The Bisimulation Proof Method: Enhancements and Open Problems

Bisimulation (and, more generally, co-induction) can be regarded as one of the most important contributions of Concurrency Theory to Computer Science. Nowadays, bisimulation and the co-inductive techniques developed from the idea of bisimulation are widely used, not only in Concurrency, but, more broadly, in Computer Science, in a number of areas: functional languages, object-oriented languages, type theory, data types, domains, databases, compiler optimisations, program analysis, verification tools, etc.. For instance, in type theory bisimulation and co-inductive techniques have been used: to prove soundness of type systems; to define the meaning of equality between (recursive) types and then to axiomatise and prove such equalities; to define co-inductive types and manipulate infinite proofs in theorem provers. Also, the development of Final Semantics, an area of Mathematics based on co-algebras and category theory and that gives us a rich and deep perspective on the meaning of co-induction and its duality with induction, has been largely motivated by the interest in bisimulation. In my talk I will discuss the bisimulation proof method – an instance of the co-induction proof method – that is at the heart of the success of bisimulation. I will discuss a number of enhancements of the method and some open problems. The objective of enhancements is to relieve the work involved with the bisimulation proof method. Thus proving a bisimulation result becomes simpler. Such enhancements can sometimes be extremely important. They seem to be even essential in calculi for mobility such as the π-calculus [?,?], and in higher-order languages (that is, languages where substitutions can involve the replacement of variables with arbitrary terms of the language) such as Higher-Order π-calculus [?], Ambients [?], or even sequential languages such as the λ–calculus.