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URN: urn:nbn:de:0030-drops-96701
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Climbing up the Elementary Complexity Classes with Theories of Automatic Structures

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Abstract

Automatic structures are structures that admit a finite presentation via automata. Their most prominent feature is that their theories are decidable. In the literature, one finds automatic structures with non-elementary theory (e.g., the complete binary tree with equal-level predicate) and automatic structures whose theories are at most 3-fold exponential (e.g., Presburger arithmetic or infinite automatic graphs of bounded degree). This observation led Durand-Gasselin to the question whether there are automatic structures of arbitrary high elementary complexity. We give a positive answer to this question. Namely, we show that for every h >=0 the forest of (infinitely many copies of) all finite trees of height at most h+2 is automatic and it's theory is complete for STA(*, exp_h(n, poly(n)), poly(n)), an alternating complexity class between h-fold exponential time and space. This exact determination of the complexity of the theory of these forests might be of independent interest.

BibTeX - Entry

@InProceedings{abuzaid_et_al:LIPIcs:2018:9670,
  author =	{Faried Abu Zaid and Dietrich Kuske and Peter Lindner},
  title =	{{Climbing up the Elementary Complexity Classes with Theories of Automatic Structures}},
  booktitle =	{27th EACSL Annual Conference on Computer Science Logic  (CSL 2018)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-088-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{119},
  editor =	{Dan Ghica and Achim Jung},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9670},
  URN =		{urn:nbn:de:0030-drops-96701},
  doi =		{10.4230/LIPIcs.CSL.2018.3},
  annote =	{Keywords: Automatic Structures, Complexity Theory, Model Theory}
}

Keywords: Automatic Structures, Complexity Theory, Model Theory
Seminar: 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)
Issue date: 2018
Date of publication: 2018


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