Computation by ` While ' Programs on Topological Partial

The language of while programs is a fundamental model for imperative programming on any data type. It leads to a generalisation of the theory of computable functions on the natural numbers to the theory of computable functions on any many-sorted algebra. The language is used to express many algorithms in scientiic computing where while programs are applied to continuous data. In the theory of data, continuous data types are modelled by topological many-sorted algebras. We study both exact and approximate computations by while programs, and while programs with arrays, over topo-logical many-sorted algebras with partial operations. First, we establish that partial operations are necessary in order to compute a wide range of continuous functions. We prove basic continuity properties of our abstract com-putability: Any partial function computable over a partial topological algebra by a while-array program is continuous. Any set semicomputable, or computable , over a partial topological algebra by a while-array program is open, or clopen, respectively. Secondly, we contrast exact and approximate computations. The class of functions that can be computed exactly can be quite limited. We show that on connected total algebras, the while and while-array computable functions are precisely those that are explicitly deenable by terms. We show that for certain general classes of topological algebras, the total functions that can be approximated by while programs are precisely those that can be eeectively approximated by terms. This property we call generalised Weierstrass approximation. An application of this result is that a function on the set R of reals is computable in the sense of computable analysis if, and only if, it is while program approximable on a simple algebra based on R. 1 2 0 Introduction The theory of computable functions and sets on many-sorted algebras is a mathematical theory for the analysis of nite deterministic computation on any kind of data. It is a beautiful and useful generalisation of the theory of computable functions and sets on the natural numbers N. The theory is abstract in the sense that it does not depend on concrete representations of the algebras of data. The origins of the generalisation lie in the formalisation of owcharts in the 1950s. Subsequently, the subject has been developed by many contributions having one of essentially three motivations, namely: (i) to better understand computability theory (e. on real number computations and ruler and compass constructions). which contains historical information and a comprehensive introduction …