The duality between flow charts and circuits

Abstract

This paper contains a precise description of the duality between the formal evolutions of flow charts and of circuits. In addition, it contains a new description of the free category-with-products on a multigraph as a familially representable construction. INTRODUCTIONIf A is a set of elementary actions then the elements of the free monoid A* may be thought of as formal evolutions of the set of actions. This can be generalised. Given a graph G of elementary actions the arrows in the free category -with appropriate structure -on G may again be thought of as the formal evolutions of the elementary actions, taking into account the specificity of the graph. This paper is concerned in particular with the formal evolution of both flow charts and of digital circuits. The appropriate notion of graph in both cases is multigraph. The formal evolutions of a flow chart are arrows in the free category-with-sums on a multigraph; the formal evolutions of a circuit are arrows in the free category-withproducts on a multigraph.As a result there is a precise categorical duality between the evolution of flow charts and of circuits. Roughly speaking, this is the duality between terms and trees.In addition to this duality theorem, we give a new description of the free categorywith-products on a multigraph. Diers [2] introduced the notion of a locally representable functor, that is, a functor represented by a family of objects rather than one object. Johnson and Walters, calling the notion instead familially representable functor, found further examples (see [4], [5]). A simple example is the free-monoid functor. The family of objects in this case is the natural numbers, and the free monoid on A may be thought of as {n -+ i ; n a natural number}. In section 5 we indicate that the free categorywith-products on a multigraph is similarly familially representable -an idea suggested by the well-known description in terms of families of the free category-with-products on a category.