Strategy Logic with Imperfect Information

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, 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 SLii 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 SLii 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 SLii follows since the reduction maps hierarchical instances of SLii to hierarchical formulas of QCTL*ii

Strategy Logic with Imperfect Information

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.Comment: arXiv admin note: text overlap with arXiv:1805.1259

Knowledge and Blameworthiness

Blameworthiness of an agent or a coalition of agents is often defined in terms of the principle of alternative possibilities: for the coalition to be responsible for an outcome, the outcome must take place and the coalition should have had a strategy to prevent it. In this article we argue that in the settings with imperfect information, not only should the coalition have had a strategy, but it also should have known that it had a strategy, and it should have known what the strategy was. The main technical result of the article is a sound and complete bimodal logic that describes the interplay between knowledge and blameworthiness in strategic games with imperfect information

A Hennessy-Milner Theorem for ATL with Imperfect Information

We show that a history-based variant of alternating bisimulation with imperfect information allows it to be related to a variant of Alternating-time Temporal Logic (ATL) with imperfect information by a full Hennessy-Milner theorem. The variant of ATL we consider has a common knowledge semantics, which requires that the uniform strategy available for a coalition to accomplish some goal must be common knowledge inside the coalition, while other semantic variants of ATL with imperfect information do not accommodate a Hennessy-Milner theorem. We also show that the existence of a history-based alternating bisimulation between two finite Concurrent Game Structures with imperfect information (iCGS) is undecidable

Reasoning about Knowledge and Strategies under Hierarchical Information

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

Alternating (In)Dependence-Friendly Logic

Hintikka and Sandu originally proposed Independence Friendly Logic ([Formula presented]) as a first-order logic of imperfect information to describe game-theoretic phenomena underlying the semantics of natural language. The logic allows for expressing independence constraints among quantified variables, in a similar vein to Henkin quantifiers, and has a nice game-theoretic semantics in terms of imperfect information games. However, the [Formula presented] semantics exhibits some limitations, at least from a purely logical perspective. It treats the players asymmetrically, considering only one of the two players as having imperfect information when evaluating truth, resp., falsity, of a sentence. In addition, truth and falsity of sentences coincide with the existence of a uniform winning strategy for one of the two players in the semantic imperfect information game. As a consequence, [Formula presented] does admit undetermined sentences, which are neither true nor false, thus failing the law of excluded middle. These idiosyncrasies limit its expressive power to the existential fragment of Second Order Logic ([Formula presented]). In this paper, we investigate an extension of [Formula presented], called Alternating Dependence/Independence Friendly Logic ([Formula presented]), tailored to overcome these limitations. To this end, we introduce a novel compositional semantics, generalising the one based on trumps proposed by Hodges for [Formula presented]. The new semantics (i) allows for meaningfully restricting both players at the same time, (ii) enjoys the property of game-theoretic determinacy, (iii) recovers the law of excluded middle for sentences, and (iv) grants [Formula presented] the full descriptive power of [Formula presented]. We also provide an equivalent Herbrand-Skolem semantics and a game-theoretic semantics for the prenex fragment of [Formula presented], the latter being defined in terms of a determined infinite-duration game that precisely captures the other two semantics on finite structures

Dependency Matrices for Multiplayer Strategic Dependencies

In multi-player games, players take their decisions on the basis of their knowledge about what other players have done, or currently do, or even, in some cases, will do. An ability to reason in games with temporal dependencies between players\u27 decisions is a challenging topic, in particular because it involves imperfect information. In this work, we propose a theoretical framework based on dependency matrices that includes many instances of strategic dependencies in multi-player imperfect information games. For our framework to be well-defined, we get inspiration from quantified linear-time logic where each player has to label the timeline with truth values of the propositional variable she owns. We study the problem of the existence of a winning strategy for a coalition of players, show it is undecidable in general, and exhibit an interesting subclass of dependency matrices that makes the problem decidable: the class of perfect-information dependency matrices