Strategy logic with imperfect information

Berthon, Raphaël; Maubert, Bastien; Murano, Aniello; Rubin, Sasha; Vardi, Moshe Y.

doi:10.1145/3427955

Title

Strategy logic with imperfect information

Author

Berthon, Raphaël

Maubert, Bastien

Murano, Aniello

Rubin, Sasha

Vardi, Moshe Y.

Abstract

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. 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 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.

Language

English

Source (journal)

ACM transactions on computational logic. - New York, N.Y.

Publication

New York, N.Y. : 2021

ISSN

1529-3785

DOI

10.1145/3427955

Volume/pages

22 :1 (2021) , 51 p.