Strategy logic with imperfect information

Abstract

We introduce an extension of Strategy logic for the imperfect-information setting, called SL ii , and study its modelchecking 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 automatatheoretic 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 .

Games on Graphs with a Public Signal Monitoring

“…defined using solely the signal (hence forgetting private actions). While this does not affect the recall for distributed synthesis (existence of a strategy for the grand coalition), nor for Nash equilibria (existence of a strategy profile as well), we believe this is important for more complex interactions between players (for instance for multi-agent logics like in [3]).…”

“…While playing, if h ∼ A h , A will not be able to know whether h or h has been played. We also define the 3 We extend all the above notions to infinite sequences in a straightforward way and to the notion of full play. We write Plays G (v 0 ) (or simply Plays(v 0 ) if G is clear in the context) for the set of full plays in G that start at v 0 .…”

“…Otherwise they are said invisible. Private visibility of payoffs, while not always assumed (see for instance [19,3]), are reasonable assumptions: using only her knowledge, a player knows her payoff. Public visibility is more restrictive, but will be required for some of the results.…”

“…We then use the abstraction to develop algorithms for some payoff functions. * This work has been supported by ERC project EQualIS (FP7-308087).1 This can be expressed by payoff player i = Rwhere γ i is the signal-to-interferenceand-noise ratio for player i, R is the rate at which the wireless system transmits the information and L is the size of the packets [29].Recently imperfect information has also been introduced in the setting of multi-agent temporal logics [20,21,2,3], and the main decidability results assume hierarchical information. However, while those logics allow to express rich interactions, it can somehow only consider qualitative properties.…”

“…Recently imperfect information has also been introduced in the setting of multi-agent temporal logics [20,21,2,3], and the main decidability results assume hierarchical information. However, while those logics allow to express rich interactions, it can somehow only consider qualitative properties.…”