Graphical representation of covariant-contravariant modal formulae

Covariant-contravariant simulation is a combination of standard (covariant) simulation, its contravariant counterpart and bisimulation. We have previously studied its logical characterization by means of the covariant-contravariant modal logic. Moreover, we have investigated the relationships between this model and that of modal transition systems, where two kinds of transitions (the so-called may and must transitions) were combined in order to obtain a simple framework to express a notion of refinement over state-transition models. In a classic paper, Boudol and Larsen established a precise connection between the graphical approach, by means of modal transition systems, and the logical approach, based on Hennessy-Milner logic without negation, to system specification. They obtained a (graphical) representation theorem proving that a formula can be represented by a term if, and only if, it is consistent and prime. We show in this paper that the formulae from the covariant-contravariant modal logic that admit a “graphical” representation by means of processes, modulo the covariant-contravariant simulation preorder, are also the consistent and prime ones. In order to obtain the desired graphical representation result, we first restrict ourselves to the case of covariant-contravariant systems without bivariant actions. Bivariant actions can be incorporated later by means of an encoding that splits each bivariant action into its covariant and its contravariant parts.

💡 Research Summary

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The paper investigates the relationship between a newly defined behavioural preorder—covariant‑contravariant simulation (cc‑simulation)—and a corresponding modal logic (cc‑modal logic). Covariant‑contravariant simulation extends the classic notions of simulation, reverse‑simulation, and bisimulation by partitioning the set of actions A into three disjoint subsets: covariant actions (A_r), contravariant actions (A_l), and optionally bivariant actions (A_bi). For covariant actions the usual forward simulation condition is required, while for contravariant actions the reverse condition holds; bivariant actions are treated as ordinary bisimulation steps.

The authors first restrict attention to systems without bivariant actions. They introduce a simple process algebra P consisting of the deadlock 0, a distinguished least element ω, action prefix a·p, and nondeterministic choice p+q, together with operational rules that make ω a universal sink for all contravariant actions. Over this algebra, the cc‑simulation relation is shown to be a preorder.

The cc‑modal logic L is defined with two modal operators: ⟨a⟩φ for a∈A_r (existential “may” modality) and