Degrees of unsolvability of continuous functions

Abstract

We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0,1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f∈𝒞[0,1] computes a non-computable subset of ℕ; there is a non-total degree between Turing degrees a<_T b iff b is a PA degree relative to a; 𝒮⊆ 2^ℕ is a Scott set iff it is the collection of f-computable subsets of ℕ for some f∈𝒞[0,1] of non-total degree; and there are computably incomparable f,g∈𝒞[0,1] which compute exactly the same subsets of ℕ. Proofs draw from classical analysis and constructive analysis as well as from computability theory.