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**Published in:** LIPIcs, Volume 10, 22nd International Conference on Rewriting Techniques and Applications (RTA'11) (2011)

Bounded rewriting for linear term rewriting systems has been defined in (I. Durand, G. Sénizergues, M. Sylvestre. Termination of linear bounded term rewriting systems. Proceedings of the 21st International Conference on Rewriting Techniques and Applications) as a restriction of the usual notion of rewriting. We extend here this notion to the whole class of left-linear term rewriting systems, and we show that bounded rewriting is effectively inverse-recognizability preserving. The bounded class (BO) is, by definition, the set of left-linear systems for which every derivation can be replaced by a bottom-up derivation. The class BO contains (strictly) several classes of systems which were already known to be inverse-recognizability preserving: the left-linear growing systems, and the inverse right-linear finite-path overlapping systems.

Irène Durand and Marc Sylvestre. Left-linear Bounded TRSs are Inverse Recognizability Preserving. In 22nd International Conference on Rewriting Techniques and Applications (RTA'11). Leibniz International Proceedings in Informatics (LIPIcs), Volume 10, pp. 361-376, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{durand_et_al:LIPIcs.RTA.2011.361, author = {Durand, Ir\`{e}ne and Sylvestre, Marc}, title = {{Left-linear Bounded TRSs are Inverse Recognizability Preserving}}, booktitle = {22nd International Conference on Rewriting Techniques and Applications (RTA'11)}, pages = {361--376}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-30-9}, ISSN = {1868-8969}, year = {2011}, volume = {10}, editor = {Schmidt-Schauss, Manfred}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2011.361}, URN = {urn:nbn:de:0030-drops-31350}, doi = {10.4230/LIPIcs.RTA.2011.361}, annote = {Keywords: Term Rewriting, Preservation of Recognizability, Rewriting Strategies} }

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**Published in:** LIPIcs, Volume 6, Proceedings of the 21st International Conference on Rewriting Techniques and Applications (2010)

For the whole class of linear term rewriting systems and for each integer k, we define k-bounded rewriting as a restriction of the usual notion of rewriting.
We show that the k-bounded uniform termination, the k-bounded termination, the inverse k-bounded uniform, and the inverse k-bounded problems are decidable.
The k-bounded class (BO(k)) is, by definition, the set of linear systems for
which every derivation can be replaced by a k-bounded derivation. In general, for BO(k) systems, the uniform (respectively inverse uniform) k-bounded termination problem is not equivalent to the uniform (resp. inverse uniform) termination problem, and the k-bounded (respectively inverse k-bounded) termination problem is not equivalent to the termination (respectively inverse termination) problem.
This leads us to
define more restricted classes for which these problems are equivalent: the classes BOLP(k) of
k-bounded systems that have the length preservation property. By definition, a system is BOLP(k) if every derivation of length n can be replaced by a k-bounded derivation of length n.
We define the class BOLP of bounded systems that have the length preservation property as the union of all the BOLP(k) classes.
The class BOLP contains (strictly) several already known classes of systems:
the inverse left-basic semi-Thue systems,
the linear growing term rewriting systems, the inverse Linear-Finite-Path-Ordering systems, the strongly bottom-up systems.

Irène Durand, Géraud Sénizergues, and Marc Sylvestre. Termination of linear bounded term rewriting systems. In Proceedings of the 21st International Conference on Rewriting Techniques and Applications. Leibniz International Proceedings in Informatics (LIPIcs), Volume 6, pp. 341-356, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{durand_et_al:LIPIcs.RTA.2010.341, author = {Durand, Ir\`{e}ne and S\'{e}nizergues, G\'{e}raud and Sylvestre, Marc}, title = {{Termination of linear bounded term rewriting systems}}, booktitle = {Proceedings of the 21st International Conference on Rewriting Techniques and Applications}, pages = {341--356}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-18-7}, ISSN = {1868-8969}, year = {2010}, volume = {6}, editor = {Lynch, Christopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2010.341}, URN = {urn:nbn:de:0030-drops-26625}, doi = {10.4230/LIPIcs.RTA.2010.341}, annote = {Keywords: Term rewriting, termination, rewriting strategy} }

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