Programming interfaces and basic topology

Peter Hancock (1) , Pierre Hyvernat (2, 3)

1 Computer and Information Sciences, Stracthclyde university
2 IML - Institut de mathématiques de Luminy
3 LAMA - Laboratoire de Mathématiques

A pattern of interaction that arises again and again in programming is a "handshake", in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent--the client or Angel--and concluded by the other--the server or Demon. We present a category in which the objects--called interaction structures in the paper--serve as descriptions of services provided across such handshaken interfaces. The morphisms--called (general) simulations--model components that provide one such service, relying on another. The morphisms are relations between the underlying sets of the interaction structures. The proof that a relation is a simulation can serve (in principle) as an executable program, whose specification is that it provides the service described by its domain, given an implementation of the service described by its codomain. This category is then shown to coincide with the subcategory of "generated" basic topologies in Sambin's terminology, where a basic topology is given by a closure operator whose induced sup-lattice structure need not be distributive; and moreover, this operator is inductively generated from a basic cover relation. This coincidence provides topologists with a natural source of examples for non-distributive formal topology. It raises a number of questions of interest both for formal topology and programming. The extra structure needed to make such a basic topology into a real formal topology is then interpreted in the context of interaction structures.

Mots clés: Constructive type theory; Predicate transformers; Simulation; Formal topology

Domaines: Logique en informatique [cs.LO]; Logique [math.LO]

Peter Hancock, Pierre Hyvernat. Programming interfaces and basic topology. Annals of Pure and Applied Logic, 2006, 137 (1--3), pp.189--239. ⟨hal-00387603⟩