Theory of Computing Systems

, Volume 61, Issue 2, pp 494–520 | Cite as

Equations Over Free Inverse Monoids with Idempotent Variables

  • Volker Diekert
  • Florent Martin
  • Géraud Sénizergues
  • Pedro V. Silva


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.


Free inverse monoid Equation Language equation Idempotent variable One-variable equation 



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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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Volker Diekert
    • 1
  • Florent Martin
    • 2
  • Géraud Sénizergues
    • 3
  • Pedro V. Silva
    • 4
  1. 1.FMIUniversität StuttgartStuttgartGermany
  2. 2.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  3. 3.LaBRIUniversité BordeauxTalence CedexFrance
  4. 4.Faculdade de CiênciasUniversidade do PortoPortoPortugal

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