Strategy Logic with Imperfect Information

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.

Towards Partial Order Reductions for Strategic Ability

“…We suspect that, unlike for synchronous systems, allowing for nondeterministic choices in strategies might make a difference in the semantics. We also plan to extend our method to a larger subset of ATL * specifications, a subset of Strategy Logic (Berthon et al, 2017a), and to sATL * with epistemic operators using possibly techniques reported by Cermák et al (2014). Further experimental evaluation of the reductions on randomly generated models is also on the list.…”

Strategic Reasoning in Automated Mechanism Design

“…Strategy Logic (SL) (Chatterjee, Henzinger, and Piterman 2010;Mogavero et al 2014) was then proposed which, by treating strategies as first-order variables, can express complex game-theoretic concepts. SL has been extended to handle imperfect information and knowledge operators (Berthon et al 2021;Belardinelli et al 2020;Maubert and Murano 2018), but none of these logics can account for quantitative aspects. Recently, SL[F ] (Bouyer et al 2019a) was introduced as a quantitative extension of SL.…”

“…However in the imperfect-information setting, we need to know with respect to which observation relation a strategy should be uniform. In (Berthon et al 2021) this is done by parameterizing strategy quantifiers with observation relations. Here we adopt a slightly less general but more intuitive notation, by parameterizing directly with the agent who will use the strategy.…”

Synthesizing Best-effort Strategies under Multiple Environment Specifications

“…Many complex synthesis problems can been expressed in a number of logics for strategic reasoning, notably in Strategy Logic and its variations (Mogavero et al 2014;Berthon et al 2021;Belardinelli et al 2020;Aminof et al 2019b). These allow one to quantify over strategies, and thus can naturally express complex strategic properties, including dominance and maximality.…”