Algebraic Theories

356 tr.; 24 cm

Partially additive categories and flow-diagram semantics

AbstractThe “semantics of flow diagrams” are used to motivate the notion of partially additive monoids and of partially additive categories as those based on the category of partially additive monoids. We show that such categories support a notion of iteration; and then axiomatize iteration in a fashion which yields other approaches as a special case. The partially additive categories generalize semiadditive categories, and we provide an alternative characterization based on the fact that coproducts +̌ Ai in a partially additive category are equipped with morphisms (prj: +̌ Ai → Aj) which enjoy many of the properties of products. A number of other approaches to flow-diagram semantics have used either the concept of partial order or of algebraic theory—we provide a systematic overview of these approaches from the perspective afforded by partially additive categories

Intertwined recursion, tree transformations and linear systems

Motivated by the way in which the recursive definition of the response of a generalized sequential machine is intertwined with that of the reachability map, we introduce an intertwined recursion principle valid for any endofunctor that admits free dynamics. This allows us to extend the Arbib-Manes definition of a machine in a category to that of a process transformation which transforms input processes to output processes. This formalization includes primitive recursion, generalized sequential machines, bottom-up tree transformations, and a generalized notion of linear systems which treats the initial state and input on a symmetric footing in its reachability theory. We also analyze the behavior of loop-free networks of process transformations