Graphical representation of covariant-contravariant modal formulae

Abstract

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 covariantcontravariant 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 covariantcontravariant 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.

Merging Process Algebra and Action-based Computation Tree Logic

“…Thus, by Lemma 7.2, ℑ contains a proper subtree with the root labelled with , where w is either p̟q or a conjunction of p̟q and a number of true. Hence p̟q ∈ F due to IH about item (1). Further, by Lemma 3.3(5), we obtain q ∈ F and p ⊙ (p̟q) ∈ F .…”

“…indeed induce a Galois connection between (Θ, ⊑) and the set of consistent prime formulae augment with some preorder [14]. In [1], L.Aceto et al address the same issue, and show that, modulo the covariantcontravariant simulation preorder, any consistent and prime formula in the covariantcontravariant modal logic also admits a representation by means of process terms.…”

“…The logic L is said to be adequate w.r.t (P, ⊲⊳) if for any process p and q, either p ⊲⊳ q iff ∀α ∈L(p |= α ⇔ q |= α ) (if ⊲⊳ is an equivalence) or p ⊲⊳ q iff ∀α ∈L(q |= α ⇒ p |= α ) (if ⊲⊳ is a preorder) This notion is considered by Hennesy and Milner in [34], where they prove that Hennesy-Milner logic (HML) is adequate w.r.t bisimilarity for image finite CCS terms. It is one of key evens that make Milner think that CCS is definitely interesting enough 1 . Following their work, the literature on concurrency theory offers a wealth of modal characterizations for various behavioral relations.…”

Refinement checking on parametric modal transition systems

“…The MTS formalism can also be viewed as a fragment of mu-calculus that is "graphically representable" [9,21]. The graphical representability of a variant of alternating simulation called covariant-contravariant simulation has been recently studied in [1].…”

Dual-Priced Modal Transition Systems with Time Durations

“…The formalism can be viewed as a fragment of a temporal logic [1,8] that at the same time offers a behavioural compositional semantics with an intuitive notion of process refinement. The formalism of MTS is essentially a labelled transition system that distinguishes two types of labelled transitions: must transitions which are required in any refinement of the system, and may transitions that are allowed to appear in a refined system but are not required.…”