Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-10T20:14:32.024Z Has data issue: false hasContentIssue false
  • English
  • Français

Equations on partial words

Published online by Cambridge University Press:  20 December 2007

Francine Blanchet-Sadri ,
D. Dakota Blair  and
Rebeca V. Lewis
Francine Blanchet-Sadri
Affiliation:
Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, North Carolina 27402–6170, USA; blanchet@uncg.edu A research assignment from the University of North Carolina at Greensboro is gratefully acknowledged. Some of this assignment was spent at the LIAFA: Laboratoire d'Informatique Algorithmique: Fondements et Applications of Université Paris 7, Paris, France, and at the University of Debrecen, Debrecen, Hungary.
D. Dakota Blair
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843–3368, USA.
Rebeca V. Lewis
Affiliation:
Department of Mathematics, Tennessee Tech University, Box 5054, Cookeville, TN 38505–0001, USA.
Get access

Abstract

It is well-known that some of the most basic properties of words, like thecommutativity (xy = yx) and the conjugacy (xz = zy), can be expressedas solutions of word equations. An important problem is to decide whetheror not a given equation on words has a solution. For instance,the equation xMyN = zP has only periodic solutions in a freemonoid, that is, if xMyN = zP holds with integers m,n,p ≥ 2,then there exists a word w such that x, y, z are powers of w.This result, which received a lot of attention, was first provedby Lyndon and Schützenberger for free groups.In this paper, we investigate equations on partial words.Partial words are sequences over a finite alphabet that may containa number of “do not know” symbols. When we speak about equationson partial words, we replace the notion of equality(=) with compatibility (↑).Among other equations, we solve xy ↑ yx,xz ↑ zy, and special cases of xmyn ↑ zp for integers m,n,p ≥ 2. 


Keywords

Equations on wordsequations on partial wordscommutativityconjugacyfree monoid.

Information

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 43 , Issue 1 , January 2009 , pp. 23 - 39
Copyright
© EDP Sciences, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Berstel, J. and Boasson, L., Partial words and a theorem of Fine and Wilf. Theoret. Comput. Sci. 218 (1999) 135–141. CrossRef
Blanchet-Sadri, F., Periodicity on partial words. Comput. Math. Appl. 47 (2004) 71–82. CrossRef
Blanchet-Sadri, F., Codes, orderings, and partial words. Theoret. Comput. Sci. 329 (2004) 177–202. CrossRef
Blanchet-Sadri, F., Primitive partial words. Discrete Appl. Math. 48 (2005) 195–213. CrossRef
Blanchet-Sadri, F. and Arundhati R. Anavekar, Testing primitivity on partial words. Discrete Appl. Math. 155 (2007) 179–287. CrossRef
Blanchet-Sadri, F., Dakota Blair, D., and Lewis, R.V., Equations on partial words, MFCS 2006 31st International Symposium on Mathematical Foundations of Computer Science. Lect. Notes Comput. Sci. 3053 (2006) 611–622.
Blanchet-Sadri, F. and Ajay Chriscoe, Local periods and binary partial words: an algorithm. Theoret. Comput. Sci. 314 (2004) 189–216. http://www.uncg.edu/mat/AlgBin/ CrossRef
Blanchet-Sadri, F. and Duncan, S., Partial words and the critical factorization theorem. J. Comb. Theory A 109 (2005) 221–245. http://www.uncg.edu/mat/cft/ CrossRef
Blanchet-Sadri, F. and Hegstrom, R.A., Partial words and a theorem of Fine and Wilf revisited. Theoret. Comput. Sci. 270 (2002) 401–419. CrossRef
Blanchet-Sadri, F. and Luhmann, D.K., Conjugacy on partial words. Theoret. Comput. Sci. 289 (2002) 297–312. CrossRef
Blanchet-Sadri, F. and Wetzler, N.D., Partial words and the critical factorization theorem revisited. Theoret. Comput. Sci. 385 (2007) 179–192. http://www.uncg.edu/mat/research/cft2/ CrossRef
Césari, Y. and Vincent, M., Une caractérisation des mots périodiques. C.R. Acad. Sci. Paris 268 (1978) 1175–1177.
C. Choffrut and J. Karhumäki, Combinatorics of Words, in Handbook of Formal Languages, Vol. 1, Ch. 6, edited by G. Rozenberg and A. Salomaa, Springer-Verlag, Berlin (1997) 329–438.
Chu, D.D. and Town, H.S., Another proof on a theorem of Lyndon and Schützenberger in a free monoid. Soochow J. Math. 4 (1978) 143–146.
M. Crochemore and W. Rytter, Text Algorithms. Oxford University Press, New York, NY (1994).
M. Crochemore and W. Rytter, Jewels of Stringology. World Scientific, NJ (2003).
Czeizler, E., The non-parametrizability of the word equation xyz = zvw : A short proof. Theoret. Comput. Sci. 345 (2005) 296–303. CrossRef
Fine, N.J. and Wilf, H.S., Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965) 109–114. CrossRef
Guibas, L.J. and Odlyzko, A.M., Periods in strings. J. Comb. Theory A 30 (1981) 19–42. CrossRef
Halava, V., Harju, T. and Ilie, L., Periods and binary words. J. Comb. Theory A 89 (2000) 298–303. CrossRef
Harju, T. and Nowotka, D., The equation xi = yjzk in a free semigroup. Semigroup Forum 68 (2004) 488–490. CrossRef
Hmelevskii, J.I., Equations in free semigroups. Proceedings of the Steklov Institute of Mathematics 107 (1971) 1–270 (American Mathematical Society, Providence, RI (1976)).
P. Leupold, Partial words: results and perspectives. GRLMC, Tarragona (2003).
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA (1983). Cambridge University Press, Cambridge (1997).
M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002).
M. Lothaire, Applied Combinatorics on Words. Cambridge University Press, Cambridge (2005).
Lyndon, R.C. and Schützenberger, M.P., The equation am = bncp in a free group. Michigan Math. J. 9 (1962) 289–298.
Makanin, G.S., The problem of solvability of equations in a free semigroup. Math. USSR Sbornik 32 (1977) 129–198. CrossRef
A.A. Markov, The theory of algorithms. Trudy Mat. Inst. Steklov 42 (1954).
Păun, G., Santean, N., Thierrin, G. and On, S. Yu the robustness of primitive words. Discrete Appl. Math. 117 (2002) 239–252. CrossRef
W. Plandowski, Satisfiability of word equations with constants is in NEXPTIME. Proceedings of the Annual ACM Symposium on Theory of Computing (1999) 721–725.
W. Plandowski, Satisfiability of word equations with constants is in PSPACE. Proceedings of the 40th Annual Symposium on Foundations of Computer Science (1999) 495–500.
Rivals, E. and Rahmann, S., Combinatorics of periods in strings. J. Comb. Theory A 104 (2003) 95–113. CrossRef
H.J. Shyr, Free Monoids and Languages. Hon Min Book Company, Taichung, Taiwan (1991).
Shyr, H.J. and Thierrin, G., Disjunctive languages and codes. Lect. Notes Comput. Sci. 56 (1977) 171–176. CrossRef