One-Rule Length-Preserving Rewrite Systems and Rational Transductions
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 48 (2014) no. 2, pp. 149-171.

We address the problem to know whether the relation induced by a one-rule length-preserving rewrite system is rational. We partially answer to a conjecture of Éric Lilin who conjectured in 1991 that a one-rule length-preserving rewrite system is a rational transduction if and only if the left-hand side u and the right-hand side v of the rule of the system are not quasi-conjugate or are equal, that means if u and v are distinct, there do not exist words x, y and z such that u = xyz and v = zyx. We prove the only if part of this conjecture and identify two non trivial cases where the if part is satisfied.

DOI: 10.1051/ita/2013044
Classification: 68Q45, 68Q42, 68R15
Keywords: string rewriting - rationality
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Latteux, Michel; Roos, Yves. One-Rule Length-Preserving Rewrite Systems and Rational Transductions. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 48 (2014) no. 2, pp. 149-171. doi : 10.1051/ita/2013044. http://www.numdam.org/articles/10.1051/ita/2013044/

[1] J. Berstel, Transductions and Context-Free Languages. Teubner Verlag (1979). | MR | Zbl

[2] M. Clerbout and Y. Roos, Semi-commutations and algebraic languages, in STACS 90, in vol. 415, edited by Christian Choffrut and Thomas Lengauer. Lect. Notes Comput. Sci. Springer Berlin/Heidelberg (1990) 82-94. | MR | Zbl

[3] N. Dershowitz, Open. closed. open, in vol. 3467 of Lect. Notes Comput. Sci. RTA, edited by J. Giesl. Springer (2005) 376-393. | MR | Zbl

[4] S. Eilenberg and B. Tilson, Automata, languages and machines, vol. B, Pure Appl. Math. Academic Press, New York, San Francisco, London (1976). | MR | Zbl

[5] A. Geser, Termination of string rewriting rules that have one pair of overlaps, in vol. 2706, Lect. Notes Comput. Sci. RTA, edited by R. Nieuwenhuis. Springer (2003) 410-423. | MR | Zbl

[6] A. Geser, D. Hofbauer, and J. Waldmann, Match-bounded string rewriting systems. Appl. Algebra Eng. Commun. Comput. 15 (2004) 149-171. | MR | Zbl

[7] W. Kurth, Termination und Konfluenz von Semi-Thue-Systemen mit nur einer Regel. Ph.D. thesis, Technische Universitt Clausthal (1990). | Zbl

[8] M. Latteux and Y. Roos, The image of a word by a one-rule semi-Thue system is not always context-free (2011). http://www.lifl.fr/ỹroos/al/one-rule-context-free.pdf.

[9] É. Lilin, Une généralisation des semi-commutations. Technical Report IT-210, Laboratoire d'Informatique Fondamentale de Lille, Université de Lille 1, France (1991). In french.

[10] Y. Métivier, Calcul de longueurs de chaînes de reé´criture dans le monoïde libre. Theor. Comput. Sci. 35 (1985) 71-87. | MR | Zbl

[11] R. Milner, The spectra of words, in vol. 3838 of Processes, Terms and Cycles: Steps on the Road to Infinity, edited by A. Middeldorp, V. van Oostrom, F. van Raamsdonk and R. de Vrijer. Lect. Notes Comput. Sci. Springer Berlin/Heidelberg (2005) 1-5. | Zbl

[12] B. Ravikumar, Peg-solitaire, string rewriting systems and finite automata. Theor. Comput. Sci. 321 (2004) 383-394. | MR | Zbl

[13] J. Sakarovitch, Elements of Automata Theory. Cambridge University Press, New York, USA (2009). | MR | Zbl

[14] J. Sakarovitch and I. Simon, Subwords. In Combinatorics on words. Cambridge Mathematical Library. Cambridge University Press (1997) 105-142.

[15] A. Terlutte and D. Simplot, Iteration of rational transductions. RAIRO: ITA 34 (2000) 99-130. | Numdam | MR | Zbl

[16] C. Wrathall, Confluence of one-rule thue systems, Word Equations and Related Topics, in vol. 572 of Lect. Notes Comput. Sci., edited by K. Schulz. Springer Berlin/Heidelberg (1992) 237-246. | MR

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