Logic Meets Algebra: Compositional Timing Analysis for Synchronous Reactive Multithreading

The intuitionistic theory of the real interval [0, 1], known as Skolem-Godel-Dummet logic (SGD), generates a well-known Heyting algebra intermediate between intuitionistic and classical logic. Originally of purely mathematical interest, it has recently received attention in Computer Science, notably for its potential applications in concurrency theory. In this paper we show how the logical operators of SGD over the discrete frame \({\mathbb Z} _\infty \), extended by the additive group structure \(({\mathbb Z}, 0, +)\), provides an expressive and yet surprisingly economic calculus to specify the quantitative stabilisation behaviour of synchronous programs. This is both a new application of SGD and a new way of looking at the semantics of synchronous programming languages. We provide the first purely algebraic semantics of timed synchronous reactions which adapts Berry’s semantics for Esterel to work on general concurrent/sequential control-flow graphs. We illustrate the power of the algebra for the modular analysis of worst-case reaction time (WCRT) characteristics for time-predictable reactive processors with hardware-supported multi-threading.