Matching Modulo Associativity and Idempotency Is NP—Complete

  • Ondřej Klíma
  • Jiří Srba
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1893)


We show that AI-matching (AI denotes the theory of an associative and idempotent function symbol), which is solving matching word equations in free idempotent semigroups, is NP-complete.


Normal Form Word Problem Function Symbol Minimal Solution Free Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Baader F.: The Theory of Idempotent Semigroups is of Unification Type Zero, J. of Automated Reasoning 2 (1986) 283–286.zbMATHCrossRefGoogle Scholar
  2. [2]
    Baader F.: Unification in Varieties of Idempotent Semigroups, Semigroup Forum 36 (1987) 127–145.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Baader F., Schulz K.U.: Unification in the Union of Disjoint Equational Theories: Combining Decision Procedures, J. Symbolic Computation 21 (1996) 211–243.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Baader F., Siekmann J.H.: Unification Theory, Handbook of Logic in Artificial Intelligence and Logic Programming (1993) Oxford University Press.Google Scholar
  5. [5]
    Book R., Otto F.: String-Rewriting Systems (1993) Springer-Verlag.Google Scholar
  6. [6]
    Černá I., Klíma O., Srba J.: Pattern Equations and Equations with Stuttering, In Proceedings of SOFSEM’99, the 26th Seminar on Current Trends in Theory and Practice of Informatics (1999) 369–378, Springer-Verlag.Google Scholar
  7. [7]
    Green J.A., Rees D.: On semigroups in which x r = x, Proc. Camb. Phil. Soc. 48 (1952) 35–40zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Klíma O., Srba J.: Matching Modulo Associativity and Idempotency is NP-complete, Technical report RS-00-13, BRICS, Aarhus University (2000).Google Scholar
  9. [9]
    Kaďourek J., Polák L.: On free semigroups satisfying x r = x, Simon Stevin 64, No. 1 (1990) 3–19MathSciNetzbMATHGoogle Scholar
  10. [10]
    Kapur D., Narendran P.: NP-completeness of the Set Unification and Matching Problems, In Proceedings of CADE’86, Springer LNCS volume 230 (1986) 489–495, Springer-Verlag.Google Scholar
  11. [11]
    Kopeček I.: Automatic Segmentation into Syllable Segments, Proc. of First International Conference on Language Resources and Evaluation (1998) 1275–1279.Google Scholar
  12. [12]
    Kopeček I., Pala K.: Prosody Modelling for Syllable-Based Speech Synthesis, Proceedings of the IASTED International Conference on Artificial Intelligence and Soft Computing, Cancun (1998) 134–137.Google Scholar
  13. [13]
    Lothaire M.: Algebraic Combinatorics on Words, Preliminary version available at
  14. [14]
    Lothaire, M.: Combinatorics on Words, Volume 17 of Encyclopedia of Mathematics and its Applications (1983) Addison-Wesley.Google Scholar
  15. [15]
    Makanin, G. S.: The Problem of Solvability of Equations in a Free Semigroup, Mat. Sbornik. 103(2) (1977) 147–236. (In Russian) English translation in: Math. USSR Sbornik 32 (1977) 129-198.MathSciNetGoogle Scholar
  16. [16]
    Papadimitriou, C.H.: Computational Complexity, Addison-Wesley Publishing Company (1994), Reading, Mass.Google Scholar
  17. [17]
    Perrin D.: Equations in Words, In H. Ait-Kaci and M. Nivat, editors, Resolution of Equations in Algebraic Structures, Vol. 2 (1989) 275–298, Academic Press.Google Scholar
  18. [18]
    Schulz, K. U.: Makanin’s Algorithm for Word Equations: Two Improvements and a Generalization, In Schulz, K.-U. (ntEd.), Proceedings of Word Equations and Related Topics, 1st International Workshop, IWW-ERT’90, Tübingen, Germany, Vol. 572 of LNCS (1992) 85–150, Berlin-Heidelberg-New York, Springer-Verlag.Google Scholar
  19. [19]
    Schmidt-Schauss M.: Unification under Associativity and Idempotence is of Type Nullary, J. of Automated Reasoning 2 (1986) 277–281.zbMATHCrossRefGoogle Scholar
  20. [20]
    Siekmann J., Szabó P.: A Noetherian and Confluent Rewrite System for Idempotent Semigroups, Semigroup Forum 25 (1982).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Ondřej Klíma
    • 1
  • Jiří Srba
    • 2
  1. 1.Faculty of Science MU Dept. of MathematicsBrnoCzech Republic
  2. 2.BRICS, Department of Computer ScienceUniversity of AarhusAarhus CDenmark

Personalised recommendations