Cylindric numbering

In computability theory a cylindric numbering is a special kind of numbering first introduced by Yuri L. Ershov in 1973.

If a numbering $\nu$ is reducible to $\mu$ then there exists a computable function $f$ with $\nu =\mu \circ f$. Usually $f$ is not injective, but if $\mu$ is a cylindric numbering we can always find an injective $f$.

Definition

A numbering $\nu$ is called cylindric if

$\nu \equiv _{1}c(\nu ).$

That is if it is one-equivalent to its cylindrification

A set $S$ is called cylindric if its indicator function

$1_{S}:\mathbb {N} \to \{0,1\}$

is a cylindric numbering.

Examples

• Every Gödel numbering is cylindric

Properties

• Cylindric numberings are idempotent: $\nu \circ \nu =\nu$

References

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