Game-theoretic semantics for ATL+ with applications to model checking

Abstract

We develop a game-theoretic semantics (GTS) for the fragment ATL + of the Alternating-time Temporal Logic ATL * , essentially extending a recently introduced GTS for ATL. We show that the new game-theoretic semantics is equivalent to the standard compositional semantics of ATL + (with perfect-recall strategies). Based on the new semantics, we provide an analysis of the memory and time resources needed for model checking ATL + and show that strategies of the verifier that use only a very limited amount of memory suffice. Furthermore, using the GTS we provide a new algorithm for model checking ATL + and identify a natural hierarchy of tractable fragments of ATL + that extend ATL.

Strategic reasoning with a bounded number of resources: The quest for tractability

“…ATL + , ATL * ) ptime-complete [1] (resp. pspace-complete [27,28], 2exptime-complete [1]), there is no difference in complexity for RB±ATL, RB±ATL + and RB±ATL * , as far as worst-case complexity is concerned.…”

“…Though establishing an exptime upper bound for MC(RB±ATL + (1)) reveals a substantial improvement in complexity compared to the 2exptime-completeness of MC(RB±ATL * ) [12], the exact complexity remains open and seems quite challenging. It is still unclear to us whether the developments in [27,Section 3.2] or in [28] about MC(ATL + ) could be adapted to obtain an optimal upper bound.…”

“…Obviously, if O is expressive enough for RB±ATL + O to embed CTL, both problems MC(RB±ATL + O ({1}, 1)) and MC(RB±ATL({1}, 1)) are ptime-complete since ptimehardness is inherited from the model-checking problem for CTL [43]. Note also that ptime fragments of ATL + have been investigated in [70].…”

“…what is done in [12] for RB±ATL * , where the number of locations is double exponential. Hence, the model-checking problem for RB±ATL + restricted to one resource is pspace-hard (inherited from ATL + [27,28]) and in exptime. Apart from the complexity results that improve our understanding of fragments of RB±ATL * , we present reductions to the parity game problem for AVASS that can be of interest for their own sake.…”

The power of clockings

“…The paper [2] provides a game-theoretic semantics for ATL with clock values declared on the fly. The paper [3] has similar motivations, but investigates the more involved case of ATL + . The article [5] concentrates on the µ-calculus.…”