Topological Concepts for Hierarchies of Variables, Types and Controls

Abstract

Let the mathematical specification of a problem be exprcssed by a function f: X → Y, X, Y being non-empty sets $X \subseteq \prod\limits_{i \in I} {{X_{[i]}}, Y \subseteq \prod\limits_{j \in J} {{Y_{[j]}}} ,f:x \mapsto y}$. For constructive computation f is usually represented as a composition of a family F = (f_k)_k∈K of given primitive functions ${f_{|f|}} \in G,G = \{ {g_{[p]}}:{X_{[p]}} \to {Y_{[p]}}|p \in P\} , P$ a finite set. All g_[q] are assumed to be computable. Composition is a concatenation of the f_k represented by a cycle free relation relating result components of some f_k with argument components of other f_l. On the resulting “algorithmic structure” algorithms can be defined. Depending on the choice of the structure and the choice of the algorithm it is expected that the result is a composite function representing f, which has to be approved by verification.