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Graded Alternating-Time Temporal Logic

Faella M., Napoli M., Parente M.

Fundamenta Informaticae

2011

Vol. 105, nr 1/2

189-210

Recently, temporal logics such as μ-calculus and Computational Tree Logic, CTL, augmented with graded modalities have received attention from the scientific community, both from a theoretical side and from an applicative perspective. In both these settings, graded modalities enrich the universal and existential quantifiers with the capability to express the concept of at least k or all but k, for a non-negative integer k. Both μ-calculus and CTL naturally apply as specification languages for closed systems: in this paper, we study how graded modalities may affect specification languages for open systems. We extend the Alternating-time Temporal Logic (ATL) introduced by Alur et al., that is a derivative of CTL interpreted on game structures, rather than transition systems. We solve the model-checking problem in the concurrent and turn-based settings, proving its PTIME-completeness. We present, and compare with each other, two different semantics: the first seems suitable to off-line synthesis applicationswhile the secondmay find application in the verification of fault-tolerant controllers. We also study the case where players can only employ memoryless strategies, showing that also in this case the model-checking problem is in PTIME.