Modal and Mixed Specifications: Key Decision Problems and their Complexities

Modal and mixed transition systems are specification formalisms that allow the mixing of over- and under-approximation. We discuss three fundamental decision problems for such specifications:

— whether a set of specifications has a common implementation;

— whether an individual specification has an implementation; and

— whether all implementations of an individual specification are implementations of another one.

For each of these decision problems we investigate the worst-case computational complexity for the modal and mixed cases. We show that the first decision problem is EXPTIME-complete for both modal and mixed specifications. We prove that the second decision problem is EXPTIME-complete for mixed specifications (it is known to be trivial for modal ones). The third decision problem is also shown to be EXPTIME-complete for mixed specifications.