An Efficient Coding Theorem via Probabilistic Representations and Its Applications

Authors Zhenjian Lu, Igor C. Oliveira

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Zhenjian Lu
  • University of Warwick, Coventry, UK
Igor C. Oliveira
  • University of Warwick, Coventry, UK


We are grateful to Lijie Chen for a suggestion that allowed us to improve the list size in Corollary 3. We also thank Rahul Santhanam and Rahul Ilango for discussions on search-to-decision reductions, and James Maynard for an email exchange on number-theoretic techniques and the problem of generating primes. Igor would like to thank Michal Koucký for asking a question about the probability threshold in the definition of rKt during the Dagstuhl Seminar "Computational Complexity of Discrete Problems" (2019). This work received support from the Royal Society University Research Fellowship URF⧵R1⧵191059.

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Zhenjian Lu and Igor C. Oliveira. An Efficient Coding Theorem via Probabilistic Representations and Its Applications. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 94:1-94:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


A probabilistic representation of a string x ∈ {0,1}ⁿ is given by the code of a randomized algorithm that outputs x with high probability [Igor C. Oliveira, 2019]. We employ probabilistic representations to establish the first unconditional Coding Theorem in time-bounded Kolmogorov complexity. More precisely, we show that if a distribution ensemble 𝒟_m can be uniformly sampled in time T(m) and generates a string x ∈ {0,1}^* with probability at least δ, then x admits a time-bounded probabilistic representation of complexity O(log(1/δ) + log (T) + log(m)). Under mild assumptions, a representation of this form can be computed from x and the code of the sampler in time polynomial in n = |x|. We derive consequences of this result relevant to the study of data compression, pseudodeterministic algorithms, time hierarchies for sampling distributions, and complexity lower bounds. In particular, we describe an instance-based search-to-decision reduction for Levin’s Kt complexity [Leonid A. Levin, 1984] and its probabilistic analogue rKt [Igor C. Oliveira, 2019]. As a consequence, if a string x admits a succinct time-bounded representation, then a near-optimal representation can be generated from x with high probability in polynomial time. This partially addresses in a time-bounded setting a question from [Leonid A. Levin, 1984] on the efficiency of computing an optimal encoding of a string.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • computational complexity
  • randomized algorithms
  • Kolmogorov complexity


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