arXiv subjects

Nathalie Bertrand

Publications and source records attributed to Nathalie Bertrand.

At least 19 recordsLinked to original sources

arXiv2026

The Stochastic Target Discounted-Sum Problem

The target discounted-sum problem (TDS) asks, given a finite integer alphabet $Σ$, a rational discount factor $λ$, and a rational target $t$, whether some infinite sequence over $Σ$ has discounted sum exactly $t$. This problem remains open and underlies several open questions in automata theory, games, and Markov decision processes. We introduce and solve its stochastic counterpart, the stochastic target discounted-sum problem, which replaces existence by computation of the probability. We show that the probability that a random sequence generated by a finite Markov chain has discounted sum $t$ is rational and computable in pseudo-polynomial time. We further show how to decide, in polynomial time, whether the discounted-sum distribution of a Markov chain is atomless, and how to approximate to an arbitrary precision the probability that the discounted sum exceeds a rational threshold. Our techniques for the stochastic TDS problem allow us to make progress on TDS objectives in stochastic games, which are known to be as hard as the TDS problem. Restricting the maximizing player to finite-memory strategies, while allowing the minimizing player to use arbitrary strategies, we reduce the value problem and the synthesis problem to corresponding problems for safety objectives in stochastic games. This yields computable optimal values and deterministic optimal strategies with pseudo-polynomially bounded memory for stochastic games, and results in pseudo-polynomial-time algorithms for special cases of Markov decision processes and deterministic two-player games.

cs.LO↗

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.

cs.LO↗

arXiv2026

Reaching as Cheap as Possible in 1-clock Robust Weighted Timed Games

The value problem for 2-player games on graph generally consists in determining the minimal value Min can ensure against any possible strategy for Max. We consider here the value problem for reachability objectives in weighted timed games (WTGs) under a robust semantics. WTGs are a modelling formalism combining real-time constraints and integer weights on transitions and locations in an adversarial setting. Robustness allows for representing timing imprecisions in the measurement of delays and clock values. Robust weighted timed games have been introduced more than a decade ago: they are undecidable in general, and were quite recently shown decidable for the subclasses of acyclic or divergent robust WTGs. This paper pursues the goal of identifying decidable subclasses and establishes the decidability of the robust value problem for 1-clock WTGs.

cs.GT↗

arXiv2026

Categorizer Automata for Discounted-Sum Payoffs

Categorizing continuous data into discrete bins is a fundamental operation in artificial intelligence. We introduce the categorizer automaton, a deterministic automaton that reads an infinite sequence of rewards and identifies which of finitely many bins contains its discounted sum. Categorizer automata generalize comparator automata, the special case of two bins, which have already proven useful in quantitative synthesis. Our main technical contribution is the construction of a categorizer automaton whose state space is linear in the number of bins, rather than exponential as obtained by a cross-product of comparator automata. We then apply categorizer automata to Markov decision processes, where they allow one to synthesize policies that maximize the expected utility of a discounted-sum payoff for utility functions that may be discontinuous. For piecewise-constant utility functions, the resulting algorithm is exact and runs in pseudo-polynomial time. For piecewise-Lipschitz utility functions, a class that includes any utility with bounded slope between finitely many jumps, it again runs in pseudo-polynomial time and yields an $\varepsilon$-optimal policy. We also show that the synthesis problem considered is PSPACE-hard already for piecewise-constant utilities.

cs.AI↗

arXiv2026

Parameterized Verification of Asynchronous Round-Based Distributed Algorithms via Reduction to Finite-Counter Systems

Traditional model-checking techniques typically verify distributed algorithms only for a fixed number of finite-state processes. Parameterized model checking generalizes this to any number of processes, while still typically assuming that each process is finite-state. In this work, we consider asynchronous round-based distributed algorithms in which each process is infinite-state since it can execute for an infinite number of rounds. We show that the parameterized verification problem for asynchronous round-based distributed algorithms is undecidable, already for simple specifications. Nevertheless, as our main contribution, we provide a reduction to LTL model checking over finite-counter systems and prove that it is sound and complete. This enables the use of off-the-shelf, mature symbolic model checkers for finite-counter systems. We demonstrate the practical applicability of this reduction by verifying safety and liveness properties of several asynchronous round-based consensus and leader-election algorithms using the nuXmv model checker.

cs.LO↗

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.

cs.GT↗

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

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.

cs.LO↗

arXiv2025

Reusable Formal Verification of DAG-based Consensus Protocols

Blockchains use consensus protocols to reach agreement, e.g., on the ordering of transactions. DAG-based consensus protocols are increasingly adopted by blockchain companies to reduce energy consumption and enhance security. These protocols collaboratively construct a partial order of blocks (DAG construction) and produce a linear sequence of blocks (DAG ordering). Given the strategic significance of blockchains, formal proofs of the correctness of key components such as consensus protocols are essential. This paper presents safety-verified specifications for five DAG-based consensus protocols. Four of these protocols -- DAG-Rider, Cordial Miners, Hashgraph, and Eventual Synchronous BullShark -- are well-established in the literature. The fifth protocol is a minor variation of Aleph, another well-established protocol. Our framework enables proof reuse, reducing proof efforts by almost half. It achieves this by providing various independent, formally verified, specifications of DAG construction and ordering variations, which can be combined to express all five protocols. We employ TLA+ for specifying the protocols and writing their proofs, and the TLAPS proof system to automatically check the proofs. Each TLA+ specification is relatively compact, and TLAPS efficiently verifies hundreds to thousands of obligations within minutes. The significance of our work is two-fold: first, it supports the adoption of DAG-based systems by providing robust safety assurances; second, it illustrates that DAG-based consensus protocols are amenable to practical, reusable, and compositional formal methods.

cs.LO↗

arXiv2022

Holistic Verification of Blockchain Consensus

Blockchain has recently attracted the attention of the industry due, in part, to its ability to automate asset transfers. It requires distributed participants to reach a consensus on a block despite the presence of malicious (a.k.a. Byzantine) participants. Malicious participants exploit regularly weaknesses of these blockchain consensus algorithms, with sometimes devastating consequences. In fact, these weaknesses are quite common and are well illustrated by the flaws in the hand-written proofs of existing blockchain consensus protocols [63]. Paradoxically, until now, no blockchain consensus has been holistically verified using model checking. In this paper, we remedy this paradox by model checking for the first time a blockchain consensus used in industry. We propose a holistic approach to verify the consensus algorithm of the Red Belly Blockchain [20], for any number $n$ of processes and any number $f<n/3$ of Byzantine processes. We decompose directly the algorithm pseudocode in two parts -- an inner broadcast algorithm and an outer decision algorithm -- each modelled as a threshold automaton [36], and we formalize their expected properties in linear-time temporal logic. We then automatically check the inner broadcasting algorithm, under a carefully identified fairness assumption. For the verification of the outer algorithm, we simplify the model of the inner algorithm by relying on its checked properties. Doing so, we formally verify not only the safety properties of the Red Belly Blockchain consensus but also its liveness in about 70 seconds.

cs.CR↗

arXiv2022

Parameterized safety verification of round-based shared-memory systems

We consider the parameterized verification problem for distributed algorithms where the goal is to develop techniques to prove the correctness of a given algorithm regardless of the number of participating processes. Motivated by an asynchronous binary consensus algorithm [J. Aspnes, Fast deterministic consensus in a noisy environment. Journal of Algorithms, 2002], we consider round-based distributed algorithms communicating with shared memory. A particular challenge in these systems is that 1) the number of processes is unbounded, and, more importantly, 2) there is a fresh set of registers at each round. A~verification algorithm thus needs to manage both sources of infinity. In this setting, we prove that the safety verification problem, which consists in deciding whether all possible executions avoid a given error state, is PSPACE-complete. For~negative instances of the safety verification problem, we~also provide exponential lower and upper bounds on the minimal number of processes needed for an error execution and on the minimal round on which the error state can be covered.

cs.FL↗

arXiv2021

Reconfiguration and Message Losses in Parameterized Broadcast Networks

Broadcast networks allow one to model networks of identical nodes communicating through message broadcasts. Their parameterized verification aims at proving a property holds for any number of nodes, under any communication topology, and on all possible executions. We focus on the coverability problem which dually asks whether there exists an execution that visits a configuration exhibiting some given state of the broadcast protocol. Coverability is known to be undecidable for static networks, i.e. when the number of nodes and communication topology is fixed along executions. In contrast, it is decidable in PTIME when the communication topology may change arbitrarily along executions, that is for reconfigurable networks. Surprisingly, no lower nor upper bounds on the minimal number of nodes, or the minimal length of covering execution in reconfigurable networks, appear in the literature. In this paper we show tight bounds for cutoff and length, which happen to be linear and quadratic, respectively, in the number of states of the protocol. We also introduce an intermediary model with static communication topology and non-deterministic message losses upon sending. We show that the same tight bounds apply to lossy networks, although, reconfigurable executions may be linearly more succinct than lossy executions. Finally, we show NP-completeness for the natural optimisation problem associated with the cutoff.

cs.LO↗

arXiv2020

Synthesizing safe coalition strategies

Concurrent games with a fixed number of agents have been thoroughly studied, with various solution concepts and objectives for the agents. In this paper, we consider concurrent games with an arbitrary number of agents, and study the problem of synthesizing a coalition strategy to achieve a global safety objective. The problem is non-trivial since the agents do not know a priori how many they are when they start the game. We prove that the existence of a safe arbitrary-large coalition strategy for safety objectives is a PSPACE-hard problem that can be decided in exponential space.

cs.LO↗

arXiv2020

Dynamic network congestion games

Congestion games are a classical type of games studied in game theory, in which n players choose a resource, and their individual cost increases with the number of other players choosing the same resource. In network congestion games (NCGs), the resources correspond to simple paths in a graph, e.g. representing routing options from a source to a target. In this paper, we introduce a variant of NCGs, referred to as dynamic NCGs: in this setting, players take transitions synchronously, they select their next transitions dynamically, and they are charged a cost that depends on the number of players simultaneously using the same transition. We study, from a complexity perspective, standard concepts of game theory in dynamic NCGs: social optima, Nash equilibria, and subgame perfect equilibria. Our contributions are the following: the existence of a strategy profile with social cost bounded by a constant is in PSPACE and NP-hard. (Pure) Nash equilibria always exist in dynamic NCGs; the existence of a Nash equilibrium with bounded cost can be decided in EXPSPACE, and computing a witnessing strategy profile can be done in doubly-exponential time. The existence of a subgame perfect equilibrium with bounded cost can be decided in 2EXPSPACE, and a witnessing strategy profile can be computed in triply-exponential time.

cs.GT↗

arXiv2019

Controlling a population

We introduce a new setting where a population of agents, each modelled by a finite-state system, are controlled uniformly: the controller applies the same action to every agent. The framework is largely inspired by the control of a biological system, namely a population of yeasts, where the controller may only change the environment common to all cells. We study a synchronisation problem for such populations: no matter how individual agents react to the actions of the controller, the controller aims at driving all agents synchronously to a target state. The agents are naturally represented by a non-deterministic finite state automaton (NFA), the same for every agent, and the whole system is encoded as a 2-player game. The first player (Controller) chooses actions, and the second player (Agents) resolves non-determinism for each agent. The game with m agents is called the m -population game. This gives rise to a parameterized control problem (where control refers to 2 player games), namely the population control problem: can Controller control the m-population game for all m in N whatever Agents does?

cs.FL↗

arXiv2018

Stochastic Shortest Paths and Weight-Bounded Properties in Markov Decision Processes

The paper deals with finite-state Markov decision processes (MDPs) with integer weights assigned to each state-action pair. New algorithms are presented to classify end components according to their limiting behavior with respect to the accumulated weights. These algorithms are used to provide solutions for two types of fundamental problems for integer-weighted MDPs. First, a polynomial-time algorithm for the classical stochastic shortest path problem is presented, generalizing known results for special classes of weighted MDPs. Second, qualitative probability constraints for weight-bounded (repeated) reachability conditions are addressed. Among others, it is shown that the problem to decide whether a disjunction of weight-bounded reachability conditions holds almost surely under some scheduler belongs to $\textrm{NP}\cap \textrm{coNP}$, is solvable in pseudo-polynomial time and is at least as hard as solving two-player mean-payoff games, while the corresponding problem for universal quantification over schedulers is solvable in polynomial time.

cs.LO↗

arXiv2018

When are Stochastic Transition Systems Tameable?

A decade ago, Abdulla, Ben Henda and Mayr introduced the elegant concept of decisiveness for denumerable Markov chains [1]. Roughly speaking, decisiveness allows one to lift most good properties from finite Markov chains to denumerable ones, and therefore to adapt existing verification algorithms to infinite-state models. Decisive Markov chains however do not encompass stochastic real-time systems, and general stochastic transition systems (STSs for short) are needed. In this article, we provide a framework to perform both the qualitative and the quantitative analysis of STSs. First, we define various notions of decisiveness (inherited from [1]), notions of fairness and of attractors for STSs, and make explicit the relationships between them. Then, we define a notion of abstraction, together with natural concepts of soundness and completeness, and we give general transfer properties, which will be central to several verification algorithms on STSs. We further design a generic construction which will be useful for the analysis of ω-regular properties, when a finite attractor exists, either in the system (if it is denumerable), or in a sound denumerable abstraction of the system. We next provide algorithms for qualitative model-checking, and generic approximation procedures for quantitative model-checking. Finally, we instantiate our framework with stochastic timed automata (STA), generalized semi-Markov processes (GSMPs) and stochastic time Petri nets (STPNs), three models combining dense-time and probabilities. This allows us to derive decidability and approximability results for the verification of these models. Some of these results were known from the literature, but our generic approach permits to view them in a unified framework, and to obtain them with less effort. We also derive interesting new approximability results for STA, GSMPs and STPNs.

cs.LO↗

arXiv2018

Parameterized verification of synchronization in constrained reconfigurable broadcast networks

Reconfigurable broadcast networks provide a convenient formalism for modelling and reasoning about networks of mobile agents broadcasting messages to other agents following some (evolving) communication topology. The parameterized verification of such models aims at checking whether a given property holds irrespective of the initial configuration (number of agents, initial states and initial communication topology). We focus here on the synchronization property, asking whether all agents converge to a set of target states after some execution. This problem is known to be decidable in polynomial time when no constraints are imposed on the evolution of the communication topology (while it is undecidable for static broadcast networks). In this paper we investigate how various constraints on reconfigurations affect the decidability and complexity of the synchronization problem. In particular, we show that when bounding the number of reconfigured links between two communications steps by a constant, synchronization becomes undecidable; on the other hand, synchronization remains decidable in PTIME when the bound grows with the number of agents.

cs.LO↗