VERIFICATION OF FAIR TRANSITION-SYSTEMS

In program verification, we check that an implementation meets its spe cification. Both the specification and the implementation describe the possible behaviors of the program, although at different levels of ab straction. We distinguish between two approaches to implementation of specifications. The first approach is trace-based implementation, wher e we require every computation of the implementation to correlate to s ome computation of the specification. The second approach is tree-base d implementation, where we require every computation tree embodied in the implementation to correlate to some computation tree embodied in t he specification. The two approaches to implementation are strongly re lated to the linear-time versus branching-time dichotomy in temporal l ogic. In this work, we examine the trace-based and the tree-based appr oaches from a complexity-theoretic point of view. We consider and comp are the complexity of verification of fair transition systems, modelin g both the implementation and the specification in the two approaches. We consider unconditional, weak, and strong fairnesses. For the trace -based approach, the corresponding problem is fair containment. For th e tree-based approach, the corresponding problem is fair simulation. W e show that while both problems are PSPACE-complete, their complexitie s in terms of the size of the implementation do not coincide, and the trace-based approach is easier. As the implementation is normally much bigger than the specification, we see this as an advantage of the tra ce-based approach. Our results are at variance with the known results for the case of transition systems with no fairness, where no approach is evidently advantageous.