Open Maps (at) Work

Abstract

The notion of bisimilarity, as defined by Park and Milner,<br />has turned out to be one of the most fundamental notions of operational<br />equivalences in the field of process algebras. Not only does it induce<br />a congruence (largest bisimulation) in CCS which has nice equational<br />properties, it has also proven itself applicable for numerous models of<br />parallel computation and settings such as Petri Nets and semantics of<br />functional languages. In an attempt to understand the relationships and<br />differences between the extensive amount of research within the field,<br />Joyal, Nielsen, and Winskel recently presented an abstract category-theoretic <br />definition of bisimulation. They identify spans of morphisms satisfying certain "path lifting"<br /> properties, so-called open maps, as a possible abstract definition of bisimilarity.<br /> In [JNW93] they show, that they can capture Park and Milner's bisimulation. <br />The aim of this paper is to show that the abstract definition of bisimilarity is applicable<br />"in practice" by showing how a representative selection of well-known<br />bisimulations and equivalences, such as e.g. Hennessy's testing equivalence,<br />Milner and Sangiorgi's barbed bisimulation, and Larsen and Skou's<br />probabilistic bisimulation, are captured in the setting of open maps and<br />hence, that the proposed notion of open maps seems successful. Hence,<br />we confirm that the treatment of strong bisimulation in [JNW93] is not<br />a one-off application of open maps.