Transposing partial components - An exercise on coalgebraic refinement

A partial component is a process which fails or dies at some stage, thus exhibiting a finite, more ephemeral behaviour than expected (e.g. operating system crash). Partiality — which is the rule rather than exception in formal modelling — can be treated mathematically via totalization techniques. In the case of partial functions, totalization involves error values and exceptions. In the context of a coalgebraic approach to component semantics, this paper argues that the behavioural counterpart to such functional techniques should extend behaviour with try-again cycles preventing from component collapse, thus extending totalization or transposition from the algebraic to the coalgebraic context. We show that a refinement relationship holds between original and totalized components which is reasoned about in a coalgebraic approach to component refinement expressed in the pointfree binary relation calculus. As part of the pragmatic aims of this research, we also address the factorization of every such totalized coalgebra into two coalgebraic components — the original one and an added front-end — which cooperate in a client-server style.