Generalizations of the Recursion Theorem

We consider two generalizations of the recursion theorem, namely Visser's ADN theorem and Arslanov's completeness criterion, and we prove a joint generalization of these theorems

Effective Genericity and Differentiability

We prove that a real x is 1-generic if and only if every differentiable computable function has continuous derivative at x. This provides a counterpart to recent results connecting effective notions of randomness with differentiability. We also consider multiply differentiable computable functions and polynomial time computable functions.Comment: Revision: added sections 6-8; minor correction

The complexity of completions in partial combinatory algebra

We discuss the complexity of completions of partial combinatory algebras, in particular of Kleene's first model. Various completions of this model exist in the literature, but all of them have high complexity. We show that although there do not exist computable completions, there exists completions of low Turing degree. We use this construction to relate completions of Kleene's first model to complete extensions of PA. We also discuss the complexity of pcas defined from nonstandard models of PA

Fixpoints and relative precompleteness

We study relative precompleteness in the context of the theory of numberings, and relate this to a notion of lowness. We introduce a notion of divisibility for numberings, and use it to show that for the class of divisible numberings, lowness and relative precompleteness coincide with being computable. We also study the complexity of Skolem functions arising from Arslanov's completeness criterion with parameters. We show that for suitably divisible numberings, these Skolem functions have the maximal possible Turing degree. In particular this holds for the standard numberings of the partial computable functions and the c.e. sets.Comment: 12 page

Intuitionistic logic and Muchnik degrees

We prove that there is a factor of the Muchnik lattice that captures intuitionistic propositional logic. This complements a now classic result of Skvortsova for the Medvedev lattice