We prove that for every countable ordinal one cannot decide whether a given infinitary rational relation is in the Borel class (respectively ). Furthermore one cannot decide whether a given infinitary rational relation is a Borel set or a -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.
Classification : 68Q45, 03D05, 03D55, 03E15
Mots clés : infinitary rational relations, topological properties, Borel and analytic sets, arithmetical properties, decision problems
@article{ITA_2003__37_2_115_0,
author = {Finkel, Olivier},
title = {Undecidability of topological and arithmetical properties of infinitary rational relations},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {115--126},
publisher = {EDP-Sciences},
volume = {37},
number = {2},
year = {2003},
doi = {10.1051/ita:2003013},
zbl = {1112.03312},
mrnumber = {2015687},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ita:2003013/}
}
TY - JOUR AU - Finkel, Olivier TI - Undecidability of topological and arithmetical properties of infinitary rational relations JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2003 DA - 2003/// SP - 115 EP - 126 VL - 37 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ita:2003013/ UR - https://zbmath.org/?q=an%3A1112.03312 UR - https://www.ams.org/mathscinet-getitem?mr=2015687 UR - https://doi.org/10.1051/ita:2003013 DO - 10.1051/ita:2003013 LA - en ID - ITA_2003__37_2_115_0 ER -
Finkel, Olivier. Undecidability of topological and arithmetical properties of infinitary rational relations. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 2, pp. 115-126. doi : 10.1051/ita:2003013. http://www.numdam.org/articles/10.1051/ita:2003013/
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