On the computational structure of the connected components of a hard problem

Abstract

The study of sparse sets has tremendous importance in Turing complexity theory. Thus it is a natural task to work out a related notion for real number models of computation. This has not fully been done so far. In the present paper we suggest such a notion based on the computational structure of the connected components a set has. Even though our notion of well-structured sets is different in spirit from the classical sparseness property we will show that it shares some important features with the latter. We are going to analyze the (non-)existence of well-structured complete sets within the Blum–Shub–Smale model of computation over the reals with linear operations and equality respectively inequality. Relations to exponential time classes are also drawn.