Abstract
In 1977 Makanin stated that the solvability problem for word equation systems is decidable ([10]). Makanin’s algorithm is very complicated and the solvability problem for word equations remains NP-hard ([1]). We show that testing solvability of word equation systems is a NP-complete problem if we look for solutions of length bounded by some given constant greater than or equal to two over some single letter alphabet. Up to this moment several evolutionary strategies have been proposed for other NP-complete problems, like 3-SAT, with a remarkable success. Following this direction we introduce here an evolutionary local search algorithm for solving word equation systems provided that some upper bound for the length of the solutions is given. We present some empirical results derived from our algorithm which indicate that our approach to this problem becomes a promising strategy. Our experimental results also certify that our local optimization technique clearly outperforms a simple genetic approach.
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References
Angluin, D.: Finding patterns common to a set of strings. J. C. S. S. 21(1), 46–62 (1980)
Eiben, A., van der Hauw, J.: Solving 3-SAT with adaptive Genetic Algorithms. In: 4th IEEE Conference on Evolutionary Computation, pp. 81–86. IEEE Press, Los Alamitos (1997)
Goldbert, D.E. (ed.): Genetic Algorithms in Search Optimization & Machine Learning. Addison Wesley Longmann, Inn., Reading (1989)
Gottlieb, J., Marchiori, E., Rossi, C.: Evolutionary Algorithms for the Satisfiability Problem. Evolutionary Computation 10(1) (2002)
Gutiérrez, C.: Solving Equations in Strings: On Makanin’s Algorithm. In: Lucchesi, C.L., Moura, A.V. (eds.) LATIN 1998. LNCS, vol. 1380, pp. 358–373. Springer, Heidelberg (1998)
Hmlevskiĭ, J.L.: Equations in Free Semigroups. Trudy Mat. Inst. Stelov 107 (1971)
Karhumaki, J., Mignosi, F., Plandowski, W.: The expressibility of languages and relations by word equations. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 98–109. Springer, Heidelberg (1997)
Koscielski, A., Pacholski, L.: Complexity of Makanin’s algorithm. J. ACM 43(4), 670–684 (1996)
Lentin, A.: Equations in Free Monoids. In: Nivat, M. (ed.) Automata Languages and Programming, pp. 67–85. North Holland, Amsterdam (1972)
Makanin, G.S.: The Problem of Solvability of Equations in a Free Semigroup. Math. USSR Sbornik 32(2), 129–198 (1977)
Plandowski, W., Rytter, W.: Application of Lempel-Ziv encodings to the Solution of Words Equations. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 731–742. Springer, Heidelberg (1998)
Plotkin, G.D.: Building-in Equational Theories. Mach. Int. 7, 73–90 (1972)
Robson, J.M., Diekert, V.: On quadratic Word Equations. In: Meinel, C., Tison, S., et al. (eds.) STACS 1999. LNCS, vol. 1563, pp. 217–226. Springer, Heidelberg (1999)
Siekmann, J.: A Modification of Robinson’s Unification Procedure. M. Sc. Thesis (1972)
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Alonso, C.L., Drubi, F., Montaña, J.L. (2004). An Evolutionary Algorithm for Solving Word Equation Systems. In: Conejo, R., Urretavizcaya, M., Pérez-de-la-Cruz, JL. (eds) Current Topics in Artificial Intelligence. TTIA 2003. Lecture Notes in Computer Science(), vol 3040. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25945-9_15
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DOI: https://doi.org/10.1007/978-3-540-25945-9_15
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