arXiv subjects

Olivier Idir

Publications and source records attributed to Olivier Idir.

6 recordsLinked to original sources

arXiv2026

Explorable Parity Automata

We define the class of explorable automata on finite or infinite words. This is a generalization of History-Deterministic (HD) automata, where this time non-deterministic choices can be resolved by building finitely many simultaneous runs instead of just one. We show that recognizing HD parity automata of fixed index among explorable ones is in PTime, thereby giving a strong link between the two notions. We then show that recognizing explorable automata is ExpTime-complete, in the case of finite words or parity automata up to index [0, 2]. Additionally, we define the notion of ω-explorable automata on infinite words, where countably many runs can be used to resolve the non-deterministic choices. We show ExpTime-completeness for ω-explorability of automata on infinite words for the safety and coBüchi acceptance conditions. We finally characterize the expressivity of (ω-)explorable automata with respect to the parity index hierarchy.

cs.FL↗

arXiv2026

Eve-positional languages: putting order into Büchi automata

An $ω$-regular language is Eve-positional if, in all games with this language as objective, the existential player can play optimally without keeping any information from the previous moves. This notion plays a crucial role in verification, automata theory and synthesis. Casares and Ohlmann recently gave several characterisations of Eve-positionality of $ω$-regular languages. For this, they introduce the notion of $\varepsilon$-complete parity automaton and show (among other results) that an $ω$-regular language is Eve-positional if and only if it can be recognised by some $\varepsilon$-completion of a deterministic parity automaton. Colcombet and Idir built on their work, and obtained a more direct algebraic characterisation of Eve-positionality. We introduce a new formalism that characterises the Eve-positional languages, consisting of a restriction of non-deterministic Büchi automata. This allows us to complete a missing implication in Casares and Ohlmann's work. We then use this formalism to describe a determinization procedure for non-deterministic Büchi automata recognising such languages, with size blow-up at most factorial. We also show that this construction is state-wise optimal for languages over sufficiently complete alphabets.

cs.FL↗

arXiv2026

An algebraic characterisation of Eve-positional languages

We present a new algebraic characterisation of Eve-positionality for $ω$-regular languages. It involves only a limited number of elementary local properties to be checked. An $ω$-regular language is Eve-positional if, in all games with this language as objective, the existential player (Eve) can play optimally without keeping any information concerning the history of the moves seen so far. This notion plays a crucial role in verification, automata theory and synthesis. Our proof heavily relies on a recent result of Casares and Ohlmann which states several characterisations of Eve-positionality for $ω$-regular languages. More precisely, it relies on their a 1-to-2 player lift result: for an $ω$-regular language, being Eve-positionally over all finite Eve-only arenas suffices for being Eve-positional over all two-player arenas.

cs.FL↗

arXiv2025

Using games and universal trees to characterise the nondeterministic index of tree languages

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 $n\in \mathbb{N}$ such that all trees in the language admit an accepting run with an attractor decomposition of width bounded by $n$, or, equivalently, if and only $A$ admits a \textit{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.

cs.FL↗

arXiv2024

Mostowski Index via extended register games

The parity index problem of tree automata asks, given a regular tree language L, what is the least number of priorities of a nondeterministic parity tree automaton that recognises L. This is a long-standing open problem, also known as the Mostowski or Rabin-Mostowski index problem, of which only a few sub-cases and variations are known to be decidable. In a significant step, Colcombet and Löding reduced the problem to the uniform universality of distance-parity automata. In this brief note, we present a similar result, with a simplified proof, based on on the games in Lehtinen's quasipolynomial algorithm for parity games. We define an extended version of these games, which we call parity transduction games, which take as parameters a parity index J and an integer bound N. We show that the language of a guidable automaton A is recognised by a nondeterministic automaton of index J if and only if there is a bound N such that the parity transduction game with parameters J and N captures membership of the language, that is, for all trees t, Eve wins the parity transduction game on the acceptance parity game of t in A if and only in t is in L(A).

cs.FL↗

arXiv2024

On the Minimisation of Deterministic and History-Deterministic Generalised (co)Büchi Automata

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.

cs.FL↗