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Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations

Olivier Finkel

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 37, Issue: 2, page 115-126
  • ISSN: 0988-3754

Abstract

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We prove that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class Σ α 0 (respectively Π α 0 ). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a Σ 1 1 -complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is an arithmetical set. We then deduce from the proof of these results some other ones, like: one cannot decide whether the complement of an infinitary rational relation is also an infinitary rational relation.

How to cite

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Finkel, Olivier. "Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations." RAIRO - Theoretical Informatics and Applications 37.2 (2010): 115-126. <http://eudml.org/doc/92717>.

@article{Finkel2010,
abstract = { We prove that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class $\{\bf \Sigma_\{\alpha\}^0\}$ (respectively $\{\bf \Pi_\{\alpha\}^0\}$). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a $\{\bf \Sigma_\{1\}^1\}$-complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is an arithmetical set. We then deduce from the proof of these results some other ones, like: one cannot decide whether the complement of an infinitary rational relation is also an infinitary rational relation. },
author = {Finkel, Olivier},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Infinitary rational relations; topological properties; Borel and analytic sets; arithmetical properties; decision problems.},
language = {eng},
month = {3},
number = {2},
pages = {115-126},
publisher = {EDP Sciences},
title = {Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations},
url = {http://eudml.org/doc/92717},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Finkel, Olivier
TI - Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 2
SP - 115
EP - 126
AB - We prove that for every countable ordinal α one cannot decide whether a given infinitary rational relation is in the Borel class ${\bf \Sigma_{\alpha}^0}$ (respectively ${\bf \Pi_{\alpha}^0}$). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a ${\bf \Sigma_{1}^1}$-complete set. We prove some recursive analogues to these properties. In particular one cannot decide whether an infinitary rational relation is an arithmetical set. We then deduce from the proof of these results some other ones, like: one cannot decide whether the complement of an infinitary rational relation is also an infinitary rational relation.
LA - eng
KW - Infinitary rational relations; topological properties; Borel and analytic sets; arithmetical properties; decision problems.
UR - http://eudml.org/doc/92717
ER -

References

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