The Extension of Bisimulation Quantified Modal Logic Based on Covariant-Contravariant Refinement

The notion of covariant-contravariant refinement (CC-refinement, for short) is a generalization of the notions of bisimulation and refinement. This paper interprets semantically a CC-refinement as bisimulation plus model restriction, that is, a CC-refinement model of a given model may be obtained from one bisimilar duplicate of this model by adding some transitions labelled by covariant actions and removing some transitions labelled by contravariant actions. By using certain proposition letter to witness a contravariant action, the standard bisimulation quantified modal logic is able to capture the characterization of this action, however, this fails for covariant actions. This paper, based on the notion of CC-refinement, introduces an extended bisimulation quantified modal logic with the universal modality (EBQML), describes syntactically CC-refinement quantification as the extended bisimulation quantification plus relativization, and establishes a translation from the language of CC-refinement modal μ-calculus to the language of EBQML such that every CC-refinement modal μ-formula is equivalent to its translation. The language of EBQML may be considered as a specification language for describing the properties of a system referring to reactive and generative actions, which are represented respectively by covariant and contravariant actions, and may be used to formalize some interesting problems in the field of formal methods.