Regularity and Context-Freeness over Word Rewriting Systems

  • Didier Caucal
  • Trong Hieu Dinh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6604)


We describe a general decomposition mechanism to express the derivation relation of a word rewriting system R as the composition of a (regular) substitution followed by the derivation relation of a system R′ ∪ D, where R′ is a strict sub-system of R and D is the Dyck rewriting system. From this decomposition, we deduce that the system R (resp. R − 1) preserves regular (resp. context-free) languages whenever R′ ∪ D (resp. its inverse) does. From this we can deduce regularity and context-freeness preservation properties for a generalization of tagged bifix systems.


  1. 1.
    Altenbernd, J.: On bifix systems and generalizations. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 40–51. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Altenbernd, J.: Reachability over word rewriting systems. Ph.D. Thesis, RWTH Aachen, Germany (2009)Google Scholar
  3. 3.
    Benois, M.: Parties rationnelles du groupe libre. C.R. Académie des Sciences, Série A 269, 1188–1190 (1969)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Berstel, J.: Transductions and context-free languages. Teubner, Stuttgart (1979)CrossRefGoogle Scholar
  5. 5.
    Book, R., Jantzen, M., Wrathall, C.: Monadic thue systems. Theoretical Computer Science 19, 231–251 (1982)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Book, R., Otto, F.: String-rewriting systems. Texts and Monographs in Computer Science. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  7. 7.
    Bouajjani, A., Müller-Olm, M., Touili, T.: Regular symbolic analysis of dynamic networks of pushdown systems. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 473–487. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Büchi, R.: Regular canonical systems. Archiv für Mathematische Logik und Grundlagenforschung 6, 91–111 (1964)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Caucal, D.: On the regular structure of prefix rewriting. Theoretical Computer Science 106, 61–86 (1992); originally published In: Arnold, A. (ed.) CAAP 1990. LNCS, vol. 431, pp. 61–86. Springer, Heidelberg (1990)Google Scholar
  10. 10.
    Endrullis, J., Hofbauer, D., Waldmann, J.: Decomposing terminating rewrite relations. In: Geser, A., Sondergaard, H. (eds.) Proc. 8th WST, pp. 39–43 (2006),, Computing Research Repository
  11. 11.
    Geser, A., Hofbauer, D., Waldmann, J.: Match-bounded string rewriting systems. Applicable Algebra in Engineering, Communication and Computing 15, 149–171 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hibbard, T.: Context-limited grammars. JACM 21(3), 446–453 (1974)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hofbauer, D., Waldmann, J.: Deleting string rewriting systems preserve regularity. Theoretical Computer Science 327, 301–317 (2004); originally published In: Ésik, Z., Fülöp, Z. (eds.) DLT 2003. LNCS, vol. 2710, pp. 301–317. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Jantzen, M., Kudlek, M., Lange, K.J., Petersen, H.: Dyck1-reductions of context-free languages. In: Budach, L., Bakharajev, R., Lipanov, O. (eds.) FCT 1987. LNCS, vol. 278, pp. 218–227. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  15. 15.
    Karhumäki, J., Kunc, M., Okhotin, A.: Computing by commuting. Theoretical Computer Science 356, 200–211 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Didier Caucal
    • 1
  • Trong Hieu Dinh
    • 1
  1. 1.LIGM, UMR CNRS 8049Université Paris-EstMarne-la-ValleFrance

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