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Cylindric numbering

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

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

Definition

A numbering ν {\displaystyle \nu } is called cylindric if

ν 1 c ( ν ) . {\displaystyle \nu \equiv _{1}c(\nu ).}

That is if it is one-equivalent to its cylindrification

A set S {\displaystyle S} is called cylindric if its indicator function

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

is a cylindric numbering.

Examples

Properties

  • Cylindric numberings are idempotent: ν ν = ν {\displaystyle \nu \circ \nu =\nu }

References

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