arXiv subjects

Antonio Casares

Publications and source records attributed to Antonio Casares.

18 recordsLinked to original sources

arXiv2026

History-Deterministic Büchi Automata are Succinct

We describe a history-deterministic Büchi automaton that has strictly less states than every language-equivalent deterministic Büchi automaton. This solves a problem that had been open since the introduction of history-determinism and actively investigated for over a decade. Our example automaton has 65 states, and proving its succinctness requires the combination of theoretical insights together with the use of solvers for NP-complete problems.

cs.FL↗

arXiv2026

A positional $\mathbfΠ^0_3$-complete objective

We study zero-sum, turn-based games on graphs. In this note, we show the existence of a game objective that is $\mathbfΠ^0_3$-complete for the Borel hierarchy and that is positional, i.e., for which positional strategies suffice for the first player to win over game graphs of arbitrary cardinality. To the best of our knowledge, this is the first known such objective; all previously known positional objectives are in $\mathbfΣ^0_3$. The objective in question is a qualitative variant of the well-studied total-payoff objective, which deals with the behaviour of the sequence of partial sums of weights seen along plays.

cs.CC↗

arXiv2026

The memory of $ω$-regular and BC($Σ_2^0$) objectives

In the context of 2-player zero-sum infinite-duration games played on (potentially infinite) graphs, the memory of an objective is the smallest integer k such that in any game won by Eve, she has a strategy with <= k states of memory. For omega-regular objectives, checking whether the memory equals a given number k was not known to be decidable. In this work, we focus on objectives in BC(Sigma0^2), i.e. recognised by a potentially infinite deterministic parity automaton. We provide a class of automata that recognise objectives with memory <= k, leading to the following results: (1) For omega-regular objectives, the memory over finite and infinite games coincides and can be computed in NP. (2) Given two objectives W1 and W2 in BC(Sigma0^2) and assuming W1 is prefix-independent, the memory of W1 U W2 is at most the product of the memories of W1 and W2. Our results also apply to chromatic memory, the variant where strategies can update their memory state only depending on which colour is seen.

cs.LO↗

arXiv2026

Infinite lexicographic products of positional objectives

This paper contributes to the study of positional determinacy of infinite duration games played on potentially infinite graphs with neutral transitions. Recently, [Ohlmann, TheoretiCS 2023] established that positionality of prefix-independent objectives is preserved by finite lexicographic products. We propose two different notions of infinite lexicographic products indexed by arbitrary ordinals, and extend Ohlmann's result by proving that they also preserve positionality. In the context of one-player positionality, this extends positional determinacy results of [Grädel and Walukiewicz, Logical Methods in Computer Science 2006] to edge-labelled games and arbitrarily many priorities for both Max-Parity and Min-Parity. Moreover, we show that the Max-Parity objectives over countable ordinals are complete for the infinite levels of the difference hierarchy over $Σ^0_2$ and that Min-Parity is complete for the class $Σ^0_3$. We obtain therefore positional languages that are complete for all those levels, as well as new insights about closure under unions and neutral letters.

cs.GT↗

arXiv2026

Positional $ω$-regular languages

In the context of two-player games over graphs, a language $L$ is called positional if, in all games using $L$ as winning objective, the protagonist can play optimally using positional strategies, that is, strategies that do not depend on the history of the play. In this work, we describe the class of parity automata recognising positional languages, providing a complete characterisation of positionality for $ω$-regular languages. As corollaries, we establish decidability of positionality in polynomial time, finite-to-infinite and 1-to-2-players lifts, and show the closure under union of prefix-independent positional objectives, answering a conjecture by Kopczyński in the $ω$-regular case.

cs.FL↗

arXiv2026

Layered automata: A canonical model for automata over infinite words

We introduce layered automata, a subclass of alternating parity automata that generalises deterministic automata. Assuming a consistency property, these automata are history deterministic and 0-1 probabilistic. We show that every omega-regular language is recognised by a unique minimal consistent layered automaton, and that this canonical form can be computed in polynomial time from every layered or deterministic automaton. We further establish that for layered automata both consistency checking and inclusion testing can be performed in polynomial time. Much like deterministic finite automata, minimal consistent layered automata admit a characterisation based on congruences.

cs.FL↗

arXiv2025

Transition-based vs stated-based acceptance for automata over infinite words

Automata over infinite objects are a well-established model with applications in logic and formal verification. Traditionally, acceptance in such automata is defined based on the set of states visited infinitely often during a run. However, there is a growing trend towards defining acceptance based on transitions rather than states. In this survey, we analyse the reasons for this shift and advocate using transition-based acceptance in the context of automata over infinite words. We present a collection of problems where the choice of formalism has a major impact and discuss the causes of these differences.

cs.FL↗

arXiv2025

Games on Graphs: From Logic and Automata to Algorithms

The objective of this book is to give a comprehensive presentation of the research field concerned with infinite duration games on graphs. Historically, these game models appeared in the study of automata and logic, and they later became important for program verification and synthesis. They have many more applications, in particular some of the models investigated in this book were introduced and studied in neighbouring research communities such as optimisation, reinforcement learning, model theory, and set theory.

cs.GT↗

arXiv2025

Characterising memory in infinite games

This paper is concerned with games of infinite duration played over potentially infinite graphs. Recently, Ohlmann (LICS 2022) presented a characterisation of objectives admitting optimal positional strategies, by means of universal graphs: an objective is positional if and only if it admits well-ordered monotone universal graphs. We extend Ohlmann's characterisation to encompass (finite or infinite) memory upper bounds. We prove that objectives admitting optimal strategies with $\varepsilon$-memory less than $m$ (a memory that cannot be updated when reading an $\varepsilon$-edge) are exactly those which admit well-founded monotone universal graphs whose antichains have size bounded by $m$. We also give a characterisation of chromatic memory by means of appropriate universal structures. Our results apply to finite as well as infinite memory bounds (for instance, to objectives with finite but unbounded memory, or with countable memory strategies). We illustrate the applicability of our framework by carrying out a few case studies, we provide examples witnessing limitations of our approach, and we discuss general closure properties which follow from our results.

cs.FL↗

arXiv2025

Fast value iteration: A uniform approach to efficient algorithms for energy games

We study algorithms for solving parity, mean-payoff and energy games. We propose a systematic framework, which we call Fast value iteration, for describing, comparing, and proving correctness of such algorithms. The approach is based on potential reductions, as introduced by Gurvich, Karzanov and Khachiyan (1988). This framework allows us to provide simple presentations and correctness proofs of known algorithms, unifying the Optimal strategy improvement algorithm by Schewe (2008) and the quasi dominions approach by Benerecetti et al. (2020), amongst others. The new approach also leads to novel symmetric versions of these algorithms, highly efficient in practice, but for which we are unable to prove termination. We report on empirical evaluation, comparing the different fast value iteration algorithms, and showing that they are competitive even to top parity game solvers.

cs.GT↗

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↗

arXiv2024

Half-Positional Objectives Recognized by Deterministic Büchi Automata

In two-player games on graphs, the simplest possible strategies are those that can be implemented without any memory. These are called positional strategies. In this paper, we characterize objectives recognizable by deterministic Büchi automata (a subclass of omega-regular objectives) that are half-positional, that is, for which the protagonist can always play optimally using positional strategies (both over finite and infinite graphs). Our characterization consists of three natural conditions linked to the language-theoretic notion of right congruence. Furthermore, this characterization yields a polynomial-time algorithm to decide half-positionality of an objective recognized by a given deterministic Büchi automaton.

cs.GT↗

arXiv2024

The Complexity of Simplifying $ω$-Automata through the Alternating Cycle Decomposition

In 2021, Casares, Colcombet and Fijalkow introduced the Alternating Cycle Decomposition (ACD), a structure used to define optimal transformations of Muller into parity automata and to obtain theoretical results about the possibility of relabelling automata with different acceptance conditions. In this work, we study the complexity of computing the ACD and its DAG-version, proving that this can be done in polynomial time for suitable representations of the acceptance condition of the Muller automaton. As corollaries, we obtain that we can decide typeness of Muller automata in polynomial time, as well as the parity index of the languages they recognise. Furthermore, we show that we can minimise in polynomial time the number of colours (resp. Rabin pairs) defining a Muller (resp. Rabin) acceptance condition, but that these problems become NP-complete when taking into account the structure of an automaton using such a condition.

cs.FL↗

arXiv2024

From Muller to Parity and Rabin Automata: Optimal Transformations Preserving (History) Determinism

We study transformations of automata and games using Muller conditions into equivalent ones using parity or Rabin conditions. We present two transformations, one that turns a deterministic Muller automaton into an equivalent deterministic parity automaton, and another that provides an equivalent history-deterministic Rabin automaton. We show a strong optimality result: the obtained automata are minimal amongst those that can be derived from the original automaton by duplication of states. We introduce the notions of locally bijective morphisms and history-deterministic mappings to formalise the correctness and optimality of these transformations. The proposed transformations are based on a novel structure, called the alternating cycle decomposition, inspired by and extending Zielonka trees. In addition to providing optimal transformations of automata, the alternating cycle decomposition offers fundamental information on their structure. We use this information to give crisp characterisations on the possibility of relabelling automata with different acceptance conditions and to perform a systematic study of a normal form for parity automata.

cs.FL↗

arXiv2023

Simple and tight complexity lower bounds for solving Rabin games

We give a simple proof that assuming the Exponential Time Hypothesis (ETH), determining the winner of a Rabin game cannot be done in time $2^{o(k \log k)} \cdot n^{O(1)}$, where $k$ is the number of pairs of vertex subsets involved in the winning condition and $n$ is the vertex count of the game graph. While this result follows from the lower bounds provided by Calude et al [SIAM J. Comp. 2022], our reduction is simpler and arguably provides more insight into the complexity of the problem. In fact, the analogous lower bounds discussed by Calude et al, for solving Muller games and multidimensional parity games, follow as simple corollaries of our approach. Our reduction also highlights the usefulness of a certain pivot problem -- Permutation SAT -- which may be of independent interest.

cs.FL↗

arXiv2023

Optimal Transformations of Muller Conditions

In this paper, we are interested in automata over infinite words and infinite duration games, that we view as general transition systems. We study transformations of systems using a Muller condition into ones using a parity condition, extending Zielonka's construction. We introduce the alternating cycle decomposition transformation, and we prove a strong optimality result: for any given deterministic Muller automaton, the obtained parity automaton is minimal both in size and number of priorities among those automata admitting a morphism into the original Muller automaton. We give two applications. The first is an improvement in the process of determinisation of Büchi automata into parity automata by Piterman and Schewe. The second is to present characterisations on the possibility of relabelling automata with different acceptance conditions.

cs.FL↗

arXiv2022

On the size of good-for-games Rabin automata and its link with the memory in Muller games

In this paper, we look at good-for-games Rabin automata that recognise a Muller language (a language that is entirely characterised by the set of letters that appear infinitely often in each word). We establish that minimal such automata are exactly of the same size as the minimal memory required for winning Muller games that have this language as their winning condition. We show how to effectively construct such minimal automata. Finally, we establish that these automata can be exponentially more succinct than equivalent deterministic ones, thus proving as a consequence that chromatic memory for winning a Muller game can be exponentially larger than unconstrained memory.

cs.FL↗

arXiv2021

On the Minimisation of Transition-Based Rabin Automata and the Chromatic Memory Requirements of Muller Conditions

In this paper, we relate the problem of determining the chromatic memory requirements of Muller conditions with the minimisation of transition-based Rabin automata. Our first contribution is a proof of the NP-completeness of the minimisation of transition-based Rabin automata. Our second contribution concerns the memory requirements of games over graphs using Muller conditions. A memory structure is a finite state machine that implements a strategy and is updated after reading the edges of the game; the special case of chromatic memories being those structures whose update function only consider the colours of the edges. We prove that the minimal amount of chromatic memory required in games using a given Muller condition is exactly the size of a minimal Rabin automaton recognising this condition. Combining these two results, we deduce that finding the chromatic memory requirements of a Muller condition is NP-complete. This characterisation also allows us to prove that chromatic memories cannot be optimal in general, disproving a conjecture by Kopczyński.

cs.FL↗