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Documents authored by Idir, Olivier

Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.160
Using Games and Universal Trees to Characterise the Nondeterministic Index of Tree Languages

Authors: Olivier Idir and Karoliina Lehtinen

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
The parity index problem of tree automata asks, given a regular tree language L and a set of priorities J, is L J-feasible, that is, recognised by a nondeterministic parity automaton with priorities J? This is a long-standing open problem, of which only a few sub-cases and variations are known to be decidable. In a significant but technically difficult step, Colcombet and Löding reduced the problem to the uniform universality of distance-parity automata. In this article, we revisit the index problem using tools from the parity game literature. We add some counters to Lehtinen’s register game, originally used to solve parity games in quasipolynomial time, and use this novel game to characterise J-feasibility. This provides a alternative proof to Colcombet and Löding’s reduction. We then provide a second characterisation, based on the notion of attractor decompositions and the complexity of their structure, as measured by a parameterised version of their Strahler number, which we call n-Strahler number. Finally, we rephrase this result using the notion of universal tree extended to automata: a guidable automaton recognises a [1,2j]-feasible language if and only if it admits a universal tree with n-Strahler number j, for some n. In particular, a language recognised by a guidable automaton {A} is Büchi-feasible if and only if there is a uniform bound n ∈ ℕ such that all trees in the language admit an accepting run with an attractor decomposition of width bounded by n. Equivalently, the language is Büchi-feasible if and only if {A} admits a finite universal tree. While we do not solve the decidability of the index problem, our work makes the state-of-the-art more accessible and brings to light the deep relationships between the J-feasibility of a language and attractor decompositions, universal trees and Lehtinen’s register game.
Cite as

Olivier Idir and Karoliina Lehtinen. Using Games and Universal Trees to Characterise the Nondeterministic Index of Tree Languages. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 160:1-160:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{idir_et_al:LIPIcs.ICALP.2025.160,
  author =	{Idir, Olivier and Lehtinen, Karoliina},
  title =	{{Using Games and Universal Trees to Characterise the Nondeterministic Index of Tree Languages}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{160:1--160:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l 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.2025.160},
  URN =		{urn:nbn:de:0030-drops-235377},
  doi =		{10.4230/LIPIcs.ICALP.2025.160},
  annote =	{Keywords: Tree automata, parity automata, Mostowski index, Strahler number, attractor decomposition, universal trees}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.22
On the Minimisation of Deterministic and History-Deterministic Generalised (Co)Büchi Automata

Authors: Antonio Casares, Olivier Idir, Denis Kuperberg, Corto Mascle, and Aditya Prakash

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
We present a polynomial-time algorithm minimising the number of states of history-deterministic generalised coBüchi automata, building on the work of Abu Radi and Kupferman on coBüchi automata. On the other hand, we establish that the minimisation problem for both deterministic and history-deterministic generalised Büchi automata is NP-complete, as well as the problem of minimising at the same time the number of states and colours of history-deterministic generalised coBüchi automata.
Cite as

Antonio Casares, Olivier Idir, Denis Kuperberg, Corto Mascle, and Aditya Prakash. On the Minimisation of Deterministic and History-Deterministic Generalised (Co)Büchi Automata. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{casares_et_al:LIPIcs.CSL.2025.22,
  author =	{Casares, Antonio and Idir, Olivier and Kuperberg, Denis and Mascle, Corto and Prakash, Aditya},
  title =	{{On the Minimisation of Deterministic and History-Deterministic Generalised (Co)B\"{u}chi Automata}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{22:1--22:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.22},
  URN =		{urn:nbn:de:0030-drops-227798},
  doi =		{10.4230/LIPIcs.CSL.2025.22},
  annote =	{Keywords: Automata minimisation, omega-regular languages, good-for-games automata}
}
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