Two-player games on finite graphs provide a rigorous foundation for modeling the strategic interaction between reactive systems and their environment. While concurrent game semantics naturally capture the synchronous interactions characteristic of many cyber-physical systems (CPS), their adoption in CPS design remains limited. Building on the concept of permissive strategy templates (PeSTels) for turn-based games, we introduce concurrent (permissive) strategy templates (ConSTels) – a novel representation for sets of randomized winning strategies in concurrent games with Safety, Büchi, and Co-Büchi objectives. ConSTels compactly encode infinite families of strategies, thereby supporting both offline and online adaptation. Offline, we exploit compositionality to enable incremental synthesis: combining ConSTels for simpler objectives into non-conflicting templates for more complex combined objectives. Online, we demonstrate how ConSTels facilitate runtime adaptation, adjusting action probabilities in response to observed opponent behavior to optimize performance while preserving correctness. We implemented ConSTel synthesis and adaptation in a prototype tool and experimentally show its potential.

Abstract We examine two-player games over finite weighted graphs with quantitative (mean-payoff or energy) objective, where one of the players additionally needs to satisfy a fairness objective. The specific fairness we consider is called strong transition fairness, given by a subset of edges of one of the players, which asks the player to take fair edges infinitely often if their source nodes are visited infinitely often. We show that when fairness is imposed on player 1, these games fall within the class of previously studied $$\omega $$ ω -regular mean-payoff and energy games. On the other hand, when the fairness is on player 2, to the best of our knowledge, these games have not been previously studied. We provide gadget-based algorithms for fair mean-payoff games where fairness is imposed on either player, and for fair energy games where the fairness is imposed on player 1. For all variants of fair mean-payoff and fair energy (under unknown initial credit) games, we give pseudo-polynomial algorithms to compute the winning regions of both players. Additionally, we analyze the strategy complexities required for these games. Our work is the first to extend the study of strong transition fairness, as well as gadget-based approaches, to the quantitative setting. We thereby demonstrate that the simplicity of strong transition fairness, as well as the applicability of gadget-based techniques, can be leveraged beyond the $$\omega $$ ω -regular domain.

We consider the problem of computing distributed logical controllers for two interacting system components via a novel sound and complete contract-based synthesis framework. Based on a discrete abstraction of component interactions as a two-player game over a finite graph and specifications for both components given as ω -regular (e.g. LTL) properties over this graph, we co-synthesize contract and controller candidates locally for each component and propose a negotiation mechanism which iteratively refines these candidates until a solution to the given distributed synthesis problem is found. Our framework relies on the recently introduced concept of permissive templates which collect an infinite number of controller candidates in a concise data structure. We utilize the efficient computability, adaptability and compositionality of such templates to obtain an efficient, yet sound and complete negotiation framework for contract-based distributed logical control. We showcase the superior performance of our approach by comparing our prototype tool CoSMo to the state-of-the-art tool on a robot motion planning benchmark suite.

Well-structured transition systems (WSTS) are an abstract family of systems that encompasses a vast landscape of infinite-state systems. By requiring a well-quasi-ordering (wqo) on the set of states, a WSTS enables generic algorithms for classic verification tasks such as coverability and termination. However, even for systems that are WSTS like vector addition systems (VAS), the framework is notoriously ill-equipped to analyse reachability (as opposed to coverability). Moreover, some important types of infinite-state systems fall out of WSTS' scope entirely, such as pushdown systems (PDS).

Abstract We present a novel method to compute permissive winning strategies in two-player games over finite graphs with $$ \omega $$ ω -regular winning conditions. Given a game graph G and a parity winning condition $$\varPhi $$ Φ , we compute a winning strategy template $$\varPsi $$ Ψ that collects an infinite number of winning strategies for objective $$\varPhi $$ Φ in a concise data structure. We use this new representation of sets of winning strategies to tackle two problems arising from applications of two-player games in the context of cyber-physical system design – (i) incremental synthesis, i.e., adapting strategies to newly arriving, additional $$\omega $$ ω -regular objectives $$\varPhi '$$ Φ ′ , and (ii) fault-tolerant control, i.e., adapting strategies to the occasional or persistent unavailability of actuators. The main features of our strategy templates – which we utilize for solving these challenges – are their easy computability, adaptability, and compositionality. For incremental synthesis, we empirically show on a large set of benchmarks that our technique vastly outperforms existing approaches if the number of added specifications increases. While our method is not complete, our prototype implementation returns the full winning region in all 1400 benchmark instances, i.e. handling a large problem class efficiently in practice.

When a system sends messages through a lossy channel, then the language encoding all sequences of messages can be abstracted by its downward closure, i.e. the set of all (not necessarily contiguous) subwords. This is useful because even if the system has infinitely many states, its downward closure is a regular language. However, if the channel has congestion control based on priorities assigned to the messages, then we need a finer abstraction: The downward closure with respect to the priority embedding. As for subword-based downward closures, one can also show that these priority downward closures are always regular. While computing finite automata for the subword-based downward closure is well understood, nothing is known in the case of priorities. We initiate the study of this problem and provide algorithms to compute priority downward closures for regular languages, one-counter languages, and context-free languages.

This paper presents a new method to automatically compute permissive strategies and permissive assumptions in ω -regular two-player games on graphs to enable strategy adaptation both during synthesis and execution of distributed symbolic controllers.

Abstract We automatically compute a new class of environment assumptions in two-player turn-based finite graph games which characterize an “adequate cooperation” needed from the environment to allow the system player to win. Given an $$\omega $$ -regular winning condition $$\varPhi $$ for the system player, we compute an $$\omega $$ -regular assumption $$\varPsi $$ for the environment player, such that (i) every environment strategy compliant with $$\varPsi $$ allows the system to fulfill $$\varPhi $$ (sufficiency), (ii) $$\varPsi $$ can be fulfilled by the environment for every strategy of the system (implementability), and (iii) $$\varPsi $$ does not prevent any cooperative strategy choice (permissiveness). For parity games, which are canonical representations of $$\omega $$ -regular games, we present a polynomial-time algorithm for the symbolic computation of adequately permissive assumptions and show that our algorithm runs faster and produces better assumptions than existing approaches—both theoretically and empirically. To the best of our knowledge, for $$\omega $$ -regular games, we provide the first algorithm to compute sufficient and implementable environment assumptions that are also permissive.