Deriving Bisimulation Congruences: 2-Categories Vs Precategories

G-relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Leifer and Milner’s approach to deriving labelled bisimulation congruences from reduction systems. This paper develops the theory of GRPOs further, arguing that they provide a simple and powerful basis towards a comprehensive solution. As an example, we construct GRPOs in a category of ‘bunches and wirings.’ We then examine the approach based on Milner’s precategories and Leifer’s functorial reactive systems, and show that it can be recast in a much simpler way into the 2-categorical theory of GRPOs. Introduction It is increasingly common for foundational calculi to be presented as reduction systems. Starting from their common ancestor, the λ calculus, most recent calculi consist of a reduction system together with a contextual equivalence (built out of basic observations, viz. barbs). The strength of such an approach resides in its intuitiveness. In particular, we need not invent labels to describe the interactions between systems and their possible environments, a procedure that has a degree of arbitrariness (cf. early and late semantics of the π calculus) and may prove quite complex (cf. [5, 4, 3, 1]). By contrast, reduction semantics suffer at times by their lack of compositionality, and have complex semantic theories because of their contextual equivalences. Labelled bisimulation congruences based on labelled transition systems (LTS) may in such cases provide fruitful proof techniques; in particular, bisimulations provide the power and manageability of coinduction, while the closure properties of congruences provide for compositional reasoning. To associate an LTS with a reduction system involves synthesising a compositional system of labels, so that silent moves (or τ-actions) reflect the original reductions, labels describe potential external interactions, and all together they yield a LTS bisimulation which is a congruence included in the original contextual reduction equivalence. Proving bisimulation is then enough to prove reduction equivalence. Sewell [19] and Leifer and Milner [13, 11] set out to develop a theory to perform such derivations using general criteria; a meta-theory of deriving bisimulation congruences. The basic idea behind their construction is to use contexts as labels. To exemplify the idea, in a CCS-like calculus one would for instance derive a transition Research supported by ‘DisCo: Semantic Foundations of Distributed Computation’, EU IHP ‘Marie Curie’ contract HPMT-CT-2001-00290, and BRICS, Basic Research in Computer Science, funded by the Danish National Research Foundation. A.D. Gordon (Ed.): FOSSACS 2003, LNCS 2620, pp. 409–424, 2003. c © Springer-Verlag Berlin Heidelberg 2003 410 Vladimiro Sassone and Paweł Sobociński a.P −|ā.Q P | Q because term a.P in context − | ā.Q reacts to become P | Q; in other words, the context is a trigger for the reduction. The first hot spot of the theory is the selection of the right triggers to use as labels. The intuition is to take only the “smallest” contexts which allow a given reaction to occur. As well as reducing the size of the LTS, this often makes the resulting bisimulation equivalence finer. Sewell’s method is based on dissection lemmas which provide a deep analysis of a term’s structure. A generalised, more scalable approach was later developed in [13], where the notion of “smallest” is formalised in categorical terms as a relative-pushout (RPOs). Both theories, however, do not seem to scale up to calculi with non trivial structural congruences. Already in the case of the monoidal rules that govern parallel composition things become rather involved. The fundamental difficulty brought about by a structural congruence≡ is that working up to ≡ gives up too much information about terms for the RPO approach to work as expected. RPOs do not usually exist in such cases, because the fundamental indication of exactly which occurrences of a term constructor belong to the redex becomes blurred. A very simple, yet significant example of this is the category Bun of bunch contexts [13], and the same problems arise in structures such as action graphs [14] and bigraphs [15]. In [17] we therefore proposed a framework in which term structure is not explicitly quotiented, but the commutation of diagrams (i.e. equality of terms) is taken up to ≡. Precisely, to give a commuting diagram rp ≡ sq one exhibits a proof α of structural congruence, which we represent as a 2-cell (constructed from the rules generating ≡ and closed under all contexts). k p