Bisimulation is not Finitely (First Order) Equationally Axiomatisable

Abstract

This paper considers the existence of finite equational axiomatisations of bisimulation over a calculus of finite state processes. To express even simple properties such as \mu X E=\mu X E[E/X] equationally it is necessary to use some notation for substitutions. Accordingly the calculus is embedded in a simply typed lambda calculus, allowing axioms such as the above to be written as equations of higher type rather than as equation schemes. Notions of higher order transition system and bisimulation are then defined and using them the nonexistence of finite axiomatisations containing at most first order variables is shown.

The same technique is then applied to calculi of star expressions containing a zero process --- in contrast to the positive result given in [FZ93] for BPA*, which differs only in that it does not contain a zero.