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Equations Over Free Inverse Monoids with Idempotent Variables

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Abstract

We introduce the notion of idempotent variables for studying equations in inverse monoids. It is proved that it is decidable in singly exponential time (DEXPTIME) whether a system of equations in idempotent variables over a free inverse monoid has a solution. Moreover the problem becomes hard for DEXPTIME, as soon as the quotient group of the free inverse monoid has rank at least two. The upper bound is proved by a direct reduction to solve language equations with one-sided concatenation and a known complexity result by Baader and Narendran (J. Symb. Comput. 31, 277–305 2001). For the lower bound we show hardness for a restricted class of language equations. Decidability for systems of typed equations over a free inverse monoid with one irreducible variable and at least one unbalanced equation is proved with the same complexity upper-bound. Our results improve known complexity bounds by Deis et al. (IJAC 17, 761–795 2007). Our results also apply to larger families of equations where no decidability has been previously known. The lower bound confirms a conjecture made in the conference version of this paper.

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Notes

  1. Assumption 2 in Definition 2 was missing, making this previous proof incomplete.

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Acknowledgments

We would like to thank the referees for numerous comments which improved greatly the quality of the text. Florent Martin acknowledges support from Labex CEMPI (ANR-11-LABX-0007-01) and SFB 1085 Higher invariants. Pedro Silva acknowledges support from: CNPq (Brazil) through a BJT-A grant (process 313768/2013-7); and the European Regional Development Fund through the programme COMPETE and the Portuguese Government through FCT (Fundaç ão para a Ciência e a Tecnologia) under the project PEst-C/MAT/UI0144/2013. Volker Diekert thanks the hospitality of Universidade Federal da Bahia, Salvador Brazil, where part of this work started in Spring 2014. The authors are thankful to the program committee of CSR 2015 for awarding our paper with the Yandex-best-paper award; and one of the authors is even more thankful for the memorable event of Computer Science in Russia 2015 which was held at the shores of a truly magnificent Lake Baikal.

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Authors and Affiliations

  1. FMI, Universität Stuttgart, Universitätsstr. 38, 70569, Stuttgart, Germany

    Volker Diekert

  2. Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93040, Regensburg, Germany

    Florent Martin

  3. LaBRI, Université Bordeaux, 351, cours de la Libération, 33405, Talence Cedex, France

    Géraud Sénizergues

  4. Faculdade de Ciências, Universidade do Porto, R. Campo Alegre 687, 4169-007, Porto, Portugal

    Pedro V. Silva

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  1. Volker Diekert
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  2. Florent Martin
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  3. Géraud Sénizergues
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  4. Pedro V. Silva
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Correspondence to Volker Diekert.

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Diekert, V., Martin, F., Sénizergues, G. et al. Equations Over Free Inverse Monoids with Idempotent Variables. Theory Comput Syst 61, 494–520 (2017). https://doi.org/10.1007/s00224-016-9693-1

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