Categories, Allegories, and Circuit Design

Languages based upon binary relations offer an appealing setting for constructing programs from specifications. For example, working with relations rather than functions allows specifications to be more abstract (for example, many programs have a natural specification using the converse operator on relations), and affords a natural treatment of non-determinism in specifications. In this paper we present a novel pictorial interpretation of relational terms as simple pictures of circuits, and a soundness/completeness result that allows relational equations to be proved by pictorial reasoning.

Making Functionality More General

The definition for the notion of a "function" is not cast in stone, but depends upon what we adopt as types in our language. With partial equivalence relations (pers) as types in a relational language, we show that the functional relations are precisely those satisfying the simple equation f = f o fu o f, where "o" and "u" are respectively the composition and converse operators for relations. This article forms part of "A calculational theory of pers as types".