Fixpoints and relative precompleteness

Article type: Research Article

Authors: Golov, Anton^{a; *} | Terwijn, Sebastiaan A.^b

Affiliations: [a] Department of Mathematics, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands | [b] Department of Mathematics, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands

Correspondence: [*] Corresponding author. agolov@science.ru.nl.

Abstract: 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.

Keywords: Computability theory, numberings

DOI: 10.3233/COM-210344

Journal: Computability, vol. 11, no. 2, pp. 135-146, 2022