Similarity Quotients as Final Coalgebras

Abstract

Abstract. We give a general framework relating a branching time relation on nodes of a transition system to a final coalgebra for a suitable endofunctor. Examples of relations treated by our theory include bisimilarity (a well known example), similarity, upper and lower similarity for transition systems with divergence, and nested similarity. Our results describe firstly how to characterize the relation in terms of a given final coalgebra, and secondly how to construct a final coalgebra using the relation. Our theory uses a notion of "relator" based on earlier work of Thijs. But whereas a relator must preserve binary composition in Thijs' framework, it only laxly preserves composition in ours. It is this weaker requirement that allows nested similarity to be an example.

Enhanced coalgebraic bisimulation

“…We leave as future work the study of either a more broadly applicable notion of bisimulation, as in (Gorín and Schröder, 2013), or a more general definition of relation lifting, which applies to arbitrary functors on Set, as in (Marti and Venema, 2012). While in our work relators must preserve binary composition, in (Levy, 2011) a framework has been developed which only laxly preserves composition. The combination of this theory with our framework allows for a study of up-to-techniques for simulations, rather than bisimulations.…”

Relating coalgebraic notions of bisimulation

“…Levy and Worrell [35,54] have considered endofunctors on categories of preorders, posets, and categories enriched in quantales, in their investigations of similarity.…”

“…Plotkin [44] discusses problems with coalgebraic bisimulation in categories of domains: the convex powerdomain does not even preserve monomorphisms. The endofunctors on posets that Levy considers [35] typically do not preserve monomorphisms either.…”

Preorders on Monads and Coalgebraic Simulations

“…Bilkova et al derive a natural definition of relations between preorders using Sierpinski-space enriched categories, and give relational liftings of endofunctors over Pre in this context [6]. Levy extends the characterisation of bisimilarity by final coalgebras to similarity [20].…”

“…In [20], it is shown that the preordered functor (F, ) is stable (Definition 10, [20]) if and only if F +( ) is a relational extension such that (F +( ) , F 2 ) is an endomorphism over π.…”