arXiv subjects

Patricia Bouyer

Publications and source records attributed to Patricia Bouyer.

At least 19 recordsLinked to original sources

arXiv2026

Antichains for Concurrent Parameterized Games (Long Version)

Concurrent parameterized games involve a fixed yet arbitrary number of players. They are described by finite arenas in which the edges are labeled with languages that describe the possible move combinations leading from one vertex to another (n players yield a word of length n). Previous work showed that, when edge labels are regular languages, one can decide whether a distinguished player, called Eve, has a strategy to ensure a reachability objective, against any strategy profile of her arbitrarily many opponents. This decision problem is known to be PSPACE-complete. A basic ingredient in the PSPACE-membership proof is the reduction to the exponential-size knowledge game, a 2-player game that reflects the knowledge Eve has on the number of opponents. In this paper, we provide a symbolic approach, based on antichains, to compute Eve's winning region in the knowledge game. In words, it gives the minimal knowledge Eve needs at every vertex to win the concurrent parameterized reachability game. More precisely, we propose two fixed-point algorithms that compute, as an antichain, the maximal elements of the winning region for Eve in the knowledge game. We implemented these two algorithms in C++, as well as the one initially proposed, and report on their relative performances on various benchmarks.

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arXiv2026

A Simple Obligation to Metric Interval Temporal Logic

Satisfiability of Metric Interval Temporal Logic (MITL) is a widely investigated subject. In this work, we present a new, and arguably simpler, approach for MITL satisfiability, based on an idea of tracking time-constrained obligations along a word. To check whether a Linear Temporal Logic (LTL) formula is true at a position of a word, it is natural to generate certain obligations that need to be satisfied at a later point. For instance, $a ~\mathcal{U}~ b$ (with strict Until semantics) is true at position $i$ if either $b$ or the set $\{a, a ~\mathcal{U}~ b\}$ is true at $i+1$. We enhance this idea in the context of MITL by introducing a notion of time inside these obligations. However, a naïve procedure could lead to more and more obligations getting generated along the word, with no bound on the number. We propose a simple mechanism to eliminate or merge redundant obligations. For MITL, this mechanism ensures that only a bounded number of obligations are maintained along the entire timed word. We develop this observation into a symbolic procedure for MITL satisfiability using regions.

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arXiv2026

Model-checking real-time systems: revisiting the alternating automaton route

Alternating timed automata (ATA) are an extension of timed automata, that are closed under complementation and hence amenable to logic-to-automata translations. Several timed logics, including Metric Temporal Logic (MTL), can be converted to equivalent 1-clock ATAs (1-ATAs). Satisfiability of an MTL formula reduces to checking emptiness of a 1-ATA. A straightforward modification of the 1-ATA emptiness algorithm can be applied for model-checking timed automata models against 1-ATA specifications. However, existing emptiness algorithms for 1-ATA proceed by an extended region construction, and are not suitable for implementations. Our goal in this work is to initiate the study of zone-based methods directly for 1-ATAs. We first introduce a deactivation operation on the 1-ATA syntax to allow an explicit deactivation of the clock in transitions. Using the deactivation operation, we improve the existing MTL-to-1-ATA conversion and present a fragment of MTL for which the equivalent 1-ATA generate a bounded number of variables. Secondly, we develop the idea of zones for 1-ATA and present an emptiness algorithm which explores a corresponding zone graph. For termination, a special entailment check between zones is necessary. Our main technical contributions are: (1) an algorithm for the entailment check using simple zone operations and (2) an NP-hardness for the entailment check in the general case. Finally, we adapt our methods to the problem of model-checking timed automata models against 1-ATA specifications. We observe that when the timed automaton is strongly non-Zeno or when the 1-ATA generates a bounded number of variables, a modified entailment check with quadratic complexity can be applied.

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arXiv2025

Reach together: How populations win repeated games

In repeated games, players choose actions concurrently at each step. We consider a parameterized setting of repeated games in which the players form a population of an arbitrary size. Their utility functions encode a reachability objective. The problem is whether there exists a uniform coalition strategy for the players so that they are sure to win independently of the population size. We use algebraic tools to show that the problem can be solved in polynomial space. First we exhibit a finite semigroup whose elements summarize strategies over a finite interval of population sizes. Then, we characterize the existence of winning strategies by the existence of particular elements in this semigroup. Finally, we provide a matching complexity lower bound, to conclude that repeated population games with reachability objectives are PSPACE-complete.

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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.

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arXiv2025

Decisiveness for countable MDPs and insights for NPLCSs and POMDPs

Markov chains and Markov decision processes (MDPs) are well-established probabilistic models. While finite Markov models are well-understood, analysing their infinite counterparts remains a significant challenge. Decisiveness has proven to be an elegant property for countable Markov chains: it is general enough to be satisfied by several natural classes of countable Markov chains, and it is a sufficient condition for simple qualitative and approximate quantitative model-checking algorithms to exist. In contrast, existing works on the formal analysis of countable MDPs usually rely on ad hoc techniques tailored to specific classes. We provide here a general framework to analyse countable MDPs by extending the notion of decisiveness. Compared to Markov chains, MDPs exhibit extra non-determinism that can be resolved in an adversarial or cooperative way, leading to multiple natural notions of decisiveness. We show that these notions enable the approximation of reachability and safety probabilities in countable MDPs using simple model-checking procedures. We then instantiate our generic approach to two concrete classes of models inducing countable MDPs: non-deterministic probabilistic lossy channel systems and partially observable MDPs. This leads to an algorithm to approximately compute safety probabilities in each of these classes.

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arXiv2024

Beyond Decisiveness of Infinite Markov Chains

Verification of infinite-state Markov chains is still a challenge despite several fruitful numerical or statistical approaches. For decisive Markov chains, there is a simple numerical algorithm that frames the reachability probability as accurately as required (however with an unknown complexity). On the other hand when applicable, statistical model checking is in most of the cases very efficient. Here we study the relation between these two approaches showing first that decisiveness is a necessary and sufficient condition for almost sure termination of statistical model checking. Afterwards we develop an approach with application to both methods that substitutes to a non decisive Markov chain a decisive Markov chain with the same reachability probability. This approach combines two key ingredients: abstraction and importance sampling (a technique that was formerly used for efficiency). We develop this approach on a generic formalism called layered Markov chain (LMC). Afterwards we perform an empirical study on probabilistic pushdown automata (an instance of LMC) to understand the complexity factors of the statistical and numerical algorithms. To the best of our knowledge, this prototype is the first implementation of the deterministic algorithm for decisive Markov chains and required us to solve several qualitative and numerical issues.

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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.

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arXiv2023

Arena-Independent Finite-Memory Determinacy in Stochastic Games

We study stochastic zero-sum games on graphs, which are prevalent tools to model decision-making in presence of an antagonistic opponent in a random environment. In this setting, an important question is the one of strategy complexity: what kinds of strategies are sufficient or required to play optimally (e.g., randomization or memory requirements)? Our contributions further the understanding of arena-independent finite-memory (AIFM) determinacy, i.e., the study of objectives for which memory is needed, but in a way that only depends on limited parameters of the game graphs. First, we show that objectives for which pure AIFM strategies suffice to play optimally also admit pure AIFM subgame perfect strategies. Second, we show that we can reduce the study of objectives for which pure AIFM strategies suffice in two-player stochastic games to the easier study of one-player stochastic games (i.e., Markov decision processes). Third, we characterize the sufficiency of AIFM strategies through two intuitive properties of objectives. This work extends a line of research started on deterministic games to stochastic ones.

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arXiv2023

From Local To Global Optimality in Concurrent Parity Games

We study two-player games on finite graphs. Turn-based games have many nice properties, but concurrent games are harder to tame: e.g. turn-based stochastic parity games have positional optimal strategies, whereas even basic concurrent reachability games may fail to have optimal strategies. We study concurrent stochastic parity games, and identify a local structural condition that, when satisfied at each state, guarantees existence of positional optimal strategies for both players.

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arXiv2023

How to Play Optimally for Regular Objectives?

This paper studies two-player zero-sum games played on graphs and makes contributions toward the following question: given an objective, how much memory is required to play optimally for that objective? We study regular objectives, where the goal of one of the two players is that eventually the sequence of colors along the play belongs to some regular language of finite words. We obtain different characterizations of the chromatic memory requirements for such objectives for both players, from which we derive complexity-theoretic statements: deciding whether there exist small memory structures sufficient to play optimally is NP-complete for both players. Some of our characterization results apply to a more general class of objectives: topologically closed and topologically open sets.

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arXiv2023

Sub-game optimal strategies in concurrent games with prefix-independent objectives

We investigate concurrent two-player win/lose stochastic games on finite graphs with prefix-independent objectives. We characterize subgame optimal strategies and use this characterization to show various memory transfer results: 1) For a given (prefix-independent) objective, if every game that has a subgame almost-surely winning strategy also has a positional one, then every game that has a subgame optimal strategy also has a positional one; 2) Assume that the (prefix-independent) objective has a neutral color. If every turn-based game that has a subgame almost-surely winning strategy also has a positional one, then every game that has a finite-choice (notion to be defined) subgame optimal strategy also has a positional one. We collect or design examples to show that our results are tight in several ways. We also apply our results to Büchi, co-Büchi, parity, mean-payoff objectives, thus yielding simpler statements.

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arXiv2023

Characterizing Omega-Regularity through Finite-Memory Determinacy of Games on Infinite Graphs

We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of $ω$-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is $ω$-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be $ω$-regular. This provides a game-theoretic characterization of $ω$-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which $ω$-regularity depends on the value of some parameters.

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arXiv2022

Playing (Almost-)Optimally in Concurrent Büchi and co-Büchi Games

We study two-player concurrent stochastic games on finite graphs, with Büchi and co-Büchi objectives. The goal of the first player is to maximize the probability of satisfying the given objective. Following Martin's determinacy theorem for Blackwell games, we know that such games have a value. Natural questions are then: does there exist an optimal strategy, that is, a strategy achieving the value of the game? what is the memory required for playing (almost-)optimally? The situation is rather simple to describe for turn-based games, where positional pure strategies suffice to play optimally in games with parity objectives. Concurrency makes the situation intricate and heterogeneous. For most ω-regular objectives, there do indeed not exist optimal strategies in general. For some objectives (that we will mention), infinite memory might also be required for playing optimally or almost-optimally. We also provide characterizations of local interactions of the players to ensure positionality of (almost-)optimal strategies for Büchi and co-Büchi objectives. This characterization relies on properties of game forms underpinning the formalism for defining local interactions of the two players. These well-behaved game forms are like elementary bricks which, when they behave well in isolation, can be assembled in graph games and ensure the good property for the whole game.

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arXiv2022

Zone-based verification of timed automata: extrapolations, simulations and what next?

Timed automata have been introduced by Rajeev Alur and David Dill in the early 90's. In the last decades, timed automata have become the de facto model for the verification of real-time systems. Algorithms for timed automata are based on the traversal of their state-space using zones as a symbolic representation. Since the state-space is infinite, termination relies on finite abstractions that yield a finite representation of the reachable states. The first solution to get finite abstractions was based on extrapolations of zones, and has been implemented in the industry-strength tool Uppaal. A different approach based on simulations between zones has emerged in the last ten years, and has been implemented in the fully open source tool TChecker. The simulation-based approach has led to new efficient algorithms for reachability and liveness in timed automata, and has also been extended to richer models like weighted timed automata, and timed automata with diagonal constraints and updates. In this article, we survey the extrapolation and simulation techniques, and discuss some open challenges for the future.

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arXiv2022

Games Where You Can Play Optimally with Arena-Independent Finite Memory

For decades, two-player (antagonistic) games on graphs have been a framework of choice for many important problems in theoretical computer science. A notorious one is controller synthesis, which can be rephrased through the game-theoretic metaphor as the quest for a winning strategy of the system in a game against its antagonistic environment. Depending on the specification, optimal strategies might be simple or quite complex, for example having to use (possibly infinite) memory. Hence, research strives to understand which settings allow for simple strategies. In 2005, Gimbert and Zielonka provided a complete characterization of preference relations (a formal framework to model specifications and game objectives) that admit memoryless optimal strategies for both players. In the last fifteen years however, practical applications have driven the community toward games with complex or multiple objectives, where memory -- finite or infinite -- is almost always required. Despite much effort, the exact frontiers of the class of preference relations that admit finite-memory optimal strategies still elude us. In this work, we establish a complete characterization of preference relations that admit optimal strategies using arena-independent finite memory, generalizing the work of Gimbert and Zielonka to the finite-memory case. We also prove an equivalent to their celebrated corollary of great practical interest: if both players have optimal (arena-independent-)finite-memory strategies in all one-player games, then it is also the case in all two-player games. Finally, we pinpoint the boundaries of our results with regard to the literature: our work completely covers the case of arena-independent memory (e.g., multiple parity objectives, lower- and upper-bounded energy objectives), and paves the way to the arena-dependent case (e.g., multiple lower-bounded energy objectives).

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arXiv2022

Decisiveness of Stochastic Systems and its Application to Hybrid Models (Full Version)

In [ABM07], Abdulla et al. introduced the concept of decisiveness, an interesting tool for lifting good properties of finite Markov chains to denumerable ones. Later, this concept was extended to more general stochastic transition systems (STSs), allowing the design of various verification algorithms for large classes of (infinite) STSs. We further improve the understanding and utility of decisiveness in two ways. First, we provide a general criterion for proving decisiveness of general STSs. This criterion, which is very natural but whose proof is rather technical, (strictly) generalizes all known criteria from the literature. Second, we focus on stochastic hybrid systems (SHSs), a stochastic extension of hybrid systems. We establish the decisiveness of a large class of SHSs and, under a few classical hypotheses from mathematical logic, we show how to decide reachability problems in this class, even though they are undecidable for general SHSs. This provides a decidable stochastic extension of o-minimal hybrid systems. [ABM07] Parosh A. Abdulla, Noomene Ben Henda, and Richard Mayr. 2007. Decisive Markov Chains. Log. Methods Comput. Sci. 3, 4 (2007).

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arXiv2021

Optimal strategies in concurrent reachability games

We study two-player reachability games on finite graphs. At each state the interaction between the players is concurrent and there is a stochastic Nature. Players also play stochastically. The literature tells us that 1) Player B, who wants to avoid the target state, has a positional strategy that maximizes the probability to win (uniformly from every state) and 2) from every state, for every ε > 0, Player A has a strategy that maximizes up to ε the probability to win. Our work is two-fold. First, we present a double-fixed-point procedure that says from which state Player A has a strategy that maximizes (exactly) the probability to win. This is computable if Nature's probability distributions are rational. We call these states maximizable. Moreover, we show that for every ε > 0, Player A has a positional strategy that maximizes the probability to win, exactly from maximizable states and up to ε from sub-maximizable states. Second, we consider three-state games with one main state, one target, and one bin. We characterize the local interactions at the main state that guarantee the existence of an optimal Player A strategy. In this case there is a positional one. It turns out that in many-state games, these local interactions also guarantee the existence of a uniform optimal Player A strategy. In a way, these games are well-behaved by design of their elementary bricks, the local interactions. It is decidable whether a local interaction has this desirable property.

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