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Undecidability of Topologicaland Arithmetical Propertiesof Infinitary Rational Relations

Published online by Cambridge University Press:  15 November 2003

Olivier Finkel
Olivier Finkel*
Affiliation:
Équipe de Logique Mathématique, U.F.R. de Mathématiques, Université Paris-7, 2 place Jussieu, 75251 Paris Cedex 05, France; finkel@logique.jussieu.fr.
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Abstract

We prove that for every countable ordinal α one cannot decidewhether a given infinitary rational relation is in the Borel class ${\bf \Sigma_{\alpha}^0}$ (respectively ${\bf \Pi_{\alpha}^0}$ ). Furthermoreone cannotdecide whether a given infinitary rational relation is a Borel set or a ${\bf \Sigma_{1}^1}$ -complete set. We prove some recursive analogues to theseproperties. In particular one cannot decide whether an infinitary rational relation is anarithmetical set.We then deduce from the proof ofthese results some other ones, like: one cannot decide whether thecomplement ofan infinitary rational relation is also an infinitary rational relation.

Keywords

Infinitary rational relationstopological propertiesBorel andanalytic setsarithmetical propertiesdecision problems.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 37 , Issue 2 , April 2003 , pp. 115 - 126
Copyright
© EDP Sciences, 2003

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