Match-Bounded String Rewriting Systems

  • Alfons Geser
  • Dieter Hofbauer
  • Johannes Waldmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2747)


We investigate rewriting systems on strings by annotating letters with natural numbers, so called match heights. A position in a reduct will get height h+1 if the minimal height of all positions in the redex is h. In a match-bounded system, match heights are globally bounded. Exploiting recent results on deleting systems, we prove that it is decidable whether a given rewriting system has a given match bound. Further, we show that match-bounded systems preserve regularity of languages. Our main focus, however, is on termination of rewriting. Match-bounded systems are shown to be linearly terminating, and–more interestingly–for inverses of match-bounded systems, termination is decidable. These results provide new techniques for automated proofs of termination.


Regular Language Cryptographic Protocol Minimal Height Uniform Termination Note Comp 
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  1. 1.
    Berstel, J.: Transductions and Context-Free Languages. Teubner, Stuttgart (1979)zbMATHGoogle Scholar
  2. 2.
    Book, R.V., Otto, F.: String-Rewriting Systems. Texts and Monographs in Computer Science. Springer, New York (1993)zbMATHGoogle Scholar
  3. 3.
    Coquand, T., Persson, H.: A proof-theoretical investigation of Zantema’s problem. In: Nielsen, M., Thomas, W. (eds.) CSL 1997. LNCS, vol. 1414, pp. 177–188. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  4. 4.
    Dershowitz, N., Hoot, C.: Topics in termination. In: Kirchner, C. (ed.) RTA 1993. LNCS, vol. 690, pp. 198–212. Springer, Heidelberg (1993)Google Scholar
  5. 5.
    Ferreira, M.C.F., Zantema, H.: Dummy elimination: Making termination easier. In: Reichel, H. (ed.) FCT 1995. LNCS, vol. 965, pp. 243–252. Springer, Heidelberg (1995)Google Scholar
  6. 6.
    Genet, T., Klay, F.: Rewriting for Cryptographic Protocol Verification. In: McAllester, D.A. (ed.) CADE 2000. LNCS, vol. 1831, pp. 271–290. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Geser, A., Hofbauer, D., Waldmann, J.: Match-bounded string rewriting systems and automated termination proofs. In: 6th Int. Workshop on Termination WST-03, Valencia, Spain (2003) Google Scholar
  8. 8.
    Ginsburg, S., Greibach, S.A.: Mappings which preserve context sensitive languages. Inform. and Control 9(6), 563–582 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hibbard, T.N.: Context-limited grammars. J. ACM 21(3), 446–453 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hofbauer, D., Waldmann, J.: Deleting string rewriting systems preserve regularity. In: Proc. 7th Int. Conf. Developments in Language Theory DLT-03. Lect. Notes Comp. Sci., Springer, Heidelberg (2003) (to appear)Google Scholar
  11. 11.
    Kobayashi, Y., Katsura, M., Shikishima-Tsuji, K.: Termination and derivational complexity of confluent one-rule string-rewriting systems. Theoret. Comput. Sci. 262(1-2), 583–632 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kurth, W.: Termination und Konfluenz von Semi-Thue-Systemen mit nur einer Regel. Dissertation, Technische Universität Clausthal, Germany (1990) Google Scholar
  13. 13.
    McNaughton, R.: Semi-Thue systems with an inhibitor. J. Automat. Reason. 26, 409–431 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Moczydłowski Jr., W.: Jednoregułowe systemy przepisywania slów. Masters thesis, Warsaw University, Poland (2002) Google Scholar
  15. 15.
    Moczydłowski Jr., W., Geser, A.: Termination of single-threaded one-rule Semi-Thue systems. Technical Report TR 02-08 (273), Warsaw University (December 2002), Available at
  16. 16.
    Moore, C., Eppstein, D.: One-dimensional peg solitaire, and duotaire. In: Nowakowski, R.J. (ed.) More Games of No Chance. Cambridge Univ. Press, Cambridge (2003)Google Scholar
  17. 17.
    Ravikumar, B.: Peg-solitaire, string rewriting systems and finite automata. In: Leong, H.-W., Imai, H., Jain, S. (eds.) ISAAC 1997. LNCS, vol. 1350, pp. 233–242. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  18. 18.
    The RTA list of open problems,
  19. 19.
    Sénizergues, G.: On the termination problem for one-rule semi-Thue systems. In: Ganzinger, H. (ed.) RTA 1996. LNCS, vol. 1103, pp. 302–316. Springer, Heidelberg (1996)Google Scholar
  20. 20.
    Tahhan Bittar, E.: Complexité linéaire du problème de Zantema. C. R. Acad. Sci. Paris Sér. I Inform. Théor., t. 323, 1201–1206 (1996)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Waldmann, J.: Rewrite games. In: Tison, S. (ed.) Proc. 13th Int. Conf. Rewriting Techniques and Applications RTA-02. Lect. Notes Comp. Sci., vol. 2378, pp. 144–158. Springer, Heidelberg (2002)Google Scholar
  22. 22.
    Zantema, H., Geser, A.: A complete characterization of termination of 0p1q→1r0s. Appl. Algebra Engrg. Comm. Comput. 11(1), 1–25 (2000)zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Alfons Geser
    • 1
  • Dieter Hofbauer
    • 2
  • Johannes Waldmann
    • 3
  1. 1.National Institute of AerospaceHamptonUSA
  2. 2.Fachbereich Mathematik/InformatikUniversität KasselKasselGermany
  3. 3.Fakultät für Mathematik und InformatikUniversität LeipzigLeipzigGermany

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