Programming interfaces and basic topology

Annals of Pure and Applied Logic 137 (1--3):189-239 (2006)

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
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DOI 10.1016/j.apal.2005.05.022
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Inductively Generated Formal Topologies.Thierry Coquand, Giovanni Sambin, Jan Smith & Silvio Valentini - 2003 - Annals of Pure and Applied Logic 124 (1-3):71-106.
A Semantics of Evidence for Classical Arithmetic.Thierry Coquand - 1995 - Journal of Symbolic Logic 60 (1):325-337.
Topology Via Logic.P. T. Johnstone & Steven Vickers - 1991 - Journal of Symbolic Logic 56 (3):1101.
The Problem of the Formalization of Constructive Topology.Silvio Valentini - 2004 - Archive for Mathematical Logic 44 (1):115-129.

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