Preprint

Reasoning about Knowledge and Strategies

Authors:
Preprints and early-stage research may not have been peer reviewed yet.
If you want to read the PDF, try requesting it from the authors.

Abstract

Two distinct semantics have been considered for knowledge in the context of strategic reasoning, depending on whether players know each other’s strategy or not. The problem of distributed synthesis for epistemic temporal specifications is known to be undecidable for the latter semantics, already on systems with hierarchical information. However, for the other, uninformed semantics, the problem is decidable on such systems. In this work we generalise this result by introducing an epistemic extension of Strategy Logic with imperfect information. The semantics of knowledge operators is uninformed, and captures agents that can change observation power when they change strategies. We solve the model-checking problem on a class of "hierarchical instances", which provides a solution to a vast class of strategic problems with epistemic temporal specifications on hierarchical systems, such as distributed synthesis or rational synthesis.

No file available

Request Full-text Paper PDF

To read the file of this research,
you can request a copy directly from the authors.

ResearchGate has not been able to resolve any citations for this publication.
Conference Paper
Full-text available
We study a class of synchronous, perfect-recall multi-agent systemswith imperfect information and broadcasting (i.e., fully observableactions). We define an epistemic extension of strategy logic withincomplete information and the assumption of uniform and coherentstrategies. In this setting, we prove that the model checking problem,and thus rational synthesis, is decidable with non-elementarycomplexity. We exemplify the applicability of the framework on arational secret-sharing scenario.
Conference Paper
Full-text available
We analyse the verification problem for synchronous, perfect recall multi-agent systems with imperfect information against a specification language that includes strategic as well as epistemic operators. While the general problem is known to be undecidable we show that if the agents' actions are public then verification is decidable, and we establish that the computational complexity is 2EexpTime-complete. To illustrate the formal framework we consider two well-known epistemic and strategic puzzles with imperfect information and public actions: the muddy children puzzle and the classic game of battleships.
Article
Full-text available
Synthesis is the automated construction of a system from its specification. The system has to satisfy its specification in all possible environments. The environment often consists of agents that have objectives of their own. Thus, it makes sense to soften the universal quantification on the behavior of the environment and take the objectives of its underlying agents into an account. Fisman et al. introduced rational synthesis: the problem of synthesis in the context of rational agents. The input to the problem consists of temporal logic formulas specifying the objectives of the system and the agents that constitute the environment, and a solution concept (e.g., Nash equilibrium). The output is a profile of strategies, for the system and the agents, such that the objective of the system is satisfied in the computation that is the outcome of the strategies, and the profile is stable according to the solution concept; that is, the agents that constitute the environment have no incentive to deviate from the strategies suggested to them. In this paper we continue to study rational synthesis. First, we suggest an alternative definition to rational synthesis, in which the agents are rational but not cooperative. We call such problem strong rational synthesis. In the strong rational synthesis setting, one cannot assume that the agents that constitute the environment take into account the strategies suggested to them. Accordingly, the output is a strategy for the system only, and the objective of the system has to be satisfied in all the compositions that are the outcome of a stable profile in which the system follows this strategy. We show that strong rational synthesis is 2ExpTime-complete, thus it is not more complex than traditional synthesis or rational synthesis. Second, we study a richer specification formalism, where the objectives of the system and the agents are not Boolean but quantitative. In this setting, the objective of the system and the agents is to maximize their outcome. The quantitative setting significantly extends the scope of rational synthesis, making the game-theoretic approach much more relevant. Finally, we enrich the setting to one that allows coalitions of agents that constitute the system or the environment.
Article
Full-text available
A general concept of uniform strategies has recently been proposed as a relevant notion in game theory for computer science, which subsumes various notions from the literature. It relies on properties involving sets of plays in two-player turn-based arenas equipped with arbitrary binary relations between plays; these properties are expressed in a language based on with a quantifier over related plays. There are two semantics for our quantifier, a strict one and a full one, that we study separately. Regarding the strict semantics, the existence of a uniform strategy is undecidable for rational binary relations, but introducing jumping tree automata and restricting attention to recognizable relations allows us to establish a 2-Exptime-complete complexity – and still capture a class of two-player imperfect-information games with epistemic temporal objectives. Regarding the full semantics, relying on information set automata we establish that the existence of a uniform strategy is decidable for rational relations and we provide a nonelementary synthesis procedure. We also exhibit an essentially optimal subclass of rational relations for which the problem becomes 2-Exptime-complete. Considering rich classes of relations makes the theory of uniform strategies powerful: it directly entails various results in logics of knowledge and time, some of them already known, and others new.
Conference Paper
Full-text available
We introduce MCMAS-SLK, a BDD-based model checker for the verification of systems against specifications expressed in a novel, epistemic variant of strategy logic. We give syntax and semantics of the specification language and a introduce a labelling algorithm for epistemic and strategy logic modalities. We provide details of the checker which can also be used for synthesizing agents strategies so that a specification is satisfied by the system. We evaluate the efficiency of the implementation by discussing the results obtained for a scheduling system and the dining cryptographers protocol.
Conference Paper
Full-text available
We present a variant of ATL with distributed knowledge operators based on a synchronous and perfect recall semantics. The coalition modalities in this logic are based on partial observation of the full history, and incorporate a form of cooperation between members of the coalition in which agents issue their actions based on the distributed knowledge, for that coalition, of the system history. We show that model-checking is decidable for this logic. The technique utilizes two variants of games with imperfect information and partially observable objectives, as well as a subset construction for identifying states whose histories are indistinguishable to the considered coalition.
Article
Full-text available
We look at ways to enrich Alternating-time Temporal Logic (ATL) - a logic for specifi- cation and verification of multi-agent systems - with a notion of knowledge. Starting point of our study is a recent proposal for a system called Alternating-time Temporal Epistemic Logic (ATEL). We show that, assuming that agents act under uncertainty in some states of the system, the notion of allowable strategy should be defined with some caution. Moreover, we demonstrate a subtle differ- ence between an agent knowing that he has a suitable strategy and knowing the strategy itself. We also point out that the agents should be assumed similar epistemic capabilities in the semantics of both strategic and epistemic operators. Trying to implement these ideas, we propose two different modifications of ATEL. The first one, dubbed Alternating-time Temporal Observational Logic (ATOL), is a logic for agents with bounded recall of the past. With the second, ATEL-R*, we present a framework to reason about both perfect and imperfect recall, and in which we also incorporate operators for reasoning about the past. We identify some feasible subsystems of this expressive system.
Article
Full-text available
Branching-time temporal logics have proved to be an extraordinarily successful tool in the formal specification and verification of distributed systems. Much of their success stems from the tractability of the model checking problem for the branching time logic CTL, which has made it possible to implement tools that allow designers to automatically verify that systems satisfy requirements expressed in CTL. Recently, CTL was generalised by Alur, Henzinger, and Kupferman in a logic known as Alternating-time Temporal Logic (ATL). The key insight in ATL is that the path quantifiers of CTL could be replaced by cooperation modalities, of the form , where is a set of agents. The intended interpretation of an ATL formula is that the agents can cooperate to ensure that holds (equivalently, that have a winning strategy for ). In this paper, we extend ATL with knowledge modalities, of the kind made popular in the work of Fagin, Halpern, Moses, Vardi and colleagues. Combining these knowledge modalities with ATL, it becomes possible to express such properties as group can cooperate to bring about iff it is common knowledge in that . The resulting logic — Alternating-time Temporal Epistemic Logic (ATEL) — shares the tractability of model checking with its ATL parent, and is a succinct and expressive language for reasoning about game-like multiagent systems.
Article
Full-text available
We investigate extensions of temporal logic by connectives defined by finite automata on infinite words. We consider three different logics, corresponding to three different types of acceptance conditions (finite, looping, and repeating) for the automata. It turns out, however that these logics all have the same expressive power and that their decision problems are all PSPACE-complete. We also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem.
Article
Full-text available
In open systems verification, to formally check for reliability, one needs an appropriate formalism to model the interaction between agents and express the correctness of the system no matter how the environment behaves. An important contribution in this context is given by modal logics for strategic ability, in the setting of multi-agent games, such as ATL, ATL\star, and the like. Recently, Chatterjee, Henzinger, and Piterman introduced Strategy Logic, which we denote here by CHP-SL, with the aim of getting a powerful framework for reasoning explicitly about strategies. CHP-SL is obtained by using first-order quantifications over strategies and has been investigated in the very specific setting of two-agents turned-based games, where a non-elementary model-checking algorithm has been provided. While CHP-SL is a very expressive logic, we claim that it does not fully capture the strategic aspects of multi-agent systems. In this paper, we introduce and study a more general strategy logic, denoted SL, for reasoning about strategies in multi-agent concurrent games. We prove that SL includes CHP-SL, while maintaining a decidable model-checking problem. In particular, the algorithm we propose is computationally not harder than the best one known for CHP-SL. Moreover, we prove that such a problem for SL is NonElementarySpace-hard. This negative result has spurred us to investigate here syntactic fragments of SL, strictly subsuming ATL\star, with the hope of obtaining an elementary model-checking problem. Among the others, we study the sublogics SL[NG], SL[BG], and SL[1G]. They encompass formulas in a special prenex normal form having, respectively, nested temporal goals, Boolean combinations of goals and, a single goal at a time. About these logics, we prove that the model-checking problem for SL[1G] is 2ExpTime-complete, thus not harder than the one for ATL\star.
Conference Paper
Full-text available
The problem of synthesizing a finite-state distributed reactive system is considered. Given a distributed architecture A , which comprises several processors P <sub>1</sub>, . . ., P <sub>k</sub> and their interconnection scheme, and a propositional temporal specification φ, a solution to the synthesis problem consists of finite-state programs Π<sub>1</sub>, . . ., Π<sub>k </sub> (one for each processor), whose joint (synchronous) behavior maintains φ against all possible inputs from the environment. Such a solution is referred to as the realization of the specification φ over the architecture A . Specifically, it is shown that the problem of realizing a given propositional specification over a given architecture is undecidable, and it is nonelementarily decidable for the very restricted class of hierarchical architectures. An extensive characterization of architecture classes for which the realizability problem is elementarily decidable and of classes for which it is undecidable is given
Article
Full-text available
We study the problems of synthesizing open systems as well as controllers for them. The key aspect of our model is that it caters to reactive environments, which can disable different sets of responses when reacting with the system. We deal with specifications given as formulas in CTL* and its sub-logic CTL. We show that both these problems, with specifications in CTL (CTL*), are 2EXPTIME-complete (resp. 3EXPTIME-complete). Thus, in a sense, reactive environments constitute a provably harder setting for the synthesis of open systems and controllers for them.
Conference Paper
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, the problem turns out to be undecidable. 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. 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 previous ones, such as decidability of multi-player games with imperfect information and hierarchical observations, and decidability of distributed synthesis for hierarchical systems. To establish the decidability result, we introduce and study QCTL * ii , an extension of QCTL (itself an extension of CTL with second-order quantification over atomic propositions) by parameterising its quantifiers with observations. The simple syntax of 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. The decidability result for SL ii follows since the reduction maps hierarchical instances of SL ii to hierarchical formulas of QCTL * ii .
Article
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.
Article
We study the extension of the alternating-time temporal logic (ATL) with strategy contexts: contrary to the original semantics, in this semantics the strategy quantifiers do not reset the previously selected strategies.We show that our extension ATLsc is very expressive, but that its decision problems are quite hard: model checking is k-EXPTIME-complete when the formula has k nested strategy quantifiers; satisfiability is undecidable, but we prove that it is decidable when restricting to turn-based games. Our algorithms are obtained through a very convenient translation to QCTL (the computation-tree logic CTL extended with atomic quantification), which we show also applies to Strategy Logic, as well as when strategy quantification ranges over memoryless strategies.
Article
While it was defined long ago, the extension of CTL with quantification over atomic propositions has never been studied extensively. Considering two different semantics (depending whether propositional quantification refers to the Kripke structure or to its unwinding tree), we study its expressiveness (showing in particular that QCTL coincides with Monadic Second-Order Logic for both semantics) and characterise the complexity of its model-checking and satisfiability problems, depending on the number of nested propositional quantifiers (showing that the structure semantics populates the polynomial hierarchy while the tree semantics populates the exponential hierarchy).
Conference Paper
A propositional logic of distributed protocols is introduced which includes both the logic of knowledge and temporal logic. Phenomena in distributed computing systems such as asynchronous time, incomplete knowledge by the computing agents in the system, and game-like behavior among the computing agents are all modeled in the logic. Two versions of the logic, the linear logic of protocols (LLP) and the tree logic of protocols (TLP) are investigated. The main result is that the set of valid formulas in LLP is undecidable.
Article
A distributed computing system consists of a set of individual processors that communicate through some medium. Coordinating the actions of such processors is essential in distributed computing. Researchers have long endeavored to find efficient solutions to a variety of coordination problems. Recently, processor knowledge has been used to characterize such solutions and to derive more efficient ones. Most of this work has concentrated on the relationship between common knowledge and simultaneous coordination. This paper considers non-simultaneous coordination problems. The results of this paper add to our understanding of the relationship between knowledge and the different requirements of coordination problems. This paper considers the ideas of optimal and optimum solutions to a coordination problem and precisely characterizes the problems for which optimum solutions exist. This characterization is based on combinations of eventual common knowledge and continual common knowledge. The paper then considers more general problems, for which optimal, but no optimum, solutions exist. It defines a new form of knowledge, called extended knowledge, which combines eventual and continual knowledge, and shows how extended knowledge can be used to both characterize and construct optimal protocols for coordination.
Article
We examine a class of infinite two-person games on finitely coloured graphs. The main aim is to construct finite memory winning strategies for both players. This problem is motivated by applications to finite automata on infinite trees. A special attention is given to the exact amount of memory needed by the players for their winning strategies. Based on a previous work of Gurevich and Harrington and on subsequent improvements of McNaughton we propose a unique framework that allows to reestablish and to improve various results concerning memoryless strategies due to Emerson and Jutla, Mostowski, Klarlund.
Article
In this paper the full branching time logic (CTL*) is studied. It has basic modalities consisting of a path quantifier, either A (“for all paths”) of E (“for some path”), followed by an arbitrary linear time assertion composed of unrestricted combinations of the usual linear temporal operators F (“sometime”), G (“always”), X (“nexttime”), and U (“until”). It is shown that the problem of determining if a CTL* formula is satisfiable in a structure generated by a binary relation is decidable in triple exponential time. The decision procedure exploits the special structure of the finite state ω-automata for linear temporal formulae which allows them to be determinized with only a single exponential blowup in size. Also the expressive power of tree automata is compared with that of CTL* augmented by quantified auxillary propositions.
Conference Paper
In temporal-logic model checking, we verify the correctness of a program with respect to a desired behaviour by checking whether a structure that models the program satisfies a temporal logic formula that specifies this behaviour. One of the ways to overcome the expressiveness limitation of temporal logics is to augment them with quantification over atomic propositions. In this paper we consider the extension of branching temporal logics with existential quantification over atomic propositions. Once we add existential quantification to a branching temporal logic, it becomes sensitive to unwinding. That is, unwinding a structure into an infinite tree does not preserve the set of formulas it satisfies. Accordingly, we distinguish between two semantics, two practices as specification languages, and two versions of the model-checking problem for such a logic. One semantics refers to the structure that models the program, and the second semantics refers to the infinite computation tree that the program induces. We examine the complexity of the model-checking problem in the two semantics for the logics CTL and CTL* augmented with existential quantification over atomic propositions. Following the cheerless results that we get, we examine also the program complexity of model checking; i.e. the complexity of this problem in terms of the program, assuming the formula is fixed. We show that while fixing the formula dramatically reduces model-checking complexity in the tree semantics, its influence on the structure semantics is poor.
Conference Paper
We address the strategy problem for ω-regular two-player games with partial information, played on finite game graphs. We consider two different kinds of observability on a general model, a standard synchronous and an asynchronous one. In the asynchronous setting, moves which have no visible effect for a player are hidden completely from that player. We generalize the usual powerset construction for eliminating partial information to arbitrary, not necessarily observation based, winning conditions, both in the synchronous and in the asynchronous case, and we show that this generalized construction effectively preserves ω-regular winning conditions. From this we infer decidability of the strategy problem for arbitrary ω-regular winning conditions, in both cases. We also show that our ω-regular framework is sufficient for reasoning about synchronous and asynchronous knowledge by proving that any formula of the epistemic temporal specification formalism ETL can be effectively translated into an S1S-formula defining the same specification.
Conference Paper
We extend the branching temporal logics CTL and CTL* with quantified propositions and consider various semantic interpretations for the quantification. The use of quantificiation greatly increases the expressive power of the logics allowing us to represent, for example, tree-automata. We also show that some interpretations of quantification allow us to represent non-propositional properties of Kripke frames, such as the branching degree of trees. However this expressive power may also make the satisfiability problem for the logic undecidable. We give a proof of one such case, and also examine decidability in the less expressive semantics.
Article
We give a proof that alternating tree automata can be simulated by nondeterministic tree automata which yields new complexity results and a unified proof of the theorems of Rabin, McNaughton and Safra. We also give a simple axiomatic framework for uniformizing strategies.
Article
In program synthesis, we transform a specification into a system that is guaranteed to satisfy the specification. When the system is open, then at each moment it reads input signals and writes output signals, which depend on the input signals and the history of the computation so far. The specification considers all possible input sequences. Thus, if the specification is linear, it should hold in every computation generated by the interaction, and if the specification is branching, it should hold in the tree that embodies all possible input sequences. Often, the system cannot read all the input signals generated by its environment. For example, in a distributed setting, it might be that each process can read input signals of only part of the underlying processes. Then, we should transform a specification into a system whose output depends only on the readable parts of the input signals and the history of the computation. This is called synthesis with incomplete information . In this work we solve the problem of synthesis with incomplete information in its full generality. We consider linear and branching settings with complete and incomplete information. We claim that alternation is a suitable and helpful mechanism for coping with incomplete information. Using alternating tree automata , we show that incomplete information does not make the synthesis problem more complex, in both the linear and the branching paradigm. In particular, we prove that independently of the presence of incomplete information, the synthesis problems for CTL and CTL*. are complete for EXPTIME and 2EXPTIME, respectively.
Conference Paper
We generalize the alternation machines of Chandra, Kozen and Stockmeyer [1] and the private alternation machines of Reif [14] to model multiple person (team) games of incomplete information. The resulting classes of machines are "multiple person alternation machines". The characterization of certain time and space bounded versions of these machines demonstrate interesting relationships between ordinary time and space hierarchies (Table 1). Our results are applied to relative succintness and power questions of finite state machines and to complexity questions of parallel finite state machines. Other machine variants, including private alternating pushdown store automata and Markovian alternation machines, are discussed.
Article
Inspired by the success of the distributed computing community in applying logics of knowledge and time to reasoning about distributed protocols, we aim for a similarly powerful and high-level abstraction when reasoning about control problems involving uncertainty. This paper concentrates on robot motion planning with uncertainty in both control and sensing, a problem that has already been well studied within the robotics community. First, a new and natural problem in this domain is defined: Does there exists a sound and complete termination condition for a motion, given initial and goal locations? If yes, how to construct it? Then we define a high-level language, a logic of time and knowledge, which we use to reason about termination conditions and to state general conditions for the existence of sound and complete termination conditions in a broad domain. Finally, we show that sound termination conditions that are optimal in a precise sense provide a natural example of knowledge-base...
Article
: Reasoning about knowledge seems to play a fundamental role in distributed systems. Indeed, such reasoning is a central part of the informal intuitive arguments used in the design of distributed protocols. Communication in a distributed system can be viewed as the act of transforming the system's state of knowledge. This paper presents a general framework for formalizing and reasoning about knowledge in distributed systems. We argue that states of knowledge of groups of processors are useful concepts for the design and analysis of distributed protocols. In particular, distributed knowledge corresponds to knowledge that is "distributed" among the members of the group, while common knowledge corresponds to a fact being "publicly known". The relationship between common knowledge and a variety of desirable actions in a distributed system is illustrated. Furthermore, it is shown that, formally speaking, in practical systems common knowledge cannot be attained. A number of weaker variants...
A logic of knowledge and strategies with imperfect information
  • F Belardinelli
Belardinelli, F. 2015. A logic of knowledge and strategies with imperfect information. In LAMAS'15, 1-15.
Admissibility in games with imperfect information (invited talk)
  • R Brenguier
  • A Pauly
  • J Raskin
  • O Sankur
Brenguier, R.; Pauly, A.; Raskin, J.; and Sankur, O. 2017. Admissibility in games with imperfect information (invited talk). In CONCUR'17, 2:1-2:23.
  • K Chatterjee
  • T A Henzinger
  • N Piterman
Chatterjee, K.; Henzinger, T. A.; and Piterman, N. 2010. Strategy Logic. Inf. Comput. 208(6):677-693.
The complexity of rational synthesis
  • R Condurache
  • E Filiot
  • R Gentilini
  • J Raskin
Condurache, R.; Filiot, E.; Gentilini, R.; and Raskin, J. 2016. The complexity of rational synthesis. In ICALP'16, 121:1-121:15.
Decidability of quantifed propositional branching time logics
  • R Fagin
  • J Y Halpern
  • Y Moses
  • M Y Vardi
Fagin, R.; Halpern, J. Y.; Moses, Y.; and Vardi, M. Y. 1995. Reasoning about knowledge. MIT press Cambridge. French, T. 2001. Decidability of quantifed propositional branching time logics. In AJCAI'01, 165-176.
Dealing with imperfect information in strategy logic
  • S Knight
  • B Maubert
Knight, S., and Maubert, B. 2015. Dealing with imperfect information in strategy logic. In SR.