License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2014.530
URN: urn:nbn:de:0030-drops-44853
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2014/4485/
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Lohrey, Markus ; Zetzsche, Georg

On Boolean closed full trios and rational Kripke frames

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Abstract

A Boolean closed full trio is a class of languages that is closed under the Boolean operations (union, intersection, and complementation) and rational transductions. It is well-known that the regular languages constitute such a Boolean closed full trio. It is shown here that every such language class that contains any non-regular language already includes the whole arithmetical hierarchy (and even the one relative to this language).

A consequence of this result is that aside from the regular languages, no full trio generated by one language is closed under complementation.

Our construction also shows that there is a fixed rational Kripke frame such that assigning an arbitrary non-regular language to some variable allows the definition of any language from the arithmetical hierarchy in the corresponding Kripke structure using multimodal logic.

BibTeX - Entry

@InProceedings{lohrey_et_al:LIPIcs:2014:4485,
  author =	{Markus Lohrey and Georg Zetzsche},
  title =	{{On Boolean closed full trios and rational Kripke frames}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{530--541},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Ernst W. Mayr and Natacha Portier},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4485},
  URN =		{urn:nbn:de:0030-drops-44853},
  doi =		{10.4230/LIPIcs.STACS.2014.530},
  annote =	{Keywords: rational transductions, full trios, arithmetical hierarchy, Boolean operations}
}

Keywords: rational transductions, full trios, arithmetical hierarchy, Boolean operations
Collection: 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)
Issue Date: 2014
Date of publication: 05.03.2014

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