First-order μ-calculus over generic transition systems and applications to the situation calculus

We consider meL, meLa, and meLp, three variants of the first-order μ-calculus studied in verification of data-aware processes, that differ in the form of quantification on objects across states. Each of these three logics has a distinct notion of bisimulation. We show that the three notions collapse for generic dynamic systems, which include all state-based systems specified using a logical formalism, e.g., the situation calculus. Hence, for such systems, muL, muLa, and muLp have the same expressive power. We also show that, when the dynamic system stores only a bounded number of objects in each state (e.g., for bounded situation calculus action theories), a finite abstraction can be constructed that is faithful for muL (the most general variant), yielding decidability of verification. This contrasts with the undecidability for first-order LTL, and notably implies that first-order ltl cannot be captured by muL.