Strategy Logic with Imperfect Information

Author:

Berthon Raphaël1,Maubert Bastien2,Murano Aniello2,Rubin Sasha3,Vardi Moshe Y.4

Affiliation:

1. Université libre de Bruxelles 8 University of Antwerp, Triomphe, Ixelles, Belgium

2. Università degli Studi di Napoli “Federico II,”, Napoli, Italy

3. University of Sydney, Darlington NSW, Australia

4. Rice University, Houston, TX, USA

Abstract

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.

Funder

NSF

Maryland Procurement Office

European Union's Horizon 2020 research and innovation programme

Marie Sklodowska-Curie

Publisher

Association for Computing Machinery (ACM)

Subject

Computational Mathematics,Logic,General Computer Science,Theoretical Computer Science

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