Computable total functions on metric algebras, universal algebraic specifications and dynamical systems

Abstract

Data such as real and complex numbers, discrete and continuous time data streams, waveforms, scalar and vector fields, and many other functions, are fundamental for many kinds of computation. In the theory of data, such data types are modelled using topological, or metric, many-sorted algebras and continuous homomorphisms. A theory of such topological data types is needed to answer the general questions: 1. What are the computable functions on topological algebras? 2. What methods exist to axiomatically specify functions on topological algebras? 3. Can all computable functions be specified? Such a theory seems to be in its infancy: there are many approaches to computability theory on general and specific spaces, and few approaches to specification theory. In some earlier papers, we have studied the questions 1 and 2 with the needs of data type theory in mind, and built a bridge between computations and specifications to try to answer 3. In this paper, we extend and combine several of our results, to prove new theorems that (i) show the equivalence of some six deterministic or non-deterministic models of computation on various metric algebras and, in particular, on spaces R n of real numbers; (ii) provide finite universal algebraic specifications for all the functions that can be computably approximated on metric algebras and, in particular, on Euclidean n-space R n ; (iii) show the existence of finite universal algebraic specifications of computably approximable finite dimensional deterministic dynamical systems. ୋ The research of the second author was supported by a grant from the Natural Sciences and Engineering Research Council (Canada).

Models of computation for partial functions on the reals

“…The present paper can perhaps best be understood against the background of the paper [23]. We first summarize what was done in that paper.…”

“…In [23] all five computation models were shown to be equivalent, for functions f : R m → R (m > 0) that are (a) total or, more generally, defined on a closed interval (or product of intervals, in the case m > 1), and (b) effectively locally uniformly continuous.…”

“…Previous work on comparing models of computability (for example, [19,23]) has tended to concentrate on computability models for total functions on R. However, many of the well-known functions of elementary real analysis, which would certainly be considered as computable, are partial; for example, the rational, log and trigonometric functions. It is therefore essential that a study of models of computability on R should include such functions in its considerations.…”

“…Abstract models of computation are independent of data representations. Concrete models, on the other hand, depend on a choice of data representation, usually constructed from the natural numbers N, so that computation on an algebra is reduced to computation on N [20,22,23].…”

“…There are two familiar concrete models that we investigate: Grzegorczyk-Lacombe (GL) [7,8,12,15] and tracking computability [22,23].…”