An Approach to Computing Downward Closures

  • Georg ZetzscheEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9135)


The downward closure of a word language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of any language is regular. While the downward closure appears to be a powerful abstraction, algorithms for computing a finite automaton for the downward closure of a given language have been established only for few language classes.

This work presents a simple general method for computing downward closures. For language classes that are closed under rational transductions, it is shown that the computation of downward closures can be reduced to checking a certain unboundedness property.

This result is used to prove that downward closures are computable for (i) every language class with effectively semilinear Parikh images that are closed under rational transductions, (ii) matrix languages, and (iii) indexed languages (equivalently, languages accepted by higher-order pushdown automata of order 2).


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  1. 1.
    Abdulla, P.A., Boasson, L., Bouajjani, A.: Effective lossy queue languages. In: Proc. of ICALP 2001, pp. 639–651 (2001)Google Scholar
  2. 2.
    Abdulla, P.A., Collomb-Annichini, A., Bouajjani, A., Jonsson, B.: Using Forward Reachability Analysis for Verification of Lossy Channel Sys- tems. Form. Method. Syst. Des. 25(1), 39–65 (2004)zbMATHCrossRefGoogle Scholar
  3. 3.
    Aho, A.V.: Indexed grammars-an extension of context-free grammars. J. ACM 15(4), 647–671 (1968)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bonnet, R., Finkel, A., Leroux, J., Zeitoun, M.: Model Checking Vector Addition Systems with one zero-test. In: LMCS 8.2:11 (2012)Google Scholar
  5. 5.
    Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of push- down automata: application to model-checking. In: Proc. of CONCUR 1997, pp. 135–150 (1997)Google Scholar
  6. 6.
    Colcombet, T.: Regular cost functions, Part I: logic and algebra over words. In: LMCS 9.3 (2013)Google Scholar
  7. 7.
    Courcelle, B.: On constructing obstruction sets of words. Bulletin of the EATCS 44, 178–186 (1991)zbMATHGoogle Scholar
  8. 8.
    Czerwiński, W., Martens, W.: A Note on Decidable Separability by Piece- wise Testable Languages (2014). arXiv:1410.1042 [cs.FL]
  9. 9.
    Dassow, J., Păun, G.: Regulated rewriting in formal language theory. Springer-Verlag, Berlin (1989)CrossRefGoogle Scholar
  10. 10.
    Dassow, J., Păun, G., Salomaa, A.: Grammars with controlled derivations. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 2, pp. 101–154. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  11. 11.
    Ehrenfeucht, A., Rozenberg, G., Skyum, S.: A relationship between ET0L and EDT0L languages. Theor. Comput. Sci. 1(4), 325–330 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gilman, R.H.: A shrinking lemma for indexed languages. Theor. Comput. Sci. 163(1-2), 277–281 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Gruber, H., Holzer, M., Kutrib, M.: The size of Higman-Haines sets. Theor. Comput. Sci. 387(2), 167–176 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Habermehl, P., Meyer, R., Wimmel, H.: The downward-closure of petri net languages. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 466–477. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  15. 15.
    Haines, L.H.: On free monoids partially ordered by embedding. J. Combin. Theory 6(1), 94–98 (1969)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Hayashi, T.: On Derivation Trees of Indexed Grammars-An Extension of the uvwxy-Theorem–. Publications of the Research Institute for Mathematical Sciences 9(1), 61–92 (1973)zbMATHCrossRefGoogle Scholar
  17. 17.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979) zbMATHGoogle Scholar
  18. 18.
    Jantzen, M.: On the hierarchy of Petri net languages. RAIRO Theor. Inf. Appl. 13(1), 19–30 (1979)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Jullien, P.: Contribution à létude des types d’ordres dispersés. Université de Marseille, PhD thesis (1969)Google Scholar
  20. 20.
    Kartzow, A.: A pumping lemma for collapsible pushdown graphs of level 2. In: Proc. of CSL 2011, pp. 322–336 (2011)Google Scholar
  21. 21.
    van Leeuwen, J.: Effective constructions in well-partially-ordered free monoids. Discrete Math. 21(3), 237–252 (1978)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Maslov, A.N.: Multilevel stack automata. Problems of Information Transmission 12(1), 38–42 (1976)Google Scholar
  23. 23.
    Mayr, R.: Undecidable problems in unreliable computations. Theor. Comput. Sci. 297(1-3), 337–354 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Parys, P.: A pumping lemma for pushdown graphs of any level. In: Proc. of STACS 2012, pp. 54–65 (2012)Google Scholar
  25. 25.
    Rounds, W.C.: Tree-oriented proofs of some theorems on context-free and indexed languages. In: Proc. of STOC 1970, pp. 109–116 (1970)Google Scholar
  26. 26.
    Seki, H., Matsumura, T., Fujii, M., Kasami, T.: On multiple context-free grammars. Theor. Comput. Sci. 88(2), 191–229 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Smith, T.: On infinite words determined by indexed languages. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014, Part I. LNCS, vol. 8634, pp. 511–522. Springer, Heidelberg (2014) Google Scholar
  28. 28.
    Zetzsche, G.: An approach to computing downward closures (2015). arXiv:1503. 01068 [cs.FL]Google Scholar
  29. 29.
    Zetzsche, G.: Computing downward closures for stacked counter au tomata. In: Proc. of STACS 2015, pp. 743–756 (2015)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Technische Universität Kaiserslautern, Fachbereich Informatik, Concurrency Theory GroupKaiserslauternGermany

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