Characteristic invariants in Hennessy–Milner logic

Abstract

In this paper, we prove that Hennessy–Milner Logic (HML), despite its structural limitations, is sufficiently expressive to specify an initial property $$\varphi _0$$φ0 and a characteristic invariant $$\upchi _{_I}$$χI for an arbitrary finite-state process P such that $$\varphi _0 \wedge \mathbf{AG }(\upchi _{_I})$$φ0∧AG(χI) is a characteristic formula for P. This means that a process Q, even if infinite state, is bisimulation equivalent to P iff $$Q \models \varphi _0 \wedge \mathbf{AG }(\upchi _{_I})$$Q⊧φ0∧AG(χI). It follows, in particular, that it is sufficient to check an HML formula for each state of a finite-state process to verify that it is bisimulation equivalent to P. In addition, more complex systems such as context-free processes can be checked for bisimulation equivalence with P using corresponding model checking algorithms. Our characteristic invariant is based on so called class-distinguishing formulas that identify bisimulation equivalence classes in P and which are expressed in HML. We extend Kanellakis and Smolka’s partition refinement algorithm for bisimulation checking in order to generate concise class-distinguishing formulas for finite-state processes.

Deciding All Behavioral Equivalences at Once: A Game for Linear-Time--Branching-Time Spectroscopy

“…There are related islands like the encoding between CTL and failure traces by Bruda and Zhang [BZ10]. There is also more recent work like Jasper et al [JSS20] extending to the generation of characteristic invariant formulas for bisimulation classes, and like Wißmann et al [WMS21] about generating distinguishing formulas for bisimulation in a general coalgebraic setting. Previous algorithms for bisimulation inequivalence tend to generate formulas that alternate a and [b] ≡ ¬ b ¬ observations.…”

A Game for Linear-time–Branching-time Spectroscopy

“…There is also more recent work like Jasper et. al [15] extending to the generation of characteristic invariant formulas for bisimulation classes. Previous algorithms for bisimulation in-equivalence tend to generate formulas that alternate a and [b] observations while pushing negation to the innermost level.…”

A Game for Linear-time–Branching-time Spectroscopy

“…There is also more recent work like Jasper et. al [JSS20] extending to the generation of characteristic invariant formulas for bisimulation classes. Previous algorithms for bisimulation in-equivalence tend to generate formulas that alternate a and [b] observations while pushing negation to the innermost level.…”