Section: New Results

Genericity of weakly computable objects

Participant : Mathieu Hoyrup.

Computability theory abounds with classes of objects, defined for instance in terms of the computability content of the objects. A natural problem is then to compare these classes and separate them when possible. In order to separate two classes, one has to build an object that belongs to one class but not the other. So this object has to be computable in one sense but not the other. We show that in many cases these computability properties have a topological interpretation, and that the object to build must be at the same time computable in some weak topology (weakly computable) but generic in a stronger topology. We prove a general theorem stating the existence of such objects, thus providing a very handy tool to separate many classes. We use it in the study of the extension of computable functions (previous result) and in other situations. These results are presented in [13].