Characteristic formulae for fixed-point semantics: a general framework

Abstract

The literature on concurrency theory offers a wealth of examples of characteristic-formula constructions for various behavioural relations over finite labelled transition systems and Kripke structures that are defined in terms of fixed points of suitable functions. Such constructions and their proofs of correctness have been developed independently, but have a common underlying structure. This study provides a general view of characteristic formulae that are expressed in terms of logics with a facility for the recursive definition of formulae. It is shown how several examples of characteristic-formula constructions from the literature can be recovered as instances of the proposed general framework, and how the framework can be used to yield novel constructions. The paper also offers general results pertaining to the definition of co-characteristic formulae and of characteristic formulae expressed in terms of infinitary modal logics.

Property-dependent reductions adequate with divergence-sensitive branching bisimilarity

“…To carry out this task, we rely upon existing work on constructing characteristic formulas, namely for weak bisimilarity [21] and for strong and ready bisimilarity (among other relations) [22,23]. The approach is as follows: if a recursively defined logical formula expresses the definition of a behavioral relation, then the largest interpretation of that formula is the characteristic formula for the derived behavioral relation.…”

“…As explained in [23], we can now construct a monotonic function declaration D, by combining the translations of the individual clauses, such that the characteristic formula is given by the largest interpretation of that declaration:…”

Logical Characterization of Bisimulation Metrics

“…last part of Section 2). To capture infinite execution we will follow the approach of [49] (later generalized in [1,56]), also known as equational µ-calculus. In detail, we will extend the logic L to a modal S-indexed logic by adding the S-indexed family of variables {X s | s ∈ S}.…”

“…To verify whether the implementation proposed for a reactive concurrent system (or process) conforms the properties and behavior entailed by its specification, two main approaches have been proposed: equivalence checking [1] and model checking [16]. The former is used when both the implementation and specification of the system are modeled by a labeled transition system (LTS) [45] and it consists in verifying whether each of the observations enabled by the implementation is allowed by the specification.…”

Characteristic Formulae for Session Types

“…Since then the theory has been studied extensively [1-3, 11, 17, 33, 39] for most of the van Glabbeek's spectrum [41] and in different settings (e.g., time [4] and probabilistic [36]). See [2,3] for a detailed historical account of the field. This is the first time characteristic formulae are applied to the field of session types.…”