Rajeev Alur, Thomas A. Henzinger, Orna Kupferman, and Moshe Y. Vardi
Alternating transition systems are a general model for composite systems which allows the study of collaborative as well as adversarial relationships between individual system components. Unlike in labeled transition systems, where each transition corresponds to a possible step of the system (which may involve some or all components), in alternating transition systems, each transition corresponds to a possible move in a game between the components. In this paper, we study refinement relations between alternating transition systems, such as "Does the implementation refine the set A of specification components without constraining the components not in A?" In particular, we generalize the definitions of the simulation and trace containment preorders from labeled transition systems to alternating transition systems. The generalizations are called alternating simulation and alternating trace containment. Unlike existing refinement relations, they allow the refinement of individual components within the context of a composite system description. We show that, like ordinary simulation, alternating simulation can be checked in polynomial time using a fixpoint computation algorithm. While ordinary trace containment is PSPACE-complete, we prove alternating trace containment to be EXPTIME-complete. Finally, we present logical characterizations for the two preorders in terms of ATL, a temporal logic capable of referring to games between system components.
Proceedings of the Ninth International Conference on Concurrency Theory (CONCUR), Lecture Notes in Computer Science 1466, Springer, 1998, pp. 163-178.