Dependences in Strategy Logic

Abstract

Strategy Logic (SL) is a very expressive logic for specifying and verifying properties of multi-agent systems: in SL, one can quantify over strategies, assign them to agents, and express properties of the resulting plays. Such a powerful framework has two drawbacks: first, model checking SL has non-elementary complexity; second, the exact semantics of SL is rather intricate, and may not correspond to what is expected. In this paper, we focus on strategy dependences in SL, by tracking how existentially-quantified strategies in a formula may (or may not) depend on other strategies selected in the formula. We study different kinds of dependences, refining the approach of [Mogavero et al., Reasoning about strategies: On the model-checking problem, 2014], and prove that they give rise to different satisfaction relations. In the setting where strategies may only depend on what they have observed, we identify a large fragment of SL for which we prove model checking can be performed in 2 -EXPTIME.Strategic interactions in ATL. Strategies in ATL are handled in a very limited way, and there are no real strategic interactions in that logic (which, in return, enjoys a polynomial-time model-checking algorithm). Over the last 10 years, various extensions have been defined and studied in order to allow for more interactions [1,6,5,14,21]. Strategy Logic (SL for short) [6,14] is such a powerful approach, in which strategies are first-class objects; formulas can quantify (universally and existentially) over strategies, store those strategies in variables, assign them to players, and express properties of the resulting plays. As a simple example, the existence of a winning strategy for Player A (with objective ϕ A ) against any strategy of Player B would be written asThis makes the logic both expressive and easy to use (at first sight), at the expense of a very high complexity:SL model checking has non-elementary complexity, and satisfiability is undecidable [14,11]. Dependences in Strategy LogicStrategy dependences in SL. It has been noticed in recent works that the nice expressiveness of SL comes with unexpected phenomena. One recently-identified phenomenon [4] is induced by the separation of strategy quantification and strategy assignment: are the events between strategy quantifications and strategy assignments part of the memory of the strategy? While both options may make sense depending on the applications, only one of them makes model checking decidable [4].A second phenomenon-which is the main focus of the present paper-concerns strategy dependences [14]: in a formula such as ∀σ A . ∃σ B . ξ, the existentially-quantified strategy σ B may depend on the whole strategy σ A ; in other terms, the action returned by strategy σ B after some finite history ρ may depend on what strategy σ A would play on any other history ρ . Again, this may be desirable in some contexts, but it may also make sense to require that strategy σ B after history ρ can be computed based solely on what has been observed along ρ. This approach was initiate...

Behavioral QLTL

“…Indeed, both the formula ϕ = ∀x∃y(Gx ↔ y) and its negation are not satisfiable, that is |= B ϕ and |= B ¬ϕ 2 . This is a common phenomenon, as it also happens when considering the behavioral semantics of logic for the strategic reasoning [28,21]. It is important, however, to notice that there are three syntactic fragments for which QLTL and QLTL B are equivalent.…”

“…Such requirement is practically impossible to fulfill and does not reflect the notion of reactive systems, where the output of system variables at the k-th instant of the computation depends only on the past assignments of the environment variables. Such principle is often referred as behavioral principle in the context of strategic reasoning, see e.g., [28,21].…”

“…In addition, they proved that behavioral and non-behavioral semantics coincide for certain fragments, including the well known ATL ⋆ [2], but diverge for more interesting classes of formulas, e.g., the ones that can express game-theoretic properties such as Nash Equilibria and the like. This has started a new line of research that aims at identifying new notions of behavioral and non-behavioral quantification, as well as characterize the syntactic fragments that are invariant to these semantic variations [20,21].…”

Good-for-Game QPTL: An Alternating Hodges Semantics

“…p must return the same value at k as a reply to both χ 1 and χ 2 , i.e., F(χ 1 )(k) = F(χ 2 )(k); in other words, F(χ)(k) cannot exploit the knowledge of the values χ(p)(t), with t > k. An analogous concept has been introduced in SL [48]. A stronger notion of behavioralness, similar to one reported in [18], requires the functor F to satisfy the above equality when χ 1 and χ 2 only (possibly) differ on p for t ≥ k and leads to the concept of strongly behavioral quantification. In game-theoretic terms, the interpretation of a behavioral quantifier w.r.t.…”

“…In the case of SL, for instance, the notion of behavioral semantics has been introduced [48], which prevents the players from looking at future choices when selecting their strategy, effectively limiting the player observation ability to the current history in the game. A more liberal semantics based on timeline dependencies has been also proposed [18], [19]. While these approaches do solve the problem in the specific case, they lead to non-compositional semantics [60], in that the interpretation of a formula is not defined in terms of the interpretation of its component subformulae.…”