We address the synthesis of control policies for unknown discrete-time stochastic dynamical systems to satisfy temporal logic objectives. We present a data-driven, abstraction-based control framework that integrates online learning with novel incremental game-solving. Under appropriate continuity assumptions, our method abstracts the system dynamics into a finite stochastic (2.5-player) game graph derived from data. Given a requirement over time on this graph, we compute the winning region -- i.e., the set of initial states from which the objective is satisfiable -- in the resulting game, together with a corresponding control policy. Our main contribution is the construction of abstractions, winning regions and control policies incrementally, as data about the system dynamics accumulates. Concretely, our algorithm refines under- and over-approximations of reachable sets for each state-action pair as new data samples arrive. These refinements induce structural modifications in the game graph abstraction -- such as the addition or removal of nodes and edges -- which in turn modify the winning region. Crucially, we show that these updates are inherently monotonic: under-approximations only grow, over-approximations only shrink, and the winning region only expands. We exploit this monotonicity by defining an objective-induced ranking function on the nodes of the abstract game that increases monotonically as new data samples are incorporated. These ranks underpin our novel incremental game-solving algorithm, which employs customized gadgets (DAG-like subgames) within a rank-lifting algorithm to efficiently update the winning region. Numerical case studies demonstrate significant computational savings compared to the baseline approach, which resolves the entire game from scratch whenever new data samples arrive.

In the theory of games on infinite-state arenas, there is a stark contrast between (i) recursion-based models such as pushdown systems and extensions on one hand, and (ii) counter-based models like vector addition systems with states (VASS) on the other. For pushdown systems and extensions, there is a rich variety of decidable and well-understood games, whereas on VASS arenas, even extremely simple games are undecidable. Here, a VASS is an automaton with counters that can be incremented and decremented, but not tested for zero. Crucially, the counters can only assume non-negative values. However, certain VASS games become decidable when using energy semantics: An energy game is played on a system with counters, but the arena includes configurations with negative counters. The requirement that the counters stay non-negative is, instead, part of the winning condition of the existential player. We study an analogue of energy semantics - legality of instructions as part of the winning condition rather than arena - on a broad class of infinite-state systems, where we call them viability games. Specifically, we study viability games in the framework of valence systems over graph monoids, where (undirected, loops allowed) graphs specify various infinite-state systems, such as pushdowns, VASS counters, integer counters, and combinations thereof. In our main results, we provide a complete description of the decidability and complexity landscape of viability games across valence systems over graph monoids. Our results reveal encouraging decidability properties. For example, in certain combinations of pushdowns and counters, viability games are decidable, despite non-termination games being undecidable there. Moreover, viability games are even decidable for certain systems where (single-player) control-state reachability is undecidable.

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.

Abstract We consider two-player games over finite graphs in which both players are restricted by fairness constraints on their moves. Given a two player game graph $$G=(V,E)$$ G = ( V , E ) and a set of fair moves $$E_f\subseteq E$$ E f ⊆ E a player is said to play fair in G if they choose an edge $$e\in E_f$$ e ∈ E f infinitely often whenever the source node of e is visited infinitely often. Otherwise, they play unfair . We equip such games with two $$\omega $$ ω -regular winning conditions $$\alpha $$ α and $$\beta $$ β deciding the winner of mutually fair and mutually unfair plays, respectively. Whenever one player plays fair and the other plays unfair, the fairly playing player wins the game. The resulting games are called fair $$\alpha /\beta $$ α / β games . We formalize fair $$\alpha /\beta $$ α / β games and show that they are determined. For fair parity/parity games, i.e., fair $$\alpha /\beta $$ α / β games where $$\alpha $$ α and $$\beta $$ β are given each by a parity condition over G , we provide a polynomial reduction to (normal) parity games via a gadget construction inspired by the reduction of stochastic parity games to parity games. We further give a direct symbolic fixpoint algorithm to solve fair parity/parity games. On a conceptual level, we illustrate the translation between the gadget-based reduction and the direct symbolic algorithm which uncovers the underlying similarities of solution algorithms for fair and stochastic parity games, as well as for the recently considered class of fair games in which only one player is restricted by fair moves.

Abstract We provide an algorithm to solve Rabin and Streett games over graphs with n vertices, m edges, and k colours that runs in $$\tilde{O}\left( mn(k!)^{1+o(1)} \right) $$ O ~ m n ( k ! ) 1 + o ( 1 ) time and $$O(nk\log k \log n)$$ O ( n k log k log n ) space, where $$\tilde{O}$$ O ~ hides poly-logarithmic factors. Our algorithm is an improvement by a super quadratic dependence on k! from the currently best known run time of $$O\left( mn^2(k!)^{2+o(1)}\right) $$ O m n 2 ( k ! ) 2 + o ( 1 ) , obtained by converting a Rabin game into a parity game, while simultaneously improving its exponential space requirement. Our main technical ingredient is a characterisation of progress measures for Rabin games using colourful trees and a combinatorial construction of succinctly-represented, universal colourful trees. Colourful universal trees are generalisations of universal trees used by Jurdziński and Lazić (2017) to solve parity games, as well as of Rabin progress measures of Klarlund and Kozen (1991). Our algorithm for Rabin games is a progress measure lifting algorithm where the lifting is performed on succinct, colourful, universal trees.

This paper discusses the problem of efficiently solving parity games where player Odd has to obey an additional 'strong transition fairness constraint' on its vertices -- given that a player Odd vertex $v$ is visited infinitely often, a particular subset of the outgoing edges (called live edges) of $v$ has to be taken infinitely often. Such games, which we call 'Odd-fair parity games', naturally arise from abstractions of cyber-physical systems for planning and control. In this paper, we present a new Zielonka-type algorithm for solving Odd-fair parity games. This algorithm not only shares 'the same worst-case time complexity' as Zielonka's algorithm for (normal) parity games but also preserves the algorithmic advantage Zielonka's algorithm possesses over other parity solvers with exponential time complexity. We additionally introduce a formalization of Odd player winning strategies in such games, which were unexplored previous to this work. This formalization serves dual purposes: firstly, it enables us to prove our Zielonka-type algorithm; secondly, it stands as a noteworthy contribution in its own right, augmenting our understanding of additional fairness assumptions in two-player games.