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Games with recurring certainty
Abstract
Infinite games where several players seek to coordinate under imperfect
information are known to be intractable, unless the information flow is
severely restricted. Examples of undecidable cases typically feature a
situation where players become uncertain about the current state of the game,
and this uncertainty lasts forever. Here we consider games where the players
attain certainty about the current state over and over again along any play.
For finite-state games, we note that this kind of recurring certainty implies a
stronger condition of periodic certainty, that is, the events of state
certainty ultimately occur at uniform, regular intervals. We show that it is
decidable whether a given game presents recurring certainty, and that, if so,
the problem of synthesising coordination strategies under w-regular winning
conditions is solvable.
... We believe that the next steps are, first, to see whether the syntactical fragments studied for SL with perfect information, such as One-Goal or Boolean-Goal Strategy Logic, can be transferred to BSL and then to ESL, and see whether they enjoy better complexity properties. The second natural move would be to look at structures which are known to work well with multiple agents with imperfect information: hierarchical knowledge [7,20], recurring common knowledge of the state [6]. . . ...
We propose an extension of Strategy Logic (SL), in which one can both reason about strategizing under imperfect information and about players' knowledge. One original aspect of our approach is that we do not force strategies to be uniform, i.e. consistent with the players' information, at the semantic level; instead, one can express in the logic itself that a strategy should be uniform. To do so, we first develop a "branching-time" version of SL with perfect information, that we call BSL, in which one can quantify over the different outcomes defined by a partial assignment of strategies to the players; this contrasts with SL, where temporal operators are allowed only when all strategies are fixed, leaving only one possible play. Next, we further extend BSL by adding distributed knowledge operators, the semantics of which rely on equivalence relations on partial plays. The logic we obtain subsumes most strategic logics with imperfect information, epistemic or not.
... In addition, both the epistemic mu-calculus and imperfect-information games are decidable for perfect-recall when the knowledge of the agents is hierarchically ordered [7,21]. Recent results on games with imperfect information also signal other classes of models with several agents and perfect recall that can be handled [6]. So even for the case of several relations on paths, a logic that would encompass both the (jumping) mu-calculus and games with imperfect information may have good computational properties on interesting classes of systems. ...
We revisit Janin and Walukiewicz’s classic result on the expressive completeness of the modal mu-calculus w.r.t. MSO, when transition systems are equipped with a binary relation over paths. We obtain two natural extensions of MSO and the mu-calculus: MSO
with path relation and the jumping mu-calculus. While “bounded-memory” binary relations bring about no extra expressivity to either of the two logics, “unbounded-memory” binary relations make the bisimulation-invariant fragment of MSO with path relation more expressive than the jumping mu-calculus: the existence of winning strategies in games with imperfect-information inhabits the gap.
... The present article extends a previous report [13] presented at the Workshop on Strategic Reasoning. ...
Infinite games where several players seek to coordinate under imperfect
information are believed to be intractable, unless the information is
hierarchically ordered among the players. We identify a class of games for
which joint winning strategies can be constructed effectively without
restricting the direction of information flow. Instead, our condition requires
that the players attain common knowledge about the actual state of the game
over and over again along every play. We show that it is decidable whether a
given game satisfies the condition, and prove tight complexity bounds for the
strategy synthesis problem under parity winning conditions.
We consider the synthesis of a reactive module with input x and output y, which is specified by the linear temporal formula @@@@(x, y). We show that there exists a program satisfying @@@@ iff the branching time formula (∀x) (∃y) A@@@@(x, y) is valid over all tree models. For the restricted case that all variables range over finite domains, the validity problem is decidable, and we present an algorithm for constructing the program whenever it exists. The algorithm is based on a new procedure for checking the emptiness of Rabin automata on infinite trees in time exponential in the number of pairs, but only polynomial in the number of states. This leads to a synthesis algorithm whose complexity is double exponential in the length of the given specification.
Game quantification is an expressive concept and has been studied in model theory and descriptive set theory, especially in
relation to infinitary logics. Automatic structures on the other hand appear very often in computer science, especially in
program verification. We extend first-order logic on structures on words by allowing to use an infinite string of alternating
quantifiers on letters of a word, the game quantifier. This extended logic is decidable and preserves regularity on automatic
structures, but can be undecidable on other structures even with decidable first-order theory. We show that in the presence
of game quantifier any relation that allows to distinguish successors is enough to define all regular relations and therefore
the game quantifier is strictly more expressive than first-order logic in such cases. Conversely, if there is an automorphism
of atomic relations that swaps some successors then we prove that it can be extended to any relations definable with game
quantifier. After investigating it’s expressiveness, we use game quantification to introduce a new type of combinatorial games
with multiple players and imperfect information exchanged with respect to a hierarchical constraint. It is shown that these
games on finite arenas exactly capture the logic with game quantifier when players alternate their moves but are undecidable
and not necessarily determined in the other case. In this way we define the first model checking games with finite arenas
that can be used for model checking first-order logic on automatic structures.
We present a general construction for eliminating imperfect information from games with several players who coordinate against nature, and to transform them into two-player games with perfect information while preserving winning strategy profiles. The construction yields an infinite game tree with epistemic models associated to nodes. To obtain a more succinct representation, we define an abstraction based on homomorphic equivalence, which we prove to be sound for games with observable winning conditions. The abstraction generates finite game graphs in several relevant cases, and leads to a new semi-decision procedure for multi-player games with imperfect information.
When seeking to coordinate in a game with imperfect infor- mation, it is often relevant for a player to know what other players know. Keeping track of the information acquired in a play of innite duration may, however, lead to innite hierarchies of higher-order knowledge. We present a construction that makes explicit which higher-order knowledge is relevant in a game and allows us to describe a class of games that admit coordinated winning strategies with nite memory.
The controller synthesis problem as formalized by Ramadge and Wonham consists of finding a finite controller that when synchronized with a given plant results in a system satisfying a required property. In this setting, both a plant and a controller are deterministic fi- nite automata, while synchronization is modelled by a synchronous product. Originally, the framework was developed only for safety and some deadlock properties. More recently, Arnold et. al. have extended the setting to all mu-calculus expressible properties and proposed a reduction of the synthesis problem to the satisfiability problem of the mu-calculus. They have also presented some results on decidability of distributed synthesis problem where one requires to find several controllers that control the plant at the same time. The additional difficulty in this case is that each controller is aware of a different part of the whole system. In this paper, an extension of the setting to nondeterministic pro- cesses is studied. In other words, the case when both a system and a controller can be presented by a nondeterministic automaton. This extension is motivated by examples in control where a continuous quantity is measured and digitized thereby introducing imprecision and uncertainty. It is shown that nondeterminism of the plant can be handled at no extra cost, both for centralized and decentralized con- trol. Centralized synthesis remains decidable even for nondetermin- istic controllers. In contrast, very few cases of decentralized control are decidable when controllers are allowed to be nondeterministic. A classification of decidable/undecidable variants of this problem is given.
The distributed synthesis problem [11] is known to be undecidable. Our purpose here is to study further this undecidability.
For this, we consider distributed games [8], an infinite variant of Peterson and Reif multiplayer games with partial information [10], in which Pnueli and Rosner’s distributed synthesis problem can be encoded and, when decidable [11,6,7], uniformly solved [8].
We first prove that even the simple problem of solving 2-process distributed game with reachability conditions is undecidable (\(\Sigma^0_1\)-complete). This decision problem, equivalent to two process distributed synthesis with fairly restricted FO-specification was left open [8]. We prove then that the safety case is \(\Pi^0_1\)-complete. More generally, we establish a correspondence between 2-process distributed game with Mostowski’s weak parity conditions [9] and levels of the arithmetical hierarchy. finally, distributed games with general ω-regular infinitary conditions are shown to be highly undecidable (\(\Sigma^1_1\)-complete).
A central aim and ever-lasting dream of computer science is to put the development of hardware and software systems on a mathematical basis which is both firm and practical. Such a scientific foundation is needed especially for the construction of reactive programs, like communication protocols or control systems.
For the construction and analysis of reactive systems an elegant and powerful theory has been developed based on automata theory, logical systems for the specification of nonterminating behavior, and infinite two-person games.
The 19 chapters presented in this multi-author monograph give a consolidated overview of the research results achieved in the theory of automata, logics, and infinite games during the past 10 years. Special emphasis is placed on coherent style, complete coverage of all relevant topics, motivation, examples, justification of constructions, and exercises.
Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by system moves; branching-time temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternating-time temporal logic offers selective quantification over those paths that are possible outcomes of games, such as the game in which the system and the environment alternate moves. While linear-time and branching-time logics are natural specification languages for closed systems, alternative-time logics are natural specification languages for open systems. For example, by preceding the temporal operator "eventually" with a selective path quantifier, we can specify that in the game between the system and the environment, the system has a strategy to reach a certain state. Also, the problems of receptiveness, realizability, and controllability can be formulated as model-checking problems for alternating-time formulas.
Depending on whether we admit arbitrary nesting of selective path quantifiers and temporal operators, we obtain the two alternating-time temporal logics ATL and ATL*. We interpret the formulas of ATL and ATL* over alternating transition systems. While in ordinary transitory systems, each transition corresponds to a possible step of the system, in alternating transition systems, each transition corresponds to a possible move in the game between the system and the environment. Fair alternating transition systems can capture both synchronous and asynchronous compositions f open systems. For synchronous systems, the expressive power of ATL beyond CTL comes at no cost: the model-checking complexity of synchronous ATL is linear in the size of the system and the length of the formula. The symbolic model-checking algorithm for CTL extends with few modifications to synchronous ATL, and with some work, also to asynchronous to ATL, whose model-checking complexity is quadratic. This makes ATL an obvious candidate for the automatic verification of open systems. In the case of ATL*, the model-checking problem is closely related to the synthesis problem for linear-time formulas, and requires doubly exponential time for both synchronous and asynchronous systems.
Two-player games of incomplete information have certain portions of positions which are private to each player and cannot be viewed by the opponent. Asymptotically optimal decision algorithms for space bounded games are provided. Various games of incomplete information are presented which are shown to be universal in the sense that they are the hardest of all reasonable games of incomplete information. The problem of determining the outcome of these universal games from a given initial position is shown to be complete in doubly exponential time. “Private alternating Turing machines” are defined to be a new type of alternating Turing machines related to games of incomplete information. The space complexity S(n) of these machines is characterized in terms of the complexity of deterministic Turing machines, with time bounds doubly exponential in S(n). Blindfold games are restricted games in that the second player is not allowed to modify the common position. Asymptotically optimal decision algorithms for space bounded blindfold games are provided. Various blindfold games are also shown to have exponential space complete outcome problems and to be universal for reasonable blindfold games. “Blind alternating Turing machines” are defined to be private alternating Turing machines with restrictions similar to those in blindfold games. The space complexity of these machines is characterized in terms of the complexity of deterministic Turing machines with a single exponential increase in space bounds.
We provide a uniform solution to the problem of synthesizing a finite-state distributed system. An instance of the synthesis problem consists of a system architecture and a temporal specification. The architecture is given as a directed graph, where the nodes represent processes (including the environment as a special process) that communicate synchronously through shared variables attached to the edges. The same variable may occur on multiple outgoing edges of a single node, allowing for the broadcast of data. A solution to the synthesis problem is a collection of finite-state programs for the processes in the architecture, such that the joint behavior of the programs satisfies the specification in an unrestricted environment. We define information forks, a comprehensive criterion that characterizes all architectures with an undecidable synthesis problem. The criterion is effective: for a given architecture with n processes and v variables, it can be determined in O(n<sup>2</sup>·v) time whether the synthesis problem is decidable. We give a uniform synthesis algorithm for all decidable cases. Our algorithm works for all ω-regular tree specification languages, including the μ-calculus. The undecidability proof, on the other hand, uses only LTL or, alternatively, CTL as the specification language. Our results therefore hold for the entire range of specification languages from LTL/CTL to the μ-calculus.