We present a complete method for synthesizing lexicographic linear ranking functions (and thus proving termination), supported by inductive invariants, in the case where the transition relation of the program includes disjunctions and existentials (large block encoding of control flow). Previous work would either synthesize a ranking function at every basic block head, not just loop headers, which reduces the scope of programs that may be proved to be terminating, or expand large block transitions including tests into (exponentially many) elementary transitions, prior to computing the ranking function, resulting in a very large global constraint system. In contrast, our algorithm incrementally refines a global linear constraint system according to extremal counterexamples: only constraints that exclude spurious solutions are included. Experiments with our tool Termite show marked performance and scalability improvements compared to other systems.

In this paper, we propose a sound abstraction for an efficient static analysis of synchronous programs describing multi-clock embedded systems in Signal. This abstraction combines the Boolean theory and numeric interval approximation to adequately address clock relations defined as combinations of logical and numerical expressions. Through a few examples, we show how the proposed solution is used to determine absence of reaction captured by empty clocks; mutual exclusion captured by two or more clocks whose associated signals never occur at the same time; or hierarchical control of component activations via clock inclusion. We also show this analysis improves the quality of the code generated automatically by the Signal compiler, e.g., a code with smaller footprint, or a code executed more efficiently thanks to optimizations enabled by the new abstraction.