Diamonds Are a Girl's Best Friend: Partial Order Reduction for Timed Automata with Abstractions

A major obstacle for using partial order reduction in the context of real time verification is that the presence of clocks and clock constraints breaks the usual diamond structure of otherwise independent transitions. This is especially true when information of the relative values of clocks is preserved in the form of diagonal constraints. However, when diagonal constraints are relaxed by a suitable abstraction, some diamond structure is re-introduced in the zone graph. In this article, we introduce a variant of the stubborn set method for reducing an abstracted zone graph. Our method works with all abstractions, but especially targets situations where one abstract execution can simulate several permutations of the corresponding concrete execution, even though it might not be able to simulate the permutations of the abstract execution. We define independence relations that capture this "hidden" diamond structure, and define stubborn sets using these relations. We provide a reference implementation for verifying timed language inclusion, to demonstrate the effectiveness of our method.