The Unambiguity of Segmented Morphisms

  • Dominik D. Freydenberger
  • Daniel Reidenbach
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4588)


A segmented morphism \(\sigma_n: \Delta^* \longrightarrow \{ {\ensuremath{\mathtt{a}}}, {\ensuremath{\mathtt{b}}} \}^*\), n ∈ ℕ, maps each symbol in Δ onto a word which consists of n distinct subwords in \({\ensuremath{\mathtt{a}}} {\ensuremath{\mathtt{b}}}^+ {\ensuremath{\mathtt{a}}}\). In the present paper, we examine the impact of n on the unambiguity of σ n with respect to any α ∈ Δ  + , i. e. the question of whether there does not exist a morphism τ satisfying τ(α) = σ n (α) and, for some symbol x in α, τ(x) ≠ σ n (x). To this end, we consider the set U(σ n ) of those α ∈ Δ  +  with respect to which σ n is unambiguous, and we comprehensively describe its relation to any U(σ m ), m ≠ n. Our paper thus contributes fundamental (and, in parts, fairly counter-intuitive) results to the recently initiated research on the ambiguity of morphisms.


Formal Language Mathematical Linguistics Neighbourhood Class Injective Morphism Adjacency Graph 
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  1. 1.
    Baker, K.A., McNulty, G.F., Taylor, W.: Growth problems for avoidable words. Theor. Comput. Sci. 69, 319–345 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cassaigne, J.: Unavoidable patterns. In: Lothaire, M. (ed.) Algebraic Combinatorics on Words, pp. 111–134. Cambridge Mathematical Library (2002)Google Scholar
  3. 3.
    Choffrut, C., Karhumäki, J.: Combinatorics of words. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 329–438. Springer, Heidelberg (1997)Google Scholar
  4. 4.
    Ehrenfeucht, A., Rozenberg, G.: Finding a homomorphism between two words is NP-complete. Inform. Process. Lett. 9, 86–88 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Freydenberger, D.D., Reidenbach, D., Schneider, J.C.: Unambiguous morphic images of strings. Int. J. Found. Comput. Sci. 17, 601–628 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Halava, V., Harju, T., Karhumäki, J., Latteux, M.: Post Correspondence Problem for morphisms with unique blocks. In: Proc. Words 2005. Publications du LACIM, vol. 36, pp. 265–274 (2005)Google Scholar
  7. 7.
    Hamm, D., Shallit, J.: Characterization of finite and one-sided infinite fixed points of morphisms on free monoids. Technical Report CS-99-17, Dep. of Computer Science, University of Waterloo (1999)Google Scholar
  8. 8.
    Jiang, T., Salomaa, A., Salomaa, K., Yu, S.: Decision problems for patterns. J. Comput. Syst. Sci. 50, 53–63 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Mateescu, A., Salomaa, A.: Patterns. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 230–242. Springer, Heidelberg (1997)Google Scholar
  10. 10.
    Reidenbach, D.: A discontinuity in pattern inference. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 129–140. Springer, Heidelberg (2004)Google Scholar
  11. 11.
    Reidenbach, D.: A non-learnable class of E-pattern languages. Theor. Comput. Sci. 350, 91–102 (2006)zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Dominik D. Freydenberger
    • 1
  • Daniel Reidenbach
    • 2
  1. 1.Research Group on Mathematical Linguistics, URV, Tarragona, Spain, Institut für Informatik, J. W. Goethe-Universität, Postfach 111932, D-60054 Frankfurt am MainGermany
  2. 2.Fachbereich Informatik, Technische Universität Kaiserslautern, KaiserslauternGermany

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