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Infinitary Rewriting: Foundations Revisited Kahrs, Stefan Infinitary rewriting,equivalence Infinitary Term Rewriting allows to express infinitary terms and infinitary reductions that converge to them. As their notion of transfinite reduction in general, and as binary relations in particular two concepts have been studied in the past: strongly and weakly convergent reductions, and in the last decade research has mostly focused around the former. Finitary rewriting has a strong connection to the equational theory of its rule set: if the rewrite system is confluent this (implies consistency of the theory and) gives rise to a semi-decision procedure for the theory, and if the rewrite system is in addition terminating this becomes a decision procedure. This connection is the original reason for the study of these properties in rewriting. For infinitary rewriting there is barely an established notion of an equational theory. The reason this issue is not trivial is that such a theory would need to include some form of ``getting to limits'', and there are different options one can pursue. These options are being looked at here, as well as several alternatives for the notion of reduction relation and their relationships to these equational theories. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Stefan Kahrs 2010 Is Part Of LIPIcs, Volume 6, Proceedings of the 21st International Conference on Rewriting Techniques and Applications (2010) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.RTA.2010.161 urn:nbn:de:0030-drops-26510 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2010.161 eng https://creativecommons.org/licenses/by-nc-nd/3.0/legalcode