Skip to main content
SpringerLink
Log in

Makanin's algorithm for word equations-two improvements and a generalization

  • Conference paper
  • First Online:
Word Equations and Related Topics (IWWERT 1990)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 572))

Included in the following conference series:

Abstract

In 1977 G.S. Makanin [Mak] proved that it is decidable whether a word equation has a solution or not. Here we describe two improvements of Makanin's algorithm which bring it nearer to the area of practical applicability: a simple pre-algorithm is suggested which decides the solvability of word equations with not more than two occurrences of each variable and which partially solves and simplifies the decision procedure for all other equations. A new transformation procedure is given which applies to arbitrary position equations and has several advantages. In a separate part we generalize Makanin's result and show that the solvability of word equations with variables x1,...,xn remains decidable when we specify regular languages L1,...,Ln over the coefficient alphabet and ask for solutions where the i-th components belongs to Li.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H.Abdulrab, Résolution d'équations sur les mots: Etude et implémentation LISP de l'algorithme de MAKANIN, Thèse de doctorat-Laboratoire d'informatique, Rouen 1987.

    Google Scholar 

  2. V.K.Bulitko, Equations and Inequalities in a Free Group and a Free Semigroup, Tul. Cos. Ped. Inst. Učen. Zap. Mat. Kafedr Vyp.2, Geometr. i Algebra (1970), pp. 242–252 (Russian).

    Google Scholar 

  3. D.C.Cooper, Theorem-proving in Arithmetic without Multiplication, Machine Intelligence 7 (1972), 82–95.

    Google Scholar 

  4. Z.Farkas, Listlog — A Prolog Extension for List Processing, Proc. TAPSOFT 1987, March 1987, LNCS 250, Vol. 2, 82–95.

    Google Scholar 

  5. L.Fribourg, List Concatenation via Extended Unification, Programmation en Logique, Actes du Séminaire 1987, Trégastel 1987, 45–59.

    Google Scholar 

  6. J.I.Hmelevskii, Equations in Free Semigroups, Trudy Mat. Inst. Steklov. 107 (1971); English transl. Proc. Steklov Inst. Math. 107 (1971).

    Google Scholar 

  7. J.Jaffar, Minimal and Complete Word Unification, JACM 37 (1), 47–85.

    Google Scholar 

  8. A.Kościelski,L.Pacholski, Complexity of Unification in free groups and free semigroups, Proc. 3st Annual IEEE Symposium on Foundations of Computer Science, Los Alamitos 1990, 824–829.

    Google Scholar 

  9. A.Lentin, Equations in Free Monoids, in Automata Languages and Programming, (M. Nivat, ed.) North Holland Publishers, Amsterdam 1972, 67–85.

    Google Scholar 

  10. A.Lentin,M.P.Schützenberger, A Combinatorial Problem in the Theory of Free Monoids, Proc. of the University of North-Carolina (1967), 128–144.

    Google Scholar 

  11. M.Livesey,J.Siekmann, Termination and Decidability Results for String Unification, Essex University, Memo, 1975.

    Google Scholar 

  12. M.Lothaire, Combinatorics on words, Addison Wesley, 1983.

    Google Scholar 

  13. G.S.Makanin, The problem of solvability of equations in a free semigroup, Math. USSR Sbornik 32, 2 (1977), 129–198.

    Google Scholar 

  14. G.S.Makanin, Recognition of the Rank of Equations in a Free Semigroup, Math. USSR Izvestija 14, 3 (1980), 499–545.

    Google Scholar 

  15. J.P.Pécuchet, Equations avec constantes et algorithme de Makanin, Thèse de doctorat, Laboratoire d' informatique, Rouen 1981.

    Google Scholar 

  16. G.Plotkin, Building-in Equational Theories, Machine Intelligence 7 (1972), 73–90.

    Google Scholar 

  17. J.A.Robinson, A Machine-Oriented Logic based on the Resolution Principle, Journal of the ACM 12 (1965), 32–41.

    Google Scholar 

  18. A.Roussel, Programmation de l'algorithme de Makanin en PROLOG II, Memoire de D.E.A., Université Aix-Marseille II, 1987.

    Google Scholar 

  19. K.U.Schulz, A Guide to Makanin's Algorithm, SNS-Bericht 88-39, Seminar für natürlich-sprachliche Systeme, University of Tübingen 1989.

    Google Scholar 

  20. K.U.Schulz, A Note on Makanin's Algorithm, SNS-Bericht 89-53, Seminar für natürlich-sprachliche Systeme, University of Tübingen 1989.

    Google Scholar 

  21. K.U.Schulz, Makanin's Algorithm — Two Improvements and a Generalization, CIS-Report 91-39, Centrum für Informations-und Sprachverarbeitung, University of Munique,1991.

    Google Scholar 

  22. J.Siekmannn, A Modification of Robinson's Unification Procedure, M.Sc.Thesis, 1972.

    Google Scholar 

  23. J. Siekmannn, Unification and Matching Problems, Ph.D.Thesis, Essex University, Memo CSA-4-78, 1978.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centrum für Informations- und Sprachverarbeitung (CIS), Universität München, Leopoldstr. 139, D-8000, München, Germany

    Klaus U. Schulz

Authors
  1. Klaus U. Schulz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

K. U. Schulz

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Schulz, K.U. (1992). Makanin's algorithm for word equations-two improvements and a generalization. In: Schulz, K.U. (eds) Word Equations and Related Topics. IWWERT 1990. Lecture Notes in Computer Science, vol 572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55124-7_4

Download citation

  • DOI: https://doi.org/10.1007/3-540-55124-7_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55124-9

  • Online ISBN: 978-3-540-46737-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation