# Alternating refinement relations

## Abstract

*Alternating transition systems* are a general model for composite systems which allow 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 establish 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.

## Keywords

Transition System Temporal Logic Winning Strategy Label Transition System Tree Automaton## Preview

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