A foundation for computable analysis

Abstract

While for countable sets there is a single well established computability theory (ordinary recursion theory), Computable Analysis is still underdeveloped. Several mutually non-equivalent theories have been proposed for it, none of which, however, has been accepted by the majority of mathematicians or computer scientists. In this contribution one of these theories, TTE (Type 2 Theorie of Effectivity), is presented, which at least in the author's opinion has important advantages over the others. TTE intends to characterize and study exactly those functions, operators etc. known from Analysis, which can be realized correctly by digital computers. The paper gives a short introduction to basic concepts of TTE and shows its general applicability by some selected examples. First, Turing computability is generalized from finite to infinite sequences of symbols. Assuming that digital computers can handle (w.l.o.g.) only sequences of symbols, infinite sequences of symbols are used as names for “infinite objects” such as real numbers, open sets, compact sets or continuous functions. Naming systems are called representations. Since only very few representations are of interest in applications, a very fundamental principle for defining effective representations for To-spaces with countable bases is introduced. The concepts are applied to real numbers, compact sets, continuous functions and measures. The problem of zero-finding is considered. Computational complexity is discussed. We conclude with some remarks on other models for Computable Analysis. The paper is a shortened and revised version of [Wei97].