Functional Modeling and Mechanized Verification of Bisimulations for NFTS

Bisimulation is a behavioral equivalence approach for discrete event systems. In recent years, bisimulation has been increasingly applied to nondeterministic fuzzy transition system (NFTS). Current researchers mainly focus on mathematical definitions of fuzzy bisimulation and the development of algorithms, however, the properties of definitions and the correctness of the algorithms are manually proven. Therefore, we formally define NFTS using the basic datatype package in theorem prover Isabelle/HOL. Subsequently, we originally present functional definitions of two critical fuzzy bisimulations (based on relational closed sets and relational lifting), and then mechanically verified the equivalence of these two definitions by using Isabelle/HOL. Furthermore, a relational lifting algorithm is formalized and its correctness is mechanically verified in this article. Finally, we present two case studies to validate the practicality of our approach. Compared with the mathematical definition and manual verification, our study demonstrated that a mechanized method for formalizing and verifying fuzzy bisimulation is more trustworthy, and it could be applied to various fuzzy bisimulation algorithms.