Hierarchies of Spatially Extended Systems and Synchronous Concurrent Algorithms

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.

[1]  Stephen Wolfram,et al.  Theory and Applications of Cellular Automata , 1986 .

[2]  Viggo Stoltenberg-Hansen,et al.  Concrete Models of Computation for Topological Algebras , 1999, Theor. Comput. Sci..

[3]  Anthony C. J. Fox Algebraic models for advanced microprocessors , 1998 .

[4]  H. T. Kung Why systolic architectures? , 1982, Computer.

[5]  J. V. Tucker,et al.  Can excitable media be considered as computational systems , 1991 .

[6]  R. Aliev,et al.  A simple two-variable model of cardiac excitation , 1996 .

[7]  Denis Noble,et al.  Simulating cardiac sinus and atrial network dynamics on the Connection Machine , 1993 .

[8]  Desmond Fearnley-Sander,et al.  Universal Algebra , 1982 .

[9]  J. V. Tucker,et al.  Verification of Synchronous Concurrent Algorithms Using OBJ3: A Case Study of the Pixel-Planes Architecture , 1991 .

[10]  J. V. Tucker,et al.  Algebraic Models and the Correctness of Microprocessors , 1993, CHARME.

[11]  Henggui Zhang,et al.  Coupling CMLs and the synchronization of a multilayer neural computing system , 1994 .

[12]  J. V. Tucker,et al.  Theory of Computation and Specification over Abstract Data Types, and its Applications , 1991 .

[13]  Arun V. Holden,et al.  Computational biology of the heart , 1998, The Mathematical Gazette.

[14]  Martin Wirsing,et al.  Algebraic Specification , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[15]  Robert L. Grossman,et al.  Timed Automata , 1999, CAV.

[16]  J. V. Tucker,et al.  Reconstructing the heart , 1994 .

[17]  Jan A. Bergstra,et al.  Grid Protocols Based on Synchronous Communication , 1997, Sci. Comput. Program..

[18]  Neal A. Harman,et al.  Algebraic Models of Superscalar Microprocessor Implementations: A Case Study , 1998, Prospects for Hardware Foundations.

[19]  K. Kaneko Overview of coupled map lattices. , 1992, Chaos.

[20]  J. V. Tucker,et al.  Theory of Computation over Stream Algebras, and its Applications , 1992, MFCS.

[21]  James L. McClelland,et al.  Parallel distributed processing: explorations in the microstructure of cognition, vol. 1: foundations , 1986 .

[22]  Y. Moschovakis Recursive metric spaces , 1964 .

[23]  Jan A. Bergstra,et al.  Grid Protocol Specifications , 1998, Prospects for Hardware Foundations.

[24]  J. V. Tucker,et al.  Complete local rings as domains , 1988, Journal of Symbolic Logic (JSL).

[25]  Jens Blanck,et al.  Domain Representability of Metric Spaces , 1997, Ann. Pure Appl. Log..

[26]  J. V. Tucker,et al.  Effective algebras , 1995, Logic in Computer Science.

[27]  J. V. Tucker,et al.  Coupled map lattices as computational systems. , 1992, Chaos.

[28]  Jan A. Bergstra,et al.  The Completeness of the Algebraic Specification Methods for Computable Data Types , 1982, Inf. Control..

[29]  Karen Stephenson Towards an Algebraic Specification of the Java Virtual Machine , 1998, Prospects for Hardware Foundations.

[30]  J. V. Tucker,et al.  AN ALGORITHMIC MODEL OF THE MAMMALIAN HEART: PROPAGATION, VULNERABILITY, RE-ENTRY AND FIBRILLATION , 1996 .