Study of the computability of the functionals over the reals with the effective filter spaces.
by
Frederic De Jaeger
Ecole Normale Superieure (Paris)

We say that an element of a topological space is computable if its neighborhood filter is recursively enumerable with respect to a suitable enumerable basis of the topology. This is the natural generalisation of calculability notions that arise in effective domains or in constructive mathematics.

But because the category of topological spaces (TOP) is not cartesian closed (CCC), it is not the suitable framework to define those notions for objects with a higher type, like the functions or the functionals over the reals. Thus, in this work, we use Hyland's cartesian closed category of filter spaces (FIL) which "contains" TOP. Then we prove some decidability results in the sub-CCC generated by the reals with the euclidian topology. It gives us the necessary tools to prove the computability of the evaluation map at any type.

Date received: May 17, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagw-65.