June 2004 Degrees of unsolvability of continuous functions
Joseph S. Miller
J. Symbolic Logic 69(2): 555-584 (June 2004). DOI: 10.2178/jsl/1082418543


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 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.


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Joseph S. Miller. "Degrees of unsolvability of continuous functions." J. Symbolic Logic 69 (2) 555 - 584, June 2004. https://doi.org/10.2178/jsl/1082418543


Published: June 2004
First available in Project Euclid: 19 April 2004

zbMATH: 1070.03026
MathSciNet: MR2058189
Digital Object Identifier: 10.2178/jsl/1082418543

Rights: Copyright © 2004 Association for Symbolic Logic


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Vol.69 • No. 2 • June 2004
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