CAI 2015: Algebraic Informatics pp 22-28

# More Than 1700 Years of Word Equations

• Volker Diekert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9270)

## 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.

## Keywords

Steklov Institute Diophantine Equation Rational Constraint Free Monoids Correct Guess
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.

## References

1. 1.
Asveld, P.R.: Controlled iteration grammars and full hyper-AFL’s. Information and Control 34(3), 248–269 (1977)
2. 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)
3. 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)
4. 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. 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. 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. 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. 8.
Ferté, J., Marin, N., Sénizergues, G.: Word-mappings of level $$2$$. Theory Comput. Syst. 54, 111–148 (2014)
9. 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. 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. 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. 12.
Karp, R.M., Rabin, M.O.: Efficient randomized pattern-matching algorithms. IBM Journal of Research and Development 31, 249–260 (1987)
13. 13.
Kharlampovich, O., Myasnikov, A.: Elementary theory of free non-abelian groups. J. of Algebra 302, 451–552 (2006)
14. 14.
Kościelski, A., Pacholski, L.: Complexity of Makanin’s algorithm. Journal of the Association for Computing Machinery 43(4), 670–684 (1996)
15. 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. 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. 17.
Matiyasevich, Yu.: Hilbert’s Tenth Problem. MIT Press, Cambridge (1993)Google Scholar
18. 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. 19.
Plandowski, W.: Satisfiability of word equations with constants is in PSPACE. Journal of the Association for Computing Machinery 51, 483–496 (2004)
20. 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)
21. 21.
Razborov, A.A.: On Systems of Equations in Free Groups. PhD thesis, Steklov Institute of Mathematics (1987) (in Russian)Google Scholar
22. 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. 23.
Rozenberg, G., Salomaa, A.: The Book of L. Springer (1986)Google Scholar
24. 24.
Sela, Z.: Diophantine geometry over groups VIII: Stability. Annals of Math. 177, 787–868 (2013)

© Springer International Publishing Switzerland 2015

## Authors and Affiliations

1. 1.Institut Für Formale Methoden der InformatikUniversität StuttgartStuttgartGermany