Concurrency and continuity

Abstract

A definition of Continuity properly generalized from full to partial orders is presented. In addition to the continuity requirements of Dedekind (no gaps, no jumps) the definition includes requirements taken from physics and from engineering. All these requirements are fulfilled by the total order structure of real numbers, but at the cost of uncountably infinite cardinality. A central aspect of the proposed definition is that it admits models which are nowhere dense.

As a consequence, physical signalling structures can be modelled as a combinatorial (in this sense “discrete”) continuum. The possibilities and constraints of physical signalling belong of course to the foundations of information technology whose progress is the main concern of our work. The combinatorial continuum arises in a quite natural way when the relation of concurrency between space/time points is made explicit and its properties studied. Concurrency is interpreted as absence of a directed signalling connection, enforced e.g. by a speed limit for signalling. It turns out that the crucial point is the acknowledgement of the existence of “proper” concurrency, an indifference relation which is not transitive, as a prototype of observational indistinguishability. — Closing the gap between discrete and continuous modelling may have considerable practical consequences, some of which are indicated in the conclusion.