|x|+1]\}\).

  7. Here we assume that the universal Turing machine is efficient. If the universal Turing machine is slower and the time is polylog(n), we obtain a branching program size lower bound of n2/polylog(n).

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Acknowledgements

This work was done while Dimitrios Myrisiotis was with the Department of Computing, Imperial College London, London, UK. We would like to express our gratitude to Emanuele Viola and Osamu Watanabe for bringing to our attention the works by Kalyanasundaram and Schnitger [28] and Watanabe [43], respectively, and for helpful discussions. In particular, we thank Emanuele Viola for explaining to us his works [17, 42]. We thank Rahul Santhanam for pointing out that Nečiporuk’s method can be applied to not only MKtP but also MKTP. We thank Chin Ho Lee for answering our questions regarding his work [29]. We thank Paul Beame for bringing his work [7] 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.

Funding

Mahdi Cheraghchi: M. Cheraghchi’s research is supported in part by the NSF awards CCF-2006455 and CCF-2107345.

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Authors and Affiliations

  1. Department of EECS, University of Michigan, Ann Arbor, MI, USA

    Mahdi Cheraghchi

  2. National Institute of Informatics, Tokyo, Japan

    Shuichi Hirahara & Yuichi Yoshida

  3. School of Computing, National University of Singapore, Singapore, Singapore

    Dimitrios Myrisiotis

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  1. Mahdi Cheraghchi
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  2. Shuichi Hirahara
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Correspondence to Dimitrios Myrisiotis.

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Appendix A: Hardness Magnification for One-Tape Turing Machines

Appendix A: Hardness Magnification for One-Tape Turing Machines

In this section, we obtain the following hardness magnification result for one-tape Turing machines.

Theorem 59 (A corollary of McKay, Murray, and Williams [32]; Theorem 1, restated)

There exists a constant μ > 0 such that, if MCSP[2μn] is not in \(\mathsf {DTIME}_{1}\!\left [N^{1.01}\right ]\), then P≠NP.

Proof

McKay, Murray, and Williams [32, Theorem 1.3] showed that if P = NP, then there exists a polynomial p such that, for any time-constructible function s(n), there exists a one-pass streaming algorithm with update time p(s(n)) that computes MCSP[s(n)]. By Lemma 10, we obtain MCSP[s(n)] ∈DTIME1[N ⋅ p(s(n))], where N = 2n. Depending on p, we choose a small constant μ > 0 and set s(n) := 2μn so that N ⋅ p(s(n)) = N1+O(μ) ≤ N1.01.

To summarize, we have proved that if P = NP, then for some constant μ > 0, \(\text {MCSP}[2^{\mu n}] \in \mathsf {DTIME}_{1}[N^{1.01}]\). This statement is logically equivalent to the following: There exists a constant μ > 0 such that P = NP implies that \(\text {MCSP}[2^{\mu n}] \not \in \mathsf {DTIME}_{1}[N^{1.01}]\) (because the statement that P = NP is independent of μ). Taking its contrapositive, we obtain the desired result. □

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Cheraghchi, M., Hirahara, S., Myrisiotis, D. et al. One-Tape Turing Machine and Branching Program Lower Bounds for MCSP. Theory Comput Syst (2022). https://doi.org/10.1007/s00224-022-10113-9

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