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# Dynamic hierarchical information: when reaching v2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_2$$\end{document}, player 1 is more informed than player 2, but the order changes when the play proceeds to position v4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_4$$\end{document}

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Infinite games with imperfect information are known to be undecidable unless the information flow is severely restricted. One fundamental decidable case occurs when there is a total ordering among players, such that each player has access to all the information that the following ones receive. In this paper we consider variations of this hierarchy...

## Citations

... Like in the single-process scenario, synthesis in distributed systems can be modeled as a game, which, in this context, are partial information games played between a cooperating set of processes against the environment [17,18,19,20]. With the exception of Berwanger et al. [20], all the above approaches assume static, reliable networks. ...

... Like in the single-process scenario, synthesis in distributed systems can be modeled as a game, which, in this context, are partial information games played between a cooperating set of processes against the environment [17,18,19,20]. With the exception of Berwanger et al. [20], all the above approaches assume static, reliable networks. In [20], Berwanger et al. study games in which information that players have about histories is hierarchically ordered, and this order may change dynamically during a play. ...

... With the exception of Berwanger et al. [20], all the above approaches assume static, reliable networks. In [20], Berwanger et al. study games in which information that players have about histories is hierarchically ordered, and this order may change dynamically during a play. The main difference to our work is that we consider a memory model where messages carry the complete causal history allowing for unbounded communication messages, while [20] is based on local observations so that, at every round, a bounded amount of information is transmitted between players. ...

The problem of distributed synthesis is to automatically generate a distributed algorithm, given a target communication network and a specification of the algorithm's correct behavior.
Previous work has focused on static networks with an a priori fixed message size. This approach has two shortcomings: Recent work in distributed computing is shifting towards dynamically changing communication networks rather than static ones, and an important class of distributed algorithms are so-called full-information protocols, where nodes piggy-pack previously received messages onto current messages.
In this work, we consider the synthesis problem for a system of two nodes communicating in rounds over a dynamic link whose message size is not bounded. Given a network model, i.e., a set of link directions, in each round of the execution, the adversary choses an arbitrary link from the network model, restricted only by the specification, and delivers messages according to the current link's directions. Motivated by communication buses with direct acknowledge mechanisms, we further assume that nodes are aware of which messages have been delivered.
We show that the synthesis problem is decidable for a network model if and only if the network model does not contain the empty link that dismisses both nodes' messages. We then extend the characterization to sequences of communication links that may contain empty links. We show that the synthesis problem is decidable in this case if and only if the number of consecutive empty links in all possible sequences is uniformly bounded from above.

... Like in the single-process scenario, synthesis in distributed systems can be modeled as a game, which, in this context, are partial information games played between a cooperating set of processes against the environment [9,30,32,33]. With the exception of [9], all the above approaches assume static, reliable networks. ...

... Like in the single-process scenario, synthesis in distributed systems can be modeled as a game, which, in this context, are partial information games played between a cooperating set of processes against the environment [9,30,32,33]. With the exception of [9], all the above approaches assume static, reliable networks. In [9], Berwanger et al. study games in which information that players have about histories is hierarchically ordered, and this order may change dynamically during a play. ...

... With the exception of [9], all the above approaches assume static, reliable networks. In [9], Berwanger et al. study games in which information that players have about histories is hierarchically ordered, and this order may change dynamically during a play. The main difference to our work is that we consider a memory model where messages carry the complete causal history allowing for unbounded communication messages, while [9] is based on local observations so that, at every round, a bounded amount of information is transmitted between players. ...

... where the players can be totally ordered according to how well they observe the system. This restriction has been used to establish results on multiplayer games [PRA02,BMvdB18] and distributed synthesis [PR90,KV01,FS05], and more recently on the model-checking problem for SL iR , an extension of Strategy Logic to the imperfect-information setting [BMM + 17]. This result states that the model-checking problem for SL iR is decidable as long as strategies quantified deeper in the formula observe the system better than those higher up in the syntactic tree. ...

Strategy Logic with imperfect information (SLiR) is a very expressive logic designed to express complex properties of strategic abilities in distributed systems. Previous work on SLiR focused on finite systems, and showed that the model-checking problem is decidable when information on the control states of the system is hierarchical among the players or components of the system, meaning that the players or components can be totally ordered according to their respective knowledge of the state. We show that moving from finite to infinite systems generated by collapsible (higher-order) pushdown systems preserves decidability, under the natural restriction that the stack content is visible. The proof follows the same lines as in the case of finite systems, but requires to use (collapsible) alternating pushdown tree automata. Such automata are undecidable, but semi-alternating pushdown tree automata were introduced and proved decidable, to study a strategic problem on pushdown systems with two players. In order to tackle multiple players with hierarchical information, we refine further these automata: we define direction-guided (collapsible) pushdown tree automata, and show that they are stable under projection, nondeterminisation and narrowing. For the latter operation, used to deal with imperfect information, stability holds under some assumption that is satisfied when used for systems with visible stack. We then use these automata to prove our main result.

... In the case of multiple players/components/agents, which interests us here, the situation is even worse: the existence of distributed winning strategies is undecidable already for two players with incomparable observation trying to enforce some reachability objective in the presence of an adversarial third player [65], and a similar result was also proved in the framework of distributed synthesis [69]. Since then, the formal-methods community has spent much effort finding restrictions and variations that ensure decidability [8,31,35,50,64,66,69,74]. The common thread in these approaches is hierarchical information: players can be totally ordered according to how well they observe the game. ...

... The literature on imperfect information in formal methods and artificial intelligence is very vast. Imperfect information has been considered in two-player games [7,26,73], module checking [43,52], distributed synthesis of reactive systems [31,50,69] and strategies in multiplayer games [8,64,65], Nash equilibria [11,13,72], rational synthesis [30,38], doomsday equilibria [19], admissible strategies [14], quantitative objectives [24,62], and more, some of which we detail below. ...

... But when synthesising programs for instance, it may be enough that their behaviours enforce the desired properties, without them having the knowledge that it is enforced. Such non-observable winning conditions have been studied in, e.g., [8,16,24]. ...

We introduce an extension of Strategy Logic for the imperfect-information setting, called SLii, and study its model-checking problem. As this logic naturally captures multi-player games with imperfect information, this problem is undecidable; but we introduce a syntactical class of "hierarchical instances" for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model, and we prove that model-checking SLii restricted to hierarchical instances is decidable. To establish this result we go through QCTL, an intermediary, "low-level" logic much more adapted to automata techniques. QCTL is an extension of CTL with second-order quantification over atomic propositions. We extend it to the imperfect information setting by parameterising second-order quantifiers with observations. While the model-checking problem of QCTLii is, in general, undecidable, we identify a syntactic fragment of hierarchical formulas and prove, using an automata-theoretic approach, that it is decidable. We apply our result to solve complex strategic problems in the imperfect-information setting. We first show that the existence of Nash equilibria for deterministic strategies is decidable in games with hierarchical information. We also introduce distributed rational synthesis, a generalisation of rational synthesis to the imperfect-information setting. Because it can easily be expressed in our logic, our main result provides solution to this problem in the case of hierarchical information.

... Like in the single-process scenario, synthesis in distributed systems can be modeled as a game, which, in this context, are partial information games played between a cooperating set of processes against the environment [8, 29, 30, 39]. With the exception of Berwanger et al. ...

The problem of distributed synthesis is to automatically generate a distributed algorithm, given a target communication network and a specification of the algorithm's correct behavior. Previous work has focused on static networks with an apriori fixed message size. This approach has two shortcomings: Recent work in distributed computing is shifting towards dynamically changing communication networks rather than static ones, and an important class of distributed algorithms are so-called full-information protocols, where nodes piggy-pack previously received messages onto current messages. In this work we consider the synthesis problem for a system of two nodes communicating in rounds over a dynamic link whose message size is not bounded. Given a network model, i.e., a set of link directions, in each round of the execution, the adversary choses a link from the network model, restricted only by the specification, and delivers messages according to the current link's directions. Motivated by communication buses with direct acknowledge mechanisms we further assume that nodes are aware of which messages have been delivered. We show that the synthesis problem is decidable for a network model if and only if it does not contain the empty link that dismisses both nodes' messages.

... We consider multiplayer game arenas with imperfect information in the spirit of, e.g., [38,21,9]. Since the DEL games we define in the next section are turn-based, i.e., the agents play in turns and not concurrently, we define turn-based arenas instead of the more general concurrent ones usually considered in the aforementioned works. ...

Dynamic Epistemic Logic (DEL) is a logical framework in which one can describe in great detail how actions are perceived by the agents, and how they affect the world. DEL games were recently introduced as a way to define classes of games with imperfect information where the actions available to the players are described very precisely. This framework makes it possible to define easily, for instance, classes of games where players can only use public actions or public announcements. These games have been studied for reachability objectives, where the aim is to reach a situation satisfying some epistemic property expressed in epistemic logic; several (un)decidability results have been established. In this work we show that the decidability results obtained for reachability objectives extend to a much more general class of winning conditions, namely those expressible in the epistemic temporal logic LTLK. To do so we establish that the infinite game structures generated by DEL public actions are regular, and we describe how to obtain finite representations on which we rely to solve them.

... Games with imperfect information are computationally hard, and even undecidable for multiple players [29]. One way to tame this complexity is to make assumptions on how the knowledge of the different players compare: if all players that cooperate can be ordered in a hierarchy where one knows more than the next, a situation called hierarchical information, then the existence of distributed strategies can be decided [28,7]. Another natural approach is to consider fragments based on classes of action types, as done for instance in [32,6,11] where different kinds of public actions are considered. ...

We define reachability games based on Dynamic Epistemic Logic (DEL), where the players' actions are finely described as DEL action models. We first consider the setting where an external controller with perfect information interacts with an environment and aims at reaching some epistemic goal state regarding the passive agents of the system. We study the problem of strategy existence for the controller, which generalises the classic epistemic planning problem, and we solve it for several types of actions such as public announcements and public actions. We then consider a yet richer setting where agents themselves are players, whose strategies must be based on their observations. We establish several (un)decidability results for the problem of existence of a distributed strategy, depending on the type of actions the players can use, and relate them to results from the literature on multiplayer games with imperfect information.

... The most general decidability results in the concurrent game setting are under the This work has been supported by ERC project EQualIS (FP7-308087). assumption of hierarchical observation [6,36] (information received by the players is ordered) or more recently under recurring common knowledge [5]. ...

We study pure Nash equilibria in games on graphs with an imperfect monitoring based on a public signal. In such games, deviations and players responsible for those deviations can be hard to detect and track. We propose a generic epistemic game abstraction, which conveniently allows to represent the knowledge of the players about these deviations, and give a characterization of Nash equilibria in terms of winning strategies in the abstraction. We then use the abstraction to develop algorithms for some payoff functions.

... Distributed synthesis is a fairly hot topic, both using the formalization via concurrent games we have already described and using the formalization via an architecture of processes [26]. The most general decidability results in the concurrent game setting are under the assumption of hierarchical observation [36,7] (information received by the players is ordered) or more recently under recurring common knowledge [6]. ...

... We have that (1) S ∈ val Core S (s). 7 The set of possible improvements of those values is finite (try to assign 0 instead of 1 to every agent A ∈ S). ...

We study Nash equilibria in games on graphs with an imperfect monitoring based on a public signal. In such games, deviations and players responsible for those deviations can be hard to detect and track. We propose a generic epistemic game abstraction, which conveniently allows to represent the knowledge of the players about these deviations, and give a characterization of Nash equilibria in terms of winning strategies in the abstraction. We then use the abstraction to develop algorithms for some payoff functions.

We introduce an extension of Strategy Logic for the imperfect-information setting, called SL ii and study its model-checking problem. As this logic naturally captures multi-player games with imperfect information, this problem is undecidable; but we introduce a syntactical class of “hierarchical instances” for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model, and we prove that model-checking SL ii restricted to hierarchical instances is decidable. This result, because it allows for complex patterns of existential and universal quantification on strategies, greatly generalises the decidability of distributed synthesis for systems with hierarchical information. It allows us to easily derive new decidability results concerning strategic problems under imperfect information such as the existence of Nash equilibria or rational synthesis.
To establish this result, we go through an intermediary, “low-level” logic much more adapted to automata techniques. QCTL * is an extension of CTL * with second-order quantification over atomic propositions that has been used to study strategic logics with perfect information. We extend it to the imperfect information setting by parameterising second-order quantifiers with observations. The simple syntax of the resulting logic, QCTL * ii , allows us to provide a conceptually neat reduction of SL ii to QCTL * ii that separates concerns, allowing one to forget about strategies and players and focus solely on second-order quantification. While the model-checking problem of QCTL * ii is, in general, undecidable, we identify a syntactic fragment of hierarchical formulas and prove, using an automata-theoretic approach, that it is decidable.