The Lost Melody Phenomenon

Carl, Merlin

The Lost Melody Phenomenon

Dateien

Zu diesem Dokument gibt es keine Dateien.

Datum

2014

Autor:innen

Carl, Merlin

Sammlungen

Mathematik und Statistik

Publikationstyp

Beitrag zu einem Sammelband

Publikationsstatus

Published

Erschienen in

ED. BY STEFAN GESCHKE ..., , ed.. Infinity, Computability, and Metamathematics : Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch. London: College Publ., 2014, pp. 49-70. Tributes. 23. ISBN 978-1-84890-130-8

Zusammenfassung

A typical phenomenon for machine models of transfinite computations is the existence of so-called lost melodies, i.e. real numbers $x$ such that the characteristic function of the set {x} is computable while x itself is not (a real having the first property is called recognizable). This was first observed by J. D. Hamkins and A. Lewis for infinite time Turing machine, then demonstrated by P. Koepke and the author for ITRMs. We prove that, for unresetting infinite time register machines introduced by P. Koepke, recognizability equals computability, i.e. the lost melody phenomenon does not occur. Then, we give an overview on our results on the behaviour of recognizable reals for ITRMs. We show that there are no lost melodies for ordinal Turing machines or ordinal register machines without parameters and that this is, under the assumption that 0^{\sharp} exists, independent of $ZFC$. Then, we introduce the notions of resetting and unresetting\alpha-register machines and give some information on the question for which of these machines there are lost melodies.

Fachgebiet (DDC)

510 Mathematik

Schlagwörter

Mathematics - Logic

Zitieren

ISO 690

CARL, Merlin, 2014. The Lost Melody Phenomenon. In: ED. BY STEFAN GESCHKE ..., , ed.. Infinity, Computability, and Metamathematics : Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch. London: College Publ., 2014, pp. 49-70. Tributes. 23. ISBN 978-1-84890-130-8

BibTex

@incollection{Carl2014Melod-30531,
  year={2014},
  title={The Lost Melody Phenomenon},
  number={23},
  isbn={978-1-84890-130-8},
  publisher={College Publ.},
  address={London},
  series={Tributes},
  booktitle={Infinity, Computability, and Metamathematics : Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch},
  pages={49--70},
  editor={ed. by Stefan Geschke ...},
  author={Carl, Merlin}
}

RDF

<rdf:RDF
    xmlns:dcterms="http://purl.org/dc/terms/"
    xmlns:dc="http://purl.org/dc/elements/1.1/"
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
    xmlns:bibo="http://purl.org/ontology/bibo/"
    xmlns:dspace="http://digital-repositories.org/ontologies/dspace/0.1.0#"
    xmlns:foaf="http://xmlns.com/foaf/0.1/"
    xmlns:void="http://rdfs.org/ns/void#"
    xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > 
  <rdf:Description rdf:about="https://kops.uni-konstanz.de/server/rdf/resource/123456789/30531">
    <dcterms:issued>2014</dcterms:issued>
    <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/>
    <dcterms:title>The Lost Melody Phenomenon</dcterms:title>
    <foaf:homepage rdf:resource="http://localhost:8080/"/>
    <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2015-03-25T07:46:30Z</dc:date>
    <dc:creator>Carl, Merlin</dc:creator>
    <bibo:uri rdf:resource="http://kops.uni-konstanz.de/handle/123456789/30531"/>
    <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2015-03-25T07:46:30Z</dcterms:available>
    <dcterms:abstract xml:lang="eng">A typical phenomenon for machine models of transfinite computations is the existence of so-called lost melodies, i.e. real numbers $x$ such that the characteristic function of the set {x\} is computable while x itself is not (a real having the first property is called recognizable). This was first observed by J. D. Hamkins and A. Lewis for infinite time Turing machine, then demonstrated by P. Koepke and the author for ITRMs. We prove that, for unresetting infinite time register machines introduced by P. Koepke, recognizability equals computability, i.e. the lost melody phenomenon does not occur. Then, we give an overview on our results on the behaviour of recognizable reals for ITRMs. We show that there are no lost melodies for ordinal Turing machines or ordinal register machines without parameters and that this is, under the assumption that 0^{\sharp} exists, independent of $ZFC$. Then, we introduce the notions of resetting and unresetting\alpha-register machines and give some information on the question for which of these machines there are lost melodies.</dcterms:abstract>
    <dc:language>eng</dc:language>
    <dc:contributor>Carl, Merlin</dc:contributor>
    <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
  </rdf:Description>
</rdf:RDF>

Universitätsbibliographie

Ja