Abstract
A partial component is a process which fails or dies at some stage, thus exhibiting a finite, more ephemeral behaviour than expected. 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.

Abstract
A transition system can be presented either as a binary relation or as a coalgebra for the powerset functor, each representation being obtained from the other by transposition. More generally, a coalgebra for a functor F generalises transition systems in the sense that a shape for transitions is determined by F, typically encoding a signature of methods and observers. This paper explores such a duality to frame in purely relational terms coalgebraic refinement, showing that relational (data) refinement of transition relations, in its two variants, downward and upward (functional) simulations, is equivalent to coalgebraic refinement based on backward and forward morphisms, respectively. Going deeper, it is also shown that downward simulation provides a complete relational rule to prove coalgebraic refinement. With such a single rule the paper defines a pre-ordered calculus for refinement of coalgebras, with bisimilarity as the induced equivalence. The calculus is monotonic with respect to the main relational operators and arbitrary relator F, therefore providing a framework for structural reasoning about refinement.

Abstract
Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to software systems. 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 bisimulations and invariants. This leads to 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 to their instantiation to the classical Park-Milner definition popular in process algebra.

Abstract
This paper addresses objectification, a formal specification technique which inspects the potential for object-orientation of a declarative model and brings the 'implicit objects' explicit. Criteria for such objectification are formalized and implemented in a runnable prototype tool which embeds VDM-SL into VDM++. The paper also includes a quick presentation of a (coinductive) calculus of such generated objects, framed as generalised Moore machines.

Abstract
When modelling complex interactive systems, traditional interactor-based approaches suffer from lack of expressiveness regarding the composition of the different interactors present in the user interface model into a coherent system. In this paper we investigate an alternative approach to the composition of interactors for the specification of complex interactive systems which is based on the coordination paradigm. We layout the fundations for the work and present an illustrative example. Lines for future work are identified.

Abstract
Although presented with a variety of 'flavours', the notion of an interactor, as an abstract characterisation of an interactive component, is well-known in the area of formal modelling techniques for interactive systems. Tins paper replaces traditional, hierarchical, 'tree-like' composition of interactors in the specification of complex interactive systems, by their exogenous coordination through general-purpose software connectors which assure the flow of data and the meet of synchronisation constraints. The paper's technical contribution is twofold. First a modal logic is defined to express behavioural properties of both interactors and connectors. The logic is new in the sense that its modalities are indexed by fragments of sets of actions to cater for action co-occurrence. Then, tins logic is used in the specification of both interactors and coordination layers which orchestrate their interconnection.