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DOI: 10.4230/LIPIcs.STACS.2023.19

One Drop of Non-Determinism in a Random Deterministic Automaton

Authors: Arnaud Carayol, Philippe Duchon, Florent Koechlin, and Cyril Nicaud

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract

Every language recognized by a non-deterministic finite automaton can be recognized by a deterministic automaton, at the cost of a potential increase of the number of states, which in the worst case can go from n states to 2ⁿ states. In this article, we investigate this classical result in a probabilistic setting where we take a deterministic automaton with n states uniformly at random and add just one random transition. These automata are almost deterministic in the sense that only one state has a non-deterministic choice when reading an input letter. In our model each state has a fixed probability to be final. We prove that for any d ≥ 1, with non-negligible probability the minimal (deterministic) automaton of the language recognized by such an automaton has more than n^d states; as a byproduct, the expected size of its minimal automaton grows faster than any polynomial. Our result also holds when each state is final with some probability that depends on n, as long as it is not too close to 0 and 1, at distance at least Ω(1/√n) to be precise, therefore allowing models with a sublinear number of final states in expectation.

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Arnaud Carayol, Philippe Duchon, Florent Koechlin, and Cyril Nicaud. One Drop of Non-Determinism in a Random Deterministic Automaton. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{carayol_et_al:LIPIcs.STACS.2023.19,
  author =	{Carayol, Arnaud and Duchon, Philippe and Koechlin, Florent and Nicaud, Cyril},
  title =	{{One Drop of Non-Determinism in a Random Deterministic Automaton}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{19:1--19:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.19},
  URN =		{urn:nbn:de:0030-drops-176719},
  doi =		{10.4230/LIPIcs.STACS.2023.19},
  annote =	{Keywords: non-deterministic automaton, powerset construction, probabilistic analysis}
}

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DOI: 10.4230/LIPIcs.CPM.2017.21

Gapped Pattern Statistics

Authors: Philippe Duchon, Cyril Nicaud, and Carine Pivoteau

Published in: LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

Abstract

We give a probabilistic analysis of parameters related to alpha-gapped repeats and palindromes in random words, under both uniform and memoryless distributions (where letters have different probabilities, but are drawn independently). More precisely, we study the expected number of maximal alpha-gapped patterns, as well as the expected length of the longest alpha-gapped pattern in a random word.

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Philippe Duchon, Cyril Nicaud, and Carine Pivoteau. Gapped Pattern Statistics. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 21:1-21:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{duchon_et_al:LIPIcs.CPM.2017.21,
  author =	{Duchon, Philippe and Nicaud, Cyril and Pivoteau, Carine},
  title =	{{Gapped Pattern Statistics}},
  booktitle =	{28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)},
  pages =	{21:1--21:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-039-2},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{78},
  editor =	{K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.21},
  URN =		{urn:nbn:de:0030-drops-73309},
  doi =		{10.4230/LIPIcs.CPM.2017.21},
  annote =	{Keywords: combinatorics on words, alpha-gapped repeats, random words, memoryless sources, analytic combinatorics}
}

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One Drop of Non-Determinism in a Random Deterministic Automaton

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Gapped Pattern Statistics

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