Skip to main content
SpringerLink
Log in

More Than 1700 Years of Word Equations

  • Conference paper
  • First Online:
Algebraic Informatics (CAI 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9270))

Included in the following conference series:

Abstract

Geometry and Diophantine equations have been ever-present in mathematics. According to the existing literature the work of Diophantus of Alexandria was mentioned before 364 AD, but a systematic mathematical study of word equations began only in the 20th century. So, the title of the present article does not seem to be justified at all. However, a Diophantine equation can be viewed as a special case of a system of word equations over a unary alphabet, and, more importantly, a word equation can be viewed as a special case of a Diophantine equation. Hence, the problem WordEquations: “Is a given word equation solvable?”, is intimately related to Hilbert’s 10th problem on the solvability of Diophantine equations. This became clear to the Russian school of mathematics at the latest in the mid 1960s, after which a systematic study of that relation began.

Here, we review some recent developments which led to an amazingly simple decision procedure for WordEquations, and to the description of the set of all solutions as an EDT0L language.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Asveld, P.R.: Controlled iteration grammars and full hyper-AFL’s. Information and Control 34(3), 248–269 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  2. Boudet, A., Comon, H.: Diophantine equations, Presburger arithmetic and finite automata. In: Kirchner, H. (ed.) CAAP 1996. LNCS, vol. 1059, pp. 30–43. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  3. Ciobanu, L., Diekert, V., Elder, M.: Solution Sets for equations over free groups are EDT0L languages. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015, Part II. LNCS, vol. 9135, pp. 134–145. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  4. Ciobanu, L., Diekert, V., Elder, M.: Solution sets for equations over free groups are EDT0L languages. ArXiv e-prints, abs/1502.03426 (2015)

    Google Scholar 

  5. Diekert, V.: Makanin’s algorithm. In: Lothaire, M., (eds.) Algebraic Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 90, chapter 12, pp. 387–442. Cambridge University Press (2002)

    Google Scholar 

  6. Diekert, V., Gutiérrez, C., Hagenah, Ch.: The existential theory of equations with rational constraints in free groups is PSPACE-complete. Information and Computation 202, 105–40 (2005). Conference version in STACS 2001. LNCS 2010, pp. 170–182. Springer, Heidelberg (2001)

    Google Scholar 

  7. Diekert, V., Jeż, A., Plandowski, W.: Finding all solutions of equations in free groups and monoids with involution. In: Hirsch, E.A., Kuznetsov, S.O., Pin, J.É., Vereshchagin, N.K. (eds.) CSR 2014. LNCS, vol. 8476, pp. 1–15. Springer, Heidelberg (2014)

    Google Scholar 

  8. Ferté, J., Marin, N., Sénizergues, G.: Word-mappings of level \(2\). Theory Comput. Syst. 54, 111–148 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gutiérrez, C.: Satisfiability of word equations with constants is in exponential space. In: Proc. 39th Ann. Symp. on Foundations of Computer Science (FOCS 1998), pp. 112–119. IEEE Computer Society Press, Los Alamitos (1998)

    Google Scholar 

  10. Gutiérrez, C.: Satisfiability of equations in free groups is in PSPACE. In: Proceedings 32nd Annual ACM Symposium on Theory of Computing, STOC 2000, pp. 21–27. ACM Press (2000)

    Google Scholar 

  11. Jeż, A.: Recompression: a simple and powerful technique for word equations. In: Portier, N., Wilke, T. (eds.) STACS. LIPIcs, vol. 20, pp. 233–244. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. To appear in JACM, Dagstuhl, Germany (2013)

    Google Scholar 

  12. Karp, R.M., Rabin, M.O.: Efficient randomized pattern-matching algorithms. IBM Journal of Research and Development 31, 249–260 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kharlampovich, O., Myasnikov, A.: Elementary theory of free non-abelian groups. J. of Algebra 302, 451–552 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kościelski, A., Pacholski, L.: Complexity of Makanin’s algorithm. Journal of the Association for Computing Machinery 43(4), 670–684 (1996)

    Article  MATH  Google Scholar 

  15. Makanin, G.S.: 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. Makanin, G.S.: Decidability of the universal and positive theories of a free group. Izv. Akad. Nauk SSSR, Ser. Mat. 48 735–749 (1984) (in Russian). nglish translation. In: Math. USSR Izvestija 25(75–88) (1985)

    Google Scholar 

  17. Matiyasevich, Yu.: Hilbert’s Tenth Problem. MIT Press, Cambridge (1993)

    Google Scholar 

  18. Plandowski, W.: Satisfiability of word equations with constants is in PSPACE. In: Proc. 40th Ann. Symp. on Foundations of Computer Science, FOCS 1999, pp. 495–500. IEEE Computer Society Press (1999)

    Google Scholar 

  19. Plandowski, W.: Satisfiability of word equations with constants is in PSPACE. Journal of the Association for Computing Machinery 51, 483–496 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Plandowski, W., Rytter, W.: Application of lempel-ziv encodings to the solution of word equations. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 731–742. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  21. Razborov, A.A.: On Systems of Equations in Free Groups. PhD thesis, Steklov Institute of Mathematics (1987) (in Russian)

    Google Scholar 

  22. Razborov, A.A.: On systems of equations in free groups. In: Combinatorial and Geometric Group Theory, pp. 269–283. Cambridge University Press (1994)

    Google Scholar 

  23. Rozenberg, G., Salomaa, A.: The Book of L. Springer (1986)

    Google Scholar 

  24. Sela, Z.: Diophantine geometry over groups VIII: Stability. Annals of Math. 177, 787–868 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Für Formale Methoden der Informatik, Universität Stuttgart, Stuttgart, Germany

    Volker Diekert

Authors
  1. Volker Diekert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Volker Diekert .

Editor information

Editors and Affiliations

  1. Institut für Maschinelle, Universität Stuttgart, Stuttgart, Baden-Württemberg, Germany

    Andreas Maletti

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Diekert, V. (2015). More Than 1700 Years of Word Equations. In: Maletti, A. (eds) Algebraic Informatics. CAI 2015. Lecture Notes in Computer Science(), vol 9270. Springer, Cham. https://doi.org/10.1007/978-3-319-23021-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-23021-4_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23020-7

  • Online ISBN: 978-3-319-23021-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation