Alternating-time temporal logic

Abstract

Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by the execution of a system; branching-time temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternating-time temporal logic offers selective quantification over those paths that are possible outcomes of games, such as the game in which the system and the environment alternate moves. While linear-time and branching-time logics are natural specification languages for closed systems, alternating-time logics are natural specification languages for open systems. For example, by preceding the temporal operator "eventually" with a selective path quantifier, we can specify that in the game between the system and the environment, the system has a strategy to reach a certain state. The problems of receptiveness, realizability, and controllability can be formulated as model-checking problems for alternating-time formulas. Depending on whether or not we admit arbitrary nesting of selective path quantifiers and temporal operators, we obtain the two alternatingtime temporal logics ATL and ATL .ATL and ATL are interpreted over concurrent game structures. Every state transition of a concurrent game structure results from a choice of moves, one for each player. The players represent A preliminary version of this article appeared in the individual components and the environment of an open system. Concurrent game structures can capture various forms of synchronous composition for open systems, and if augmented with fairness constraints, also asynchronous composition. Over structures without fairness constraints, the model-checking complexity of ATL is linear in the size of the game structure and length of the formula, and the symbolic model-checking algorithm for CTL extends with few modifications to ATL. Over structures with weak-fairness constraints, ATL model checking requires the solution of 1-pair Rabin games, and can be done in polynomial time. Over structures with strong-fairness constraints, ATL model checking requires the solution of games with Boolean combinations of Büchi conditions, and can be done in PSPACE. In the case of ATL , the model-checking problem is closely related to the synthesis problem for linear-time formulas, and requires doubly exponential time.

Extended Graded Modalities in Strategy Logic

“…Sentences of GRADEDSL are interpreted over concurrent game structures, just as for ATL and SL [3,37]. Definition 2.1 A concurrent game structure (CGS) is a tuple G AP, Ag, Ac, St, s I , ap, tr , where…”

“…T, u such that T is the set of histories h of G starting with s and u(h) (f, q) where q is the last symbol in h and f : free(ϕ) → Ac is defined by f(e) χ(e)(h) for all e ∈ free(ϕ). 3 The following lemma is proved by induction on the structure of the formula ϕ, as in [37]. The idea for handling the new case, i.e., the graded quantifier x 1 , .…”

“…the size of the model. 3. If all grades are restricted to be in N, then the complexity is (k + 1)EXPTIME if k ≥ 1 is the quantifier-block rank of ϕ.…”

“…A strategy x can be quantified existentially x (read: there exists a strategy x) and, dually, universally [[x]] (read: for all strategies x). Strategies are not intrinsically glued to specific agents: the binding operator (α, x) allows one to bind an agent α to the strategy x. SL is built over the structure of the linear-time temporal-logic LTL [43] and strictly subsumes several other well-known logics for the strategic reasoning including ATL [3] and the like [1,14,24,27,28,34,49,50].…”

Deciding Epistemic and Strategic Properties of Cryptographic Protocols

“…Recently, game-based properties of cryptographic protocols have been studied [KR02]. Such properties are relevant e.g., for contract signing [BOGMR90,ASW98,GJM99] and non-repudiation [KR03] protocols, and can naturally be expressed in Alternating-Time Temporal Logic (ATL, [AHK02]), a logic explicitly designed to reason about strategies. Decidability results for such properties have been obtained in [KKT07,KKW09].…”