Compositional reasoning about projected and infinite time

Modularity is of fundamental importance in computer science. The need for a formal theory of modularity in the design and maintenance of large systems is especially pronounced. In recent work on Interval Temporal Logic (ITL) we gave an axiom system in which proofs of sequential and parallel systems can be decomposed into proofs for the syntactic subcomponents. This provides a precise framework for describing and generalizing the insights of Francez and Pnueli (1978) and Jones (1983) for modular reasoning about concurrency using what are often called assumptions and commitments. It combines the benefits of these ideas with temporal logic. We now show that such techniques can be used to analyze temporal projection operators developed by us for describing systems with multiple time granularities. In addition, we consider how to compositionally reason in ITL about the absence of deadlock in systems running for infinite time. This demonstrates that our generalization of Jones' techniques for assumptions and commitments handles not only safety properties but also liveness ones.