A SIMPLE AND POWERFUL APPROACH FOR STUDYING CONSTRUCTIVITY, COMPUTABILITY, AND COMPLEXITY

In this contribution a natural and simple as well as general and efficient frame for studying effectivity (Type 2 theory of effectivity, TTE) is presented. TTE is a straightforward ''logic free'' extension of ordinary computability theory. Three basic kinds of effectivity for functions on SIGMA* and SIGMA(omega) are distinguished: continuity, computability, and easy computability (computational complexity). As the most remarkable property, continuity in TTE can be very adequateley interpreted as a basic kind of constructivity. Effectivity is transferred from SIGMA* and SIGMA(omega) to other sets by notations, (where finite words serve as names) and by representations (where omega-words serve as names), respectively. In this contribution the structure of TTE is explained and its applicability is demonstrated by simple examples mainly from analysis. Especially it is shown how the ''effectivity gap'' between Abstract Analysis and Numerical Analysis can be closed step by step by introducing stronger and stronger effectivity requirements. It is suggested that the theory outlined in this paper is adequate to introduce effectivity in Analysis into Comupter Science curricula.