More on the Size of Higman-Haines Sets: Effective Constructions

  • Hermann Gruber
  • Markus Holzer
  • Martin Kutrib
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4664)


A not well-known result [9, Theorem 4.4] in formal language theory is that the Higman-Haines sets for any language are regular, but it is easily seen that these sets cannot be effectively computed in general. Here the Higman-Haines sets are the languages of all scattered subwords of a given language and the sets of all words that contain some word of a given language as a scattered subword. Recently, the exact level of unsolvability of Higman-Haines sets was studied in [10]. We focus on language families whose Higman-Haines sets are effectively constructible. In particular, we study the size of Higman-Haines sets for the lower classes of the Chomsky hierarchy, namely for the families of regular, linear context-free, and context-free languages, and prove upper and lower bounds on the size of these sets.


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  1. 1.
    Buntrock, G., Otto, F.: Growing context-sensitive languages and Church-Rosser languages. Inform. Comput. 141, 1–36 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Dassow, J., Păun, G.: Regulated Rewriting in Formal Language Theory. Springer, Berlin (1989)Google Scholar
  3. 3.
    Ehrenfeucht, A., Haussler, D., Rozenberg, G.: On regularity of context-free languages. Theoret. Comput. Sci. 27, 311–332 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fernau, H., Stephan, F.: Characterizations of recursively enumerable sets by programmed grammars with unconditional transfer. J. Autom., Lang. Comb. 4, 117–152 (1999)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Gilman, R.H.: A shrinking lemma for indexed languages. Theoret. Comput. Sci. 163, 277–281 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. Inform. Process. Lett. 59, 75–77 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gruber, H., Holzer, M.: Results on the average state and transition complexity of finite automata. Descriptional Complexity of Formal Systems (DCFS 2006), University of New Mexico, Technical Report NMSU-CS-2006-001, pp. 267–275 (2006)Google Scholar
  8. 8.
    Haines, L.H.: On free monoids partially ordered by embedding. J. Combinatorial Theory 6, 94–98 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Higman, G.: Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2, 326–336 (1952)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gruber, H., Holzer, M., Kutrib, M.: The size of Higman-Haines sets. Theoret. Comput. Sci. (to appear)Google Scholar
  11. 11.
    Ilie, L.: Decision problems on orders of words. Ph.D. thesis, Department of Mathematics, University of Turku, Finland (1998)Google Scholar
  12. 12.
    Kruskal, J.B.: The theory of well-quasi-ordering: A frequently discovered concept. J. Combinatorial Theory 13, 297–305 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    van Leeuwen, J.: A regularity condition for parallel rewriting systems. SIGACT News 8, 24–27 (1976)CrossRefGoogle Scholar
  14. 14.
    van Leeuwen, J.: Effective constructions in well-partially-ordered free monoids. Discrete Mathematics 21, 237–252 (1978)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Hermann Gruber
    • 1
  • Markus Holzer
    • 2
  • Martin Kutrib
    • 3
  1. 1.Institut für Informatik, Ludwig-Maximilians-Universität München, Oettingenstraße 67, D-80538 MünchenGermany
  2. 2.Institut für Informatik, Technische Universität München, Boltzmannstraße 3, D-85748 Garching bei MünchenGermany
  3. 3.Institut für Informatik, Universität Giessen, Arndtstraße 2, D-35392 GiessenGermany

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