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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.RTA.2010.311
URN: urn:nbn:de:0030-drops-26609
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2660/
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Simonsen, Jakob Grue

Weak Convergence and Uniform Normalization in Infinitary Rewriting

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Abstract

We study infinitary term rewriting systems containing finitely many rules. For these, we show that if a weakly convergent reduction is not strongly convergent, it contains a term that reduces to itself in one step (but the step itself need not be part of the reduction). Using this result, we prove
the starkly surprising result
that for any orthogonal system with finitely many rules, the system is
weakly normalizing under weak convergence if{f} it is strongly normalizing under weak convergence if{f} it is weakly normalizing under strong convergence if{f} it is strongly normalizing under strong convergence.

As further corollaries, we derive a number of new results for weakly convergent rewriting: Systems with finitely many rules enjoy unique normal forms, and acyclic orthogonal systems are confluent. Our results suggest that it may be possible to recover some of the positive results for strongly convergent rewriting in the setting of weak convergence, if systems with finitely many rules are considered. Finally, we give a number of counterexamples showing failure of most of the results when infinite sets of rules are allowed.

BibTeX - Entry

@InProceedings{simonsen:LIPIcs:2010:2660,
  author =	{Jakob Grue Simonsen},
  title =	{{Weak Convergence and Uniform Normalization in Infinitary Rewriting}},
  booktitle =	{Proceedings of the 21st International Conference on Rewriting Techniques and Applications},
  pages =	{311--324},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-18-7},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{6},
  editor =	{Christopher Lynch},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2010/2660},
  URN =		{urn:nbn:de:0030-drops-26609},
  doi =		{10.4230/LIPIcs.RTA.2010.311},
  annote =	{Keywords: Infinitary rewriting, weak convergence, uniform normalization}
}

Keywords: Infinitary rewriting, weak convergence, uniform normalization
Collection: Proceedings of the 21st International Conference on Rewriting Techniques and Applications
Issue Date: 2010
Date of publication: 06.07.2010

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