Abstract
Classically, several properties and relations of words, such as “being a power of a same word”, can be expressed by using word equations. This paper is devoted to study in general the expressive power of word equations. As main results we prove theorems which allow us to show that certain properties of words are not expressible as components of solutions of word equations. In particular, “the primitiveness” and “the equal length” are such properties, as well as being “any word over a proper subalphabet”.
Supported by Academy of Finland under grant 14047.
Supported partially by the grant KBN 8T11C01208. On leave from Instytut Informatyki, Uniwersytet Warszawski, Banacha 2, 02-097 Warszawa, Poland.
Supported by Academy of Finland under grant 14047.
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References
Albert, M.H., and Lawrence, J., A proof of Ehrenfeucht's Conjecture, Theoret. Comput. Sci. 41, 121–123, 1985.
Angluin D., Finding pattern common to a set of strings, in Proceedings of STOC'79, 130–141, 1979.
Berstel, J., and Perrin D., Theory of Codes, Academic Press, 1985.
Büchi, R. and Senger, S., Coding in the existential theory of concatenation, Arch. Math. Logik, 26, 101–106, 1986/87.
Bulitko, V.K., Equations and inequalities in a free group and a free semigroup, Tul. Gos. Ped. Inst. Ucen. Zap. Mat. Kafedr. Geometr. i Algebra, 2, 242–252, 1970 (in Russian).
Choffrut, C., and Karhumäki, J., Combinatorics of words, in G.Rozenberg and A.Salomaa (eds), Handbook of Formal Languages, Springer, 1997.
Culik II, K., and Karhumäki, J., Systems of equations and Ehrenfeucht's conjecture, Discr. Math., 43, 139–153, 1983.
Grigorieff, S., Personal comunication.
Guba, V., The equivalence of infinite systems of equations in free groups and free semigroups to their finite subsystems, Matem.Zametki, 40 (3), September 1986 (in Russian).
Harrison, M.A., Introduction to Formal Language Theory, Addison-Wesley Publishing Company, 1978.
Jiang T., Salomaa A., Salomaa K., Yu S., Decision problems for patterns, J. Cornput. Sys. Sciences 50, 53–63, 1995.
Khmelevski, Yu. I., Solution of word equations in three variables, Dokl.Akad.Nauk. SSSR, 177, 1023–1025, 1967 (in Russian).
Khmelevski, Yu. I., Equations in free semigroups, Trudy Mat. Inst. Steklov, 107, 1971 (English translation: Proc. Steklov Inst. of Mathematics 107 (1971), American Mathematical Society, 1976.)
Koscielski, A., and Pacholski, L., Complexity of Makanin's algorithm, J. ACM 43(4), 670–684, 1996.
Lentin, A., Equations dans des Monoides Libres, Gouthiers-Villars, 1972.
Lothaire, M., Combinatorics on Words, Addison-Wesley, 1993.
Makanin, G.S., The problem of solvability of equations in a free semigroup, Mat. Sb., Vol. 103,(145), 147–233, 1977. English transi, in Math. U.S.S.R. Sb. Vol 32, 1977.
Matijasevich, Y., Enumerable sets are diophantine, Soviet. Math. Doklady 11, 354–357, 1970. English transl, in Dokl. Akad. Nauk SSSR 191, 279–282, 1971.
Seibert, S., Quantifier hierarchies and word relations, Springer LNCS 626, 329–338 (1992).
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Karhumäki, J., Plandowski, W., Mignosi, F. (1997). The expressibility of languages and relations by word equations. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds) Automata, Languages and Programming. ICALP 1997. Lecture Notes in Computer Science, vol 1256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63165-8_168
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DOI: https://doi.org/10.1007/3-540-63165-8_168
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