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Sublinear-Time Probabilistic Cellular Automata

Author Augusto Modanese



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Augusto Modanese
  • Aalto University, Espoo, Finland

Acknowledgements

I would like to thank Thomas Worsch for the helpful discussions and feedback as well as the anonymous reviewers for their insightful comments.

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Augusto Modanese. Sublinear-Time Probabilistic Cellular Automata. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 47:1-47:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.47

Abstract

We propose and investigate a probabilistic model of sublinear-time one-dimensional cellular automata. In particular, we modify the model of ACA (which are cellular automata that accept if and only if all cells simultaneously accept) so that every cell changes its state not only dependent on the states it sees in its neighborhood but also on an unbiased coin toss of its own. The resulting model is dubbed probabilistic ACA (PACA). We consider one- and two-sided error versions of the model (in the same spirit as the classes RP and BPP) and establish a separation between the classes of languages they can recognize all the way up to o(√n) time. As a consequence, we have a Ω(√n) lower bound for derandomizing constant-time one-sided error PACAs (using deterministic ACAs). We also prove that derandomization of T(n)-time PACAs (to polynomial-time deterministic cellular automata) for various regimes of T(n) = ω(log n) implies non-trivial derandomization results for the class RP (e.g., P = RP). The main contribution is an almost full characterization of the constant-time PACA classes: For one-sided error, the class equals that of the deterministic model; that is, constant-time one-sided error PACAs can be fully derandomized with only a constant multiplicative overhead in time complexity. As for two-sided error, we identify a natural class we call the linearly testable languages (LLT) and prove that the languages decidable by constant-time two-sided error PACAs are "sandwiched" in-between the closure of LLT under union and intersection and the class of locally threshold testable languages (LTT).

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • Cellular automata
  • local computation
  • probabilistic models
  • subregular language classes

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