We introduce the k-variable-occurrence fragment, which is the set of terms having at most k occurrences of variables. We give a sufficient condition for the decidability of the equational theory of the k-variable-occurrence fragment using the finiteness of a monoid. As a case study, we prove that for Tarski’s calculus of relations with bounded dot-dagger alternation (an analogy of quantifier alternation in first-order logic), the equational theory of the k-variable-occurrence fragment is decidable for each k.
@InProceedings{nakamura:LIPIcs.MFCS.2023.69, author = {Nakamura, Yoshiki}, title = {{On the Finite Variable-Occurrence Fragment of the Calculus of Relations with Bounded Dot-Dagger Alternation}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {69:1--69:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.69}, URN = {urn:nbn:de:0030-drops-186030}, doi = {10.4230/LIPIcs.MFCS.2023.69}, annote = {Keywords: Relation algebra, First-order logic, Decidable fragment, Monoid} }
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