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Documents authored by Kebis, Pavol

Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.144
On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words

Authors: Pavol Kebis, Florian Luca, Joël Ouaknine, Andrew Scoones, and James Worrell

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We consider numbers of the form S_β(u): = ∑_{n=0}^∞ (u_n)/(βⁿ), where u = ⟨u_n⟩_{n=0}^∞ is an infinite word over a finite alphabet and β ∈ ℂ satisfies |β| > 1. Our main contribution is to present a combinatorial criterion on u, called echoing, that implies that S_β(u) is transcendental whenever β is algebraic. We show that every Sturmian word is echoing, as is the Tribonacci word, a leading example of an Arnoux-Rauzy word. We furthermore characterise ̅{ℚ}-linear independence of sets of the form {1, S_β(u₁),…,S_β(u_k)}, where u₁,…,u_k are Sturmian words having the same slope. Finally, we give an application of the above linear independence criterion to the theory of dynamical systems, showing that for a contracted rotation on the unit circle with algebraic slope, its limit set is either finite or consists exclusively of transcendental elements other than its endpoints 0 and 1. This confirms a conjecture of Bugeaud, Kim, Laurent, and Nogueira.
Cite as

Pavol Kebis, Florian Luca, Joël Ouaknine, Andrew Scoones, and James Worrell. On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 144:1-144:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kebis_et_al:LIPIcs.ICALP.2024.144,
  author =	{Kebis, Pavol and Luca, Florian and Ouaknine, Jo\"{e}l and Scoones, Andrew and Worrell, James},
  title =	{{On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{144:1--144:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.144},
  URN =		{urn:nbn:de:0030-drops-202873},
  doi =		{10.4230/LIPIcs.ICALP.2024.144},
  annote =	{Keywords: Transcendence, Subspace Theorem, Fibonacci Word, Tribonacci Word}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2023.129
Regular Methods for Operator Precedence Languages

Authors: Thomas A. Henzinger, Pavol Kebis, Nicolas Mazzocchi, and N. Ege Saraç

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
The operator precedence languages (OPLs) represent the largest known subclass of the context-free languages which enjoys all desirable closure and decidability properties. This includes the decidability of language inclusion, which is the ultimate verification problem. Operator precedence grammars, automata, and logics have been investigated and used, for example, to verify programs with arithmetic expressions and exceptions (both of which are deterministic pushdown but lie outside the scope of the visibly pushdown languages). In this paper, we complete the picture and give, for the first time, an algebraic characterization of the class of OPLs in the form of a syntactic congruence that has finitely many equivalence classes exactly for the operator precedence languages. This is a generalization of the celebrated Myhill-Nerode theorem for the regular languages to OPLs. As one of the consequences, we show that universality and language inclusion for nondeterministic operator precedence automata can be solved by an antichain algorithm. Antichain algorithms avoid determinization and complementation through an explicit subset construction, by leveraging a quasi-order on words, which allows the pruning of the search space for counterexample words without sacrificing completeness. Antichain algorithms can be implemented symbolically, and these implementations are today the best-performing algorithms in practice for the inclusion of finite automata. We give a generic construction of the quasi-order needed for antichain algorithms from a finite syntactic congruence. This yields the first antichain algorithm for OPLs, an algorithm that solves the ExpTime-hard language inclusion problem for OPLs in exponential time.
Cite as

Thomas A. Henzinger, Pavol Kebis, Nicolas Mazzocchi, and N. Ege Saraç. Regular Methods for Operator Precedence Languages. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 129:1-129:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{henzinger_et_al:LIPIcs.ICALP.2023.129,
  author =	{Henzinger, Thomas A. and Kebis, Pavol and Mazzocchi, Nicolas and Sara\c{c}, N. Ege},
  title =	{{Regular Methods for Operator Precedence Languages}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{129:1--129:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.129},
  URN =		{urn:nbn:de:0030-drops-181814},
  doi =		{10.4230/LIPIcs.ICALP.2023.129},
  annote =	{Keywords: operator precedence automata, syntactic congruence, antichain algorithm}
}
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