Bisimulation is Not Finitely (First Order) Equationally Axiomatisable

This paper considers the existence of nite equational axiomatisations of bisimulation over a calculus of nite state processes. To express even simple properties such as XE = XE[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 de ned and using them the nonexistence of nite axiomatisations containing at most rst 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 di ers only in that it does not contain a zero.