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  • Print publication year: 1998
  • Online publication date: May 2010

Types and dynamics in partially additive categories

Summary

Introduction

A challenging problem in studying complex systems is to model the internal structure of the system and its dynamical evolution in an integrated view. An algorithm computes a function with domain consisting of all its possible inputs and as codomain all possible outputs. In order to “compare” algorithms computing the same function (e.g. with respect to time or space complexity) it is necessary to model the “dynamic behavior” of the algorithm. The main approach for modeling the dynamic behavior Bh(P) of an algorithm is “the dynamic system paradigm”: Bh(P) is obtained as the action of a (control) monoid on a (state) space. Various mathematical structures have been considered for attacking such a problem; among others, deterministic automata and stochastic automata. In order to capture the fine structure of computation, an attempt to describe the “macroscopic” global dynamic behavior of an algorithm in terms of the “microscopic” local dynamic behavior of its components has recently been proposed within a logical approach to computing called “Geometry of Interaction” developed within C*-algebras. We propose an algebraic approach to “structured dynamics” inspired by the Geometry of Interaction and based on a “many objects” generalization of semirings: partially additive categories.

The paper is organized as follows: in the next section we present a general view of structural dynamics inspired by proof theory. In Section 3 we present the execution of pseudoalgorithms in the framework of partially additive categories as a kind of iteration of endomorphisms.