Avoidable Sets and Well Quasi-Orders

  • Flavio D’Alessandro
  • Stefano Varricchio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3340)


Let I be a finite set of words and \(\Rightarrow_{I}^{*}\) be the derivation relation generated by the set of productions {εu | uI }. Let L I ε be the set of words u such that \(\epsilon {\Rightarrow_{I}^{*}}\). We prove that the set I is unavoidable if and only if the relation \(\Rightarrow_{I}^{*}\) is a well quasi-order on the set L I ε . This result generalizes a theorem of [7]. Further generalizations are investigated.


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  1. 1.
    Bovet, D.P., Varricchio, S.: On the regularity of languages on a binary alphabet generated by copying systems. Information Processing Letters 44, 119–123 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D’Alessandro, F., Varricchio, S.: On Well Quasi-orders On Languages. In: Ésik, Z., Fülöp, Z. (eds.) DLT 2003. LNCS, vol. 2710, pp. 230–241. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    D’Alessandro, F., Varricchio, S.: Well quasi-orders and context-free grammars, Theoretical Computer Science (to appear)Google Scholar
  4. 4.
    de Luca, A., Varricchio, S.: Some regularity conditions based on well quasi-orders. LNCS, vol. 583, pp. 356–371. Springer-Verlag, Berlin (1992)Google Scholar
  5. 5.
    de Luca, A., Varricchio, S.: Well quasi-orders and regular languages. Acta Informatica 31, 539–557 (1994)CrossRefMathSciNetGoogle Scholar
  6. 6.
    de Luca, A., Varricchio, S.: Finiteness and regularity in semigroups and formal languages. In: EATCS Monographs on Theoretical Computer Science. Springer, Berlin (1999)Google Scholar
  7. 7.
    Ehrenfeucht, A., Haussler, D., Rozenberg, G.: On regularity of context-free languages. Theoretical Computer Science 27, 311–332 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Harju, T., Ilie, L.: On well quasi orders of words and the confluence property. Theoretical Computer Science 200, 205–224 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Haussler, D.: Another generalization of Higman’s well quasi-order result on Σ*. Discrete Mathematics 57, 237–243 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Higman, G.H.: Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 3, 326–336 (1952)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Ilie, L., Salomaa, A.: On well quasi orders of free monoids. Theoretical Computer Science 204, 131–152 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Intrigila, B., Varricchio, S.: On the generalization of Higman and Kruskal’s theorems to regular languages and rational trees. Acta Informatica 36, 817–835 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ito, M., Kari, L., Thierrin, G.: Shuffle and scattered deletion closure of languages. Theoretical Computer Science 245(1), 115–133 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jantzen, M.: Extending regular expressions with iterated shuffle. Theoretical Computer Science 38, 223–247 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kruskal, J.: The theory of well-quasi-ordering: a frequently discovered concept. J. Combin. Theory, Ser. A 13, 297–305 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Puel, L.: Using unavoidable sets of trees to generalize Kruskal’s theorem. J. Symbolic Comput. 8(4), 335–382 (1989)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Flavio D’Alessandro
    • 1
  • Stefano Varricchio
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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