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A parity game, where a vertex with priority i has label pi. The dotted edge in red is a co-live edge, while the dashed edges in blue are singleton live-groups. (Color figure online)
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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 $$\...
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... To identify these small and useful sub-games, we use permissive strategy templates [2] in finite-state abstracted games. They describe a potentially infinite set of winning strategies using local conditions on the transitions of the game. ...
... The main objective of this work is to identify small and useful sub-games, for which the results can enhance the symbolic game-solving process. To achieve this, we use a technique called permissive strategy templates [2], designed for finite game graphs. These templates can represent (potentially infinite) sets of winning strategies through local edge conditions. ...
... Definition 3 (Winning Strategy Template [2]). A strategy template Ψ for player p is winning if every strategy satisfying Ψ is winning for p in pG, Ωq. ...
Infinite-state games are a commonly used model for the synthesis of reactive systems with unbounded data domains. Symbolic methods for solving such games need to be able to construct intricate arguments to establish the existence of winning strategies. Often, large problem instances require prohibitively complex arguments. Therefore, techniques that identify smaller and simpler sub-problems and exploit the respective results for the given game-solving task are highly desirable. In this paper, we propose the first such technique for infinite-state games. The main idea is to enhance symbolic game-solving with the results of localized attractor computations performed in sub-games. The crux of our approach lies in identifying useful sub-games by computing permissive winning strategy templates in finite abstractions of the infinite-state game. The experimental evaluation of our method demonstrates that it outperforms existing techniques and is applicable to infinite-state games beyond the state of the art.
... This iterative refinement is enabled by a novel formalization of contracted strategy-masks (csm) (Ψ i , Π i ) for each Player i which encodes the essence of the required cooperation between players in a concise data structure. Contracted strategy-masks are based on the idea of permissive templates which were recently introduced to formalize and synthesize permissive assumptions [4] and permissive strategies [6] in (zero-sum) parity games. Within a csm, Π i is a strategy template which collects an infinite number of winning strategies for Player i under the assumption that Player 1 − i chooses any strategy from the assumption template Ψ i . ...
... Our algorithm for solving Problem 5 is introduced in Section 4-Section 6. Conceptually, this algorithm builds upon the recently introduced formalism of permissive templates from [4,6] and utilizes their efficient computability, adaptability and permissiveness to solve Problem 5. Moreover, we show that our algorithm is sound and complete, i.e., it always terminates and returns an iR-maC-specification pair. ...
... This section recalls the concept of templates from [4,6]. In principle, a template is simply an LTL formula Λ over a game graph G. ...
We present a novel method to compute $\textit{assume-guarantee contracts}$ in non-zerosum two-player games over finite graphs where each player has a different $ \omega $-regular winning condition. Given a game graph $G$ and two parity winning conditions $\Phi_0$ and $\Phi_1$ over $G$, we compute $\textit{contracted strategy-masks}$ ($\texttt{csm}$) $(\Psi_{i},\Phi_{i})$ for each Player $i$. Within a $\texttt{csm}$, $\Phi_{i}$ is a $\textit{permissive strategy template}$ which collects an infinite number of winning strategies for Player $i$ under the assumption that Player $1-i$ chooses any strategy from the $\textit{permissive assumption template}$ $\Psi_{i}$. The main feature of $\texttt{csm}$'s is their power to $\textit{fully decentralize all remaining strategy choices}$ -- if the two player's $\texttt{csm}$'s are compatible, they provide a pair of new local specifications $\Phi_0^\bullet$ and $\Phi_1^\bullet$ such that Player $i$ can locally and fully independently choose any strategy satisfying $\Phi_i^\bullet$ and the resulting strategy profile is ensured to be winning in the original two-objective game $(G,\Phi_0,\Phi_1)$. In addition, the new specifications $\Phi_i^\bullet$ are $\textit{maximally cooperative}$, i.e., allow for the distributed synthesis of any cooperative solution. Further, our algorithmic computation of $\texttt{csm}$'s is complete and ensured to terminate. We illustrate how the unique features of our synthesis framework effectively address multiple challenges in the context of \enquote{correct-by-design} logical control software synthesis for cyber-physical systems and provide empirical evidence that our approach possess desirable structural and computational properties compared to state-of-the-art techniques.
... In addition, our new game solving formalism is also related to other work in the reactive synthesis community. While strategy templates have been very recently introduced in [23,24], progress group annotations appeared previously in [25] for a restricted class of temporal specifications and only induced by uncontrolled dynamics. Further, [26] also tackles the problem of reactive control for dynamical systems via parity games, but only presents sufficient conditions for the existence of certificates and controllers, while our method is fully constructive. ...
... While it is well known how to compute a single winning strategy for a parity game G, it was recently shown that strategy templates [23], which characterize an infinite number of winning strategies in a succinct manner, are particularly useful in the context of CPS control design. They are utilized within this paper to obtain a novel translation of high-level logical control actions into low-level feedback controllers. ...
... Intuitively, the initial game G I constructed from φ via Lemma 2 reveals all logical dependencies of propositions relevant to the synthesis problem at hand. After constructing G I from φ, we can directly apply the algorithm from [23] to synthesize a winning strategy template ψ I on G I as discussed in Section 2.4. ...
We consider the problem of automatically synthesizing a hybrid controller for non-linear dynamical systems which ensures that the closed-loop fulfills an arbitrary \emph{Linear Temporal Logic} specification. Moreover, the specification may take into account logical context switches induced by an external environment or the system itself. Finally, we want to avoid classical brute-force time- and space-discretization for scalability. We achieve these goals by a novel two-layer strategy synthesis approach, where the controller generated in the lower layer provides invariant sets and basins of attraction, which are exploited at the upper logical layer in an abstract way. In order to achieve this, we provide new techniques for both the upper- and lower-level synthesis. Our new methodology allows to leverage both the computing power of state space control techniques and the intelligence of finite game solving for complex specifications, in a scalable way.
... In addition, our new game solving formalism is also related to other work in the reactive synthesis community. While strategy templates have been very recently introduced in [23], [24], progress group annotations appeared previously in [25] for a restricted class of temporal specifications and only induced by uncontrolled dynamics. Further, [26] also tackles the problem of reactive control for dynamical systems via parity games, but only presents sufficient conditions for the existence of certificates and controllers, while our method is fully constructive. ...
... While it is well known how to compute a single winning strategy for a parity game G, it was recently shown that strategy templates [23], which characterize an infinite number of winning strategies in a succinct manner, are particularly useful in the context of CPS control design. They are utilized within this paper to obtain a novel translation of high-level logical control actions into low-level feedback controllers. ...
... Intuitively, the initial game G I constructed from φ via Lemma 2 reveals all logical dependencies of propositions relevant to the synthesis problem at hand. After constructing G I from φ (i.e., going from 1 to 2 in Fig. 2), we can directly apply the algorithm from [23] to synthesize a winning strategy template ψ I on G I (i.e., going from 2 to 3 in Fig. 2) as discussed in Section II-D. This gives the following result which is a direct consequence of Lemma 2 and the definition of strategy templates. ...
We consider the problem of automatically synthesizing a hybrid controller for non-linear dynamical systems which ensures that the closed-loop fulfills an arbitrary
Linear Temporal Logic
specification. Moreover, the specification may take into account logical context switches induced by an external environment or the system itself. Finally, we want to avoid classical brute-force time- and space-discretization for scalability. We achieve these goals by a novel two-layer strategy synthesis approach, where the controller generated in the lower layer provides invariant sets and basins of attraction, which are exploited at the upper logical layer in an abstract way. In order to achieve this, we provide new techniques for both the upper- and lower-level synthesis. Our new methodology allows to leverage both the computing power of state space control techniques and the intelligence of finite game solving for complex specifications, in a scalable way.
