Solutions of Word Equations Over Partially Commutative Structures

Authors Volker Diekert, Artur Jez, Manfred Kufleitner

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Volker Diekert
Artur Jez
Manfred Kufleitner

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Volker Diekert, Artur Jez, and Manfred Kufleitner. Solutions of Word Equations Over Partially Commutative Structures. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 127:1-127:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We give NSPACE(n*log(n)) algorithms solving the following decision problems. Satisfiability: Is the given equation over a free partially commutative monoid with involution (resp. a free partially commutative group) solvable? Finiteness: Are there only finitely many solutions of such an equation? PSPACE algorithms with worse complexities for the first problem are known, but so far, a PSPACE algorithm for the second problem was out of reach. Our results are much stronger: Given such an equation, its solutions form an EDT0L language effectively representable in NSPACE(n*log(n)). In particular, we give an effective description of the set of all solutions for equations with constraints in free partially commutative monoids and groups.
  • Word equations
  • EDT0L language
  • trace monoid
  • right-angled Artin group
  • partial commutation


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  1. Peter R.J. Asveld. Controlled iteration grammars and full hyper-AFL’s. Information and Control, 34(3):248-269, 1977. Google Scholar
  2. Jean Berstel and Dominique Perrin. The origins of combinatorics on words. Eur. J. Comb., 28:996-1022, 2007. Google Scholar
  3. Montserrat Casals and Ilya Kazachkov. On systems of equations over partially commutative groups. Memoirs Amer. Math. Soc., 212:1-153, 2011. Google Scholar
  4. Laura Ciobanu, Volker Diekert, and Murray Elder. Solution sets for equations over free groups are EDT0L languages. In ICALP 2015, Proceedings, volume 9135 of LNCS, pages 134-145. Springer, 2015. Google Scholar
  5. Volker Diekert, Claudio Gutiérrez, and \Christian Hagenah. The existential theory of equations with rational constraints in free groups is PSPACE-complete. Information and Computation, 202:105-140, 2005. Google Scholar
  6. Volker Diekert, Artur Jeż, and Manfred Kufleitner. Solutions of word equations over partially commutative structures. ArXiv e-prints, 2016. URL:
  7. Volker Diekert and Markus Lohrey. Word equations over graph products. Int. J. Algebra Comput., 18:493-533, 2008. Google Scholar
  8. Volker Diekert and Anca Muscholl. Solvability of equations in free partially commutative groups is decidable. Int. J. Algebra Comput., 16:1047-1070, 2006. Google Scholar
  9. Julien Ferté, Nathalie Marin, and Géraud Sénizergues. Word-mappings of level 2. Theory Comput. Syst., 54:111-148, 2014. Google Scholar
  10. \Ju. I. Hmelevski\uı. Equations in Free Semigroups. Number 107 in Proc. Steklov Institute of Mathematics. American Mathematical Society, 1976. Translated from the Russian original: Trudy Mat. Inst. Steklov. 107, 1971. Google Scholar
  11. Sanjay Jain, Alexei Miasnikov, and Frank Stephan. The complexity of verbal languages over groups. In LICS 2012, Proceedings, pages 405-414. IEEE Computer Society, 2012. Google Scholar
  12. Artur Jeż. Recompression: a simple and powerful technique for word equations. J. ACM, 63:1-51, 2016. Conference version at STACS 2013. Google Scholar
  13. O. Kharlampovich and A. Myasnikov. Elementary theory of free non-abelian groups. J. of Algebra, 302:451-552, 2006. Google Scholar
  14. Roger C. Lyndon and Marcel-Paul Schützenberger. The equation a^M = b^Nc^P in a free group. Michigan Math. J., 9:289-298, 1962. Google Scholar
  15. Gennadií S. Makanin. The problem of solvability of equations in a free semigroup. Math. Sbornik, 103:147-236, 1977. English transl. in Math. USSR Sbornik 32 (1977). Google Scholar
  16. Gennadií S. Makanin. Equations in a free group. Izv. Akad. Nauk SSR, Ser. Math. 46:1199-1273, 1983. English transl. in Math. USSR Izv. 21 (1983). Google Scholar
  17. \Yuri Matiyasevich. Some decision problems for traces. In LFCS 1997, Proceedings, volume 1234 of LNCS, pages 248-257. Springer, 1997. Google Scholar
  18. Antoni Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977. Google Scholar
  19. Wojciech Plandowski. Satisfiability of word equations with constants is in PSPACE. J. ACM, 51:483-496, 2004. Google Scholar
  20. Wojciech Plandowski. An efficient algorithm for solving word equations. In STOC, pages 467-476. ACM, 2006. Google Scholar
  21. Wojciech Plandowski and Wojciech Rytter. Application of Lempel-Ziv encodings to the solution of word equations. In ICALP 1998, Proceedings, volume 1443 of LNCS, pages 731-742. Springer, 1998. Google Scholar
  22. Alexander A. Razborov. On Systems of Equations in Free Groups. PhD thesis, Steklov Institute of Mathematics, 1987. In Russian. Google Scholar
  23. Zlil Sela. Diophantine geometry over groups VIII: Stability. Ann. of Math., 177:787-868, 2013. Google Scholar
  24. Daniel Wise. From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry. American Mathematical Society, 2012. Google Scholar
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