On Connecting Modules Together Uniformly to Form a Modular Computer

Abstract

A modular computer may be informally defined to be a device consisting of a large (or infinite) number of identical circuit modules connected together in some uniform manner. This paper is concerned with making the concept of ``connected together in a uniform manner'' mathematically precise. Uniformity of connection in a modular device is first defined in terms of the linear graph whose vertices correspond to the modules and whose edges correspond to the cables connecting them. It is shown that the class of graphs satisfying the definition is precisely the class of group-graphs; that is, the vertices correspond to the elements of a group G, and there is a finite subset G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> of G' (1 ∉ G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ) such that {g, g'} is an edge of the graph if, and only if, there exists g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> in G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> such that g'. = gg <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . It is then shown that restricting the group G to be Abelian restricts the patterns of simultaneous activity which may occur within the computer.