Calculating Invariants as Coreflexive Bisimulations

Abstract

Abstract. Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to computer systems engineering. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as types. An outcome of such a theory is a calculus of invariants' proof obligation discharge, a fragment of which is presented in the paper. The approach has two main ingredients: one is that of adopting relations as "first class citizens" in a pointfree reasoning style; the other lies on a synergy found between a relational construct, Reynolds' relation on functions involved in the abstraction theorem on parametric polymorphism and the coalgebraic account of bisimulation and invariants. In this process, we provide an elegant proof of the equivalence between two different definitions of bisimulation found in coalgebra literature (due to B. Jacobs and Aczel & Mendler, respectively) and their instantiation to the classical ParkMilner definition popular in process algebra.

UML Model Refactoring as Refinement: A Coalgebraic Perspective

“…In this example isomorphism s : V × U ←− U × V relating the state spaces of classes p ⊠ q and q ⊠ p, is shown to be a Tcoalgebra morphism. The only non trivial part of the proof is the one related to the methods' component, which we detail as follows 3 , in a completely pointfree style:…”

Transforming Data by Calculation

“…Taking concrete invariants into account is useful because these ensure (for free) properties at target-data level which can be advantageous in the transcription of source operations. The techniques presented in section 7 and detailed in [64] are the subject of current research taking into account the PF-calculus of invariants of [8]. Moreover, ≤-rules should be able to take invariants into account (a topic suggested but little developed in [56]).…”

“…For this pattern, a succinct specification of the four CRUD operations on simple M is as follows: Space constraints preclude going further on this topic in this paper. The interested reader will find in reference [58] the application of the PF-transform in speeding-up reasoning about CRUD preservation of datatype invariants on simple relations, as a particular case of the general theory [8]. Similar gains are expected from the same approach applied to CRUD transcription.…”

“…But syntax alone is not enough: the ability to reason in such a notation is essential, otherwise different data models won't be comparable. Thus the reason why, in what follows, we choose the PF-notation as unifying framework 8 .…”

“…In the section which follows we show how such invariants help in calculating model transformations. The reader is also referred to [8] for a PF-theory of invariants in general.…”