Using Approximation to Relate Computational Classes over the Reals

The elementary computable functions over the real numbers: applying two new techniques

The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). We provide general tools for investigating these issues, using two techniques we call approximat...

Algebraic Characterization of Complexity Classes of Computable Real Functions

Several algebraic machine-independant characterizations of computable functions over the reals have been obtained recently. In particular nice connections between the class of computable functions (and some of its sub and sup-classes) over the reals and algebraically defined (suband sup-) classes of R-recursive functions à la Moore 96 have been obtained. We provide in this paper a framework tha...

Exotic Quantifiers, Complexity Classes, and Complete Problems

We introduce some operators defining new complexity classes from existing ones in the Blum-Shub-Smale theory of computation over the reals. Each one of these operators is defined with the help of a quantifier differing from the usual ones, ∀ and ∃, and yet having a precise geometric meaning. Our agenda in doing so is twofold. On the one hand, we show that a number of problems whose precise comp...

The Methods of Approximation and Lifting in Real Computation

The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). We provide general tools for investigating these issues, using a technique we call the method of...

Logics Which Capture Complexity Classes over the Reals