# Undecidability of Topological and Arithmetical Properties of Infinitary Rational Relations

RAIRO - Theoretical Informatics and Applications (2010)

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

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topFinkel, 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 -

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