Hierarchies of Spatially Extended Systems and Synchronous Concurrent Algorithms

  • Authors:

  • M. J. Poole;A. V. Holden;J. V. Tucker

  • Affiliations:

  • -;-;-

  • Venue:
  • Proceedings of the ESPRIT Working Group 8533 on Prospects for Hardware Foundations: NADA - New Hardware Design Methods, Survey Chapters
  • Year:
  • 1998

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Abstract

First, we study the general idea of a spatially extended system (SES) and argue that many mathematical models of systems in computing and natural science are examples of SESs. We examine the computability and the equational definability of SESs and show that, in the discrete case, there is a natural sense in which an SES is computable if, and only if, it is definable by equations. We look at a simple idea of hierarchical structure for SESs and, using respacings and retimings, we define how one SES abstracts, approximates, or is implemented by another SES. Secondly, we study a special kind of SES called a synchronous concurrent algorithm (SCA). We define the simplest kind of SCA with a global clock and unit delay which are computable and equationally definable by primitive recursive equations over time. We focus on two examples of SCAs: a systolic array for convolution and a non-linear model of cardiac tissue. We investigate the hierarchical structure of SCAs by applying the earlier general concepts for the hierarchical structure of SESs. We apply the resulting SCA hierarchy to the formal analysis of both the implementation of a systolic array and the approximation of a biologically detailed model of cardiac tissue.