Unifying computers and dynamical systems using the theory of synchronous concurrent algorithms

A synchronous concurrent algorithm (SCA) is a parallel deterministic algorithm based on a network of modules and channels, computing and communicating data in parallel, and synchronised by a global clock with discrete time. Many types of algorithms, computer architectures, and mathematical models of...

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Bibliographic Details
Published in:Applied mathematics and computation Vol. 215; no. 4; pp. 1386 - 1403
Main Authors: Thompson, B.C., Tucker, J.V., Zucker, J.I.
Format: Journal Article Conference Proceeding
Language:English
Published: Amsterdam Elsevier Inc 15.10.2009
Elsevier
Subjects:
Combinatorics. Ordered structures
Computability on topological algebras
Computable physical systems
Dynamical systems
Equational specifications
Exact sciences and technology
Functional analysis
Global analysis, analysis on manifolds
Many sorted algebras
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Order, lattices, ordered algebraic structures
Sciences and techniques of general use
Streams
Synchronous concurrent algorithms
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Equational specifications
Computable physical systems
Computability on topological algebras
Synchronous concurrent algorithms
Dynamical systems
Many sorted algebras
Streams
Parallel algorithm
Automaton
Systems theory
Neural network
Dynamical system
Computability
Biological system
Numerical analysis
Applied mathematics
Computer architecture
Discrete time
Computer system
Mathematical model
Computing method
Deterministic algorithms
ISSN:0096-3003, 1873-5649
Online Access:Get full text
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Summary:A synchronous concurrent algorithm (SCA) is a parallel deterministic algorithm based on a network of modules and channels, computing and communicating data in parallel, and synchronised by a global clock with discrete time. Many types of algorithms, computer architectures, and mathematical models of physical and biological systems are examples of SCAs. For example, conventional digital hardware is made from components that are SCAs and many computational models possess the essential features of SCAs, including systolic arrays, neural networks, cellular automata and coupled map lattices. In this paper we formalise the general concept of an SCA equipped with a global clock in order to analyse precisely (i) specifications of their spatio-temporal behaviour; and (ii) the senses in which the algorithms are correct. We start the mathematical study of SCA computation, specification and correctness using methods based on computation on many-sorted topological algebras and equational logic. We show that specifications can be given equationally and, hence, that the correctness of SCAs can be reduced to the validity of equations in certain computable algebras. Since the idea of an SCA is general, our methods and results apply to each of the particular classes of algorithms and dynamical systems above.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2009.04.058