Components as coalgebras

Components as coalgebras (link)
L. S. Barbosa. PhD Thesis, DI, Universidade do Minho, ISBN 972-95918-6-5, 2001.

Abstract

In the tradition of mathematical modelling in physics and chemistry, constructive formal specification methods are based on the notion of a software model, understood as a state-based abstract machine which persists and evolves in time, according to a behavioural model capturing, for example, partiality or (different degrees of) non determinism. This can be identified with the more prosaic notion of a software component advocated by the software industry as `building block' of large, often distributed, systems. Such a component typically encapsulates a number of services through a public interface which provides a limited access to a private state space, paying tribute to the nowadays widespread object-oriented programming principles.

The tradition of communicating systems formal design, by contrast, has developed the notion of a process as an abstraction of the behavioural patterns of a computing system, deliberately ignoring the data and state aspects of software systems.

Both processes and components are among the broad group of computing phenomena which are hardly definable (or simply not definable) algebraically, ie, in terms of a complete set of constructors. Their semantics is essentially observational, in the sense that all that can be traced of their evolution is their interaction with the environment. Therefore, coalgebras, whose theory has recently witnessed remarkable developments, appear as a suitable modelling tool.

The basic observation of category theory that universal constructions always come in pairs, has motivated research on the duality between algebras and coalgebras, which provides a bridge between models of static (constructive, data-oriented) and dynamical (observational, behaviour-oriented) systems. At the programming level, the intuitive symmetry between data and behaviour provides evidence of such a duality, in its canonical initial-final specialisation.

This line of thought entails both definitional and proof principles, ie, a basis for the development of program calculi directly based on (actually driven by) type specifications. Moreover, such properties can be expressed in terms of generic programming combinators which are used, not only to calculate programs, but also to program with.

Framed in this context, this thesis addresses the following main themes:

• The investigation of a semantic model for software components. These are regarded as concrete coalgebras for some Set endofunctors, with specified initial conditions, and organise themselves in a bicategorical setting. The model is able to capture both behavioural issues, which are usually left implicit in state-based specification methods, and interaction through structured data, which is usually a minor concern on process calculi. Two basic cases are considered entailing, respectively, a `functional' and an `object-oriented' shape for components. Both cases are parametrized by a model of behaviour, introduced as a strong (usually commutative) monad.
• The development of corresponding component calculi, also parametric on the behaviour model, which adds to the genericity of the approach.
• The study of processes and the `reconstruction' of classical (CCS-like) process calculi on top of their representation as inhabitants of (the carriers of) final coalgebras, in an essentially pointfree, calculational style.
• An overall concern for genericity, in the sense that models and calculi for both components and processes are parametric on the behaviour model and the interaction discipline, respectively.
• The animation of both processes and components in Charity, a functional programming language entirely based on inductive and coinductive categorical data types. In particular this leads to the development of a process calculi interpreter parametric on the interaction discipline.

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