2026-07-23T13:46:53Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16433 2024-03-06T10:57:28Z ddc:004 open_access

Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity Lu, Zhenjian Oliveira, Igor C. Zimand, Marius computational complexity randomized algorithms Kolmogorov complexity The classical coding theorem in Kolmogorov complexity states that if an n-bit string x is sampled with probability δ by an algorithm with prefix-free domain then 𝖪(x) ≤ log(1/δ) + O(1). In a recent work, Lu and Oliveira [Zhenjian Lu and Igor C. Oliveira, 2021] established an unconditional time-bounded version of this result, by showing that if x can be efficiently sampled with probability δ then rKt(x) = O(log(1/δ)) + O(log n), where rKt denotes the randomized analogue of Levin’s Kt complexity. Unfortunately, this result is often insufficient when transferring applications of the classical coding theorem to the time-bounded setting, as it achieves a O(log(1/δ)) bound instead of the information-theoretic optimal log(1/δ). Motivated by this discrepancy, we investigate optimal coding theorems in the time-bounded setting. Our main contributions can be summarised as follows. • Efficient coding theorem for rKt with a factor of 2. Addressing a question from [Zhenjian Lu and Igor C. Oliveira, 2021], we show that if x can be efficiently sampled with probability at least δ then rKt(x) ≤ (2 + o(1)) ⋅ log(1/δ) + O(log n). As in previous work, our coding theorem is efficient in the sense that it provides a polynomial-time probabilistic algorithm that, when given x, the code of the sampler, and δ, it outputs, with probability ≥ 0.99, a probabilistic representation of x that certifies this rKt complexity bound. • Optimality under a cryptographic assumption. Under a hypothesis about the security of cryptographic pseudorandom generators, we show that no efficient coding theorem can achieve a bound of the form rKt(x) ≤ (2 - o(1)) ⋅ log(1/δ) + poly(log n). Under a weaker assumption, we exhibit a gap between efficient coding theorems and existential coding theorems with near-optimal parameters. • Optimal coding theorem for pK^t and unconditional Antunes-Fortnow. We consider pK^t complexity [Halley Goldberg et al., 2022], a variant of rKt where the randomness is public and the time bound is fixed. We observe the existence of an optimal coding theorem for pK^t, and employ this result to establish an unconditional version of a theorem of Antunes and Fortnow [Luis Filipe Coelho Antunes and Lance Fortnow, 2009] which characterizes the worst-case running times of languages that are in average polynomial-time over all 𝖯-samplable distributions. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Zhenjian Lu and Igor C. Oliveira and Marius Zimand 2022 Is Part Of LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2022.92 urn:nbn:de:0030-drops-164331 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.92 eng https://creativecommons.org/licenses/by/4.0/legalcode