Abstract
We prove connections between one-unknown word equations and three-unknown constant-free word equations, and use them to prove that the number of equations in an independent system of three-unknown constant-free equations is at most logarithmic with respect to the length of the shortest equation in the system. We also study two well-known conjectures. The first conjecture claims that there is a constant c such that every one-unknown equation has either infinitely many solutions or at most c. The second conjecture claims that there is a constant c such that every independent system of three-unknown constant-free equations with a nonperiodic solution is of size at most c. We prove that the first conjecture implies the second one, possibly for a different constant.
This work has been supported by the DFG Heisenberg grant 590179 (Dirk Nowotka), the DFG research grant 614256 and the Vilho, Yrjö and Kalle Väisälä Foundation (Aleksi Saarela).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Albert, M.H., Lawrence, J.: A proof of Ehrenfeucht’s conjecture. Theoret. Comput. Sci. 41(1), 121–123 (1985)
Budkina, L.G., Markov, A.A.: \(F\)-semigroups with three generators. Mat. Zametki 14, 267–277 (1973)
Culik, K., Karhumäki, J.: Systems of equations over a free monoid and Ehrenfeucht’s conjecture. Discrete Math. 43(2–3), 139–153 (1983)
Da̧browski, R., Plandowski, W.: On word equations in one variable. Algorithmica 60(4), 819–828 (2011)
Eyono Obono, S., Goralcik, P., Maksimenko, M.: Efficient solving of the word equations in one variable. In: Privara, I., Ružička, P., Rovan, B. (eds.) MFCS 1994. LNCS, vol. 841, pp. 336–341. Springer, Heidelberg (1994)
Guba, V.S.: Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems. Mat. Zametki 40(3), 321–324 (1986)
Harju, T., Nowotka, D.: On the independence of equations in three variables. Theoret. Comput. Sci. 307(1), 139–172 (2003)
Holub, Š., Žemlička, J.: Algebraic properties of word equations. J. Algebra 434, 283–301 (2015)
Jez, A.: One-variable word equations in linear time. Algorithmica 74(1), 1–48 (2016)
Karhumäki, J., Plandowski, W.: On the defect effect of many identities in free semigroups. In: Paun, G. (ed.) Mathematical Aspects of Natural and Formal Languages, pp. 225–232. World Scientific (1994)
Karhumäki, J., Saarela, A.: On maximal chains of systems of word equations. Proc. Steklov Inst. Math. 274, 116–123 (2011)
Laine, M., Plandowski, W.: Word equations with one unknown. Int. J. Found. Comput. Sci. 22(2), 345–375 (2011)
Saarela, A.: On the complexity of Hmelevskii’s theorem and satisfiability of three unknown equations. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 443–453. Springer, Heidelberg (2009)
Saarela, A.: Systems of word equations, polynomials and linear algebra: a new approach. Eur. J. Comb. 47, 1–14 (2015)
Spehner, J.C.: Quelques problémes d’extension, de conjugaison et de présentation des sous-monoïdes d’un monoïde libre. Ph.D. thesis, Univ. Paris (1976)
Spehner, J.C.: Les systemes entiers d’équations sur un alphabet de 3 variables. In: Semigroups, pp. 342–357 (1986)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nowotka, D., Saarela, A. (2016). One-Unknown Word Equations and Three-Unknown Constant-Free Word Equations. In: Brlek, S., Reutenauer, C. (eds) Developments in Language Theory. DLT 2016. Lecture Notes in Computer Science(), vol 9840. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53132-7_27
Download citation
DOI: https://doi.org/10.1007/978-3-662-53132-7_27
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-53131-0
Online ISBN: 978-3-662-53132-7
eBook Packages: Computer ScienceComputer Science (R0)