Numberings, c.e. oracles, and fixed points

Article type: Research Article

Authors: Faizrahmanov, Marat^{; *}

Affiliations: Volga Region Scientific-Educational Centre of Mathematics, Kazan (Volga Region) Federal University, 420008, 35 Kremlevskaya Street, Kazan Russian Federation

Correspondence: [*] Corresponding author. marat.faizrahmanov@gmail.com.

Abstract: The Arslanov completeness criterion says that a c.e. set A is Turing complete if and only there exists an A-computable function f without fixed points, i.e. a function f such that Wf(x)≠Wx for each integer x. Recently, Barendregt and Terwijn proved that the completeness criterion remains true if we replace the Gödel numbering x↦Wx with an arbitrary precomplete computable numbering. In this paper, we prove criteria for noncomputability and highness of c.e. sets in terms of (pre)complete computable numberings and fixed point properties. We also find some precomplete and weakly precomplete numberings of arbitrary families computable relative to Turing complete and non-computable c.e. oracles respectively.

Keywords: Arslanov completeness criterion, fixed point, complete numbering, precomplete numbering, weakly precomplete numbering, computable numbering, minimal numbering, high set

DOI: 10.3233/COM-210387

Journal: Computability, vol. 12, no. 3, pp. 271-282, 2023