Characteristic Formulae for Fixed-Point Semantics: A General Framework

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.