Multi-relations in Z

Abstract

Both the theories of binary relations and multi-sets (or bags) in Z have been usefully applied to software specification and development. In this paper we examine a useful theory—multi-relations—which is a cross between these two theories. One way of viewing relations is as sets of pairs. Here, by analogy, we view multi-relations as multi-sets of pairs, and we define multirelation equivalents of most of the traditional operators defined on binary relations. Multi-relations can also be viewed as graphs or two-dimensional matrices (with indices over arbitrary sets). The use of multi-relations is illustrated by specifying a bill-of-materials system. This provides a good example of the paradigm of building a suitable mathematical theory first and then developing a specification in terms of the theory.

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