Cylindrification

In computability theory a cylindrification is a construction that associates a cylindric numbering to each numbering. The concept was first introduced by Yuri L. Ershov in 1973.

Definition

Given a numbering $\nu$, the cylindrification $c(\nu )$ is defined as

$\mathrm {Domain} (c(\nu )):=\{\langle n,k\rangle |n\in \mathrm {Domain} (\nu )\}$

$c(\nu )\langle n,k\rangle :=\nu (n)$

where $\langle n,k\rangle$ is the Cantor pairing function.

Note that the cylindrification operation increases the input arity by 1.

Properties

• Given two numberings $\nu$ and $\mu$ then $\nu \leq \mu \Leftrightarrow c(\nu )\leq _{1}c(\mu )$
• $\nu \leq _{1}c(\nu )$

References

• Yu. L. Ershov, "Theorie der Numerierungen I." Zeitschrift für mathematische Logik und Grundlagen der Mathematik 19, 289-388 (1973).