Refinement Modal Logic Based on Finite Approximation of Covariant-Contravariant Refinement

The notion of covariant-contravariant refinement (CC-refinement) is a generalization of the notions of bisimulation and refinement, which is coinductively defined to describe behavior relation, i.e., two related processes are required to be always able to provide matched transitions each other. The notion of CC- n -refinement, where n is a natural number, presented in this paper, finitely approximates the notion of CC-refinement through weakening the above-mentioned requirement. A process is said to CC- n -refine another process whenever they can, within n -steps, match the transitions each other, and a CC- n -refinement relation may be considered to be a CC-refinement relation if n is big enough. Based on this kind of finite approximation, this paper presents CC- n -refinement modal logic ( n CCRML) obtained from the modal system \boldsymbol {K} by adding CC- n -refinement quantifiers, establishes an axiom system for n CCRML, and explores some important properties of this axiom system: soundness, completeness, and decidability. CC- n -refinement quantifiers can be used to formalize some interesting problems in the field of formal method.