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 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.