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Further Closure Properties of Input-Driven Pushdown Automata

  • Alexander Okhotin
  • Kai Salomaa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10952)

Abstract

The paper investigates the closure of the language family defined by input-driven pushdown automata (IDPDA) under the following operations: insertion \(ins(L, K)=\{xyz \mid xz \in L, \, y \in K\}\), deletion \(del(L, K)=\{xz \mid xyz \in L, \, y \in K\}\), square root \(\sqrt{L}=\{w \mid ww \in L\}\), and the first half \(\frac{1}{2}L=\{u \mid \exists v: |u|=|v|, \, uv \in L\}\). For K and L recognized by nondeterministic IDPDA, with m and with n states, respectively, insertion requires \(mn+2m\) states, as long as K is well-nested; deletion is representable with 2n states, for well-nested K; square root requires \(n^3-O(n^2)\) states, for well-nested L; the well-nested subset of the first half is representable with \(2^{O(n^2)}\) states. Without the well-nestedness constraints, non-closure is established in each case.

References

  1. 1.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: ACM Symposium on Theory of Computing (STOC 2004), Chicago, USA, 13–16 June 2004, pp. 202–211 (2004).  https://doi.org/10.1145/1007352.1007390
  2. 2.
    Alur, R., Madhusudan, P.: Adding nesting structure to words. J. ACM 56(3) (2009).  https://doi.org/10.1145/1516512.1516518MathSciNetCrossRefGoogle Scholar
  3. 3.
    von Braunmühl, B., Verbeek, R.: Input driven languages are recognized in \(\log n\) space. Ann. Discret. Math. 24, 1–20 (1985).  https://doi.org/10.1016/S0304-0208(08)73072-XMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Domaratzki, M.: State complexity of proportional removals. J. Autom. Lang. Comb. 7(4), 455–468 (2002)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Esparza, J., Ganty, P., Kiefer, S., Luttenberger, M.: Parikh’s theorem: a simple and direct automaton construction. Inf. Process. Lett. 111(12), 614–619 (2011).  https://doi.org/10.1016/j.ipl.2011.03.019MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Goč, D., Palioudakis, A., Salomaa, K.: Nondeterministic state complexity of proportional removals. Int. J. Found. Comput. Sci. 25(7), 823–836 (2014).  https://doi.org/10.1142/S0129054114400103MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Haines, L.H.: On free monoids partially ordered by embedding. J. Comb. Theory 6, 94–98 (1969).  https://doi.org/10.1016/S0021-9800(69)80111-0MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Han, Y.-S., Ko, S.-K., Ng, T., Salomaa, K.: State complexity of insertion. Int. J. Found. Comput. Sci. 27(7), 863–878 (2016).  https://doi.org/10.1142/S0129054116500349MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Han, Y.-S., Ko, S.-K., Salomaa, K.: State complexity of deletion and bipolar deletion. Acta Informatica 53(1), 67–85 (2016).  https://doi.org/10.1007/s00236-015-0245-yMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Han, Y.-S., Salomaa, K.: Nondeterministic state complexity of nested word automata. Theor. Comput. Sci. 410, 2961–2971 (2009).  https://doi.org/10.1016/j.tcs.2009.01.004MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Higman, G.: Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. s3–2(1), 326–336 (1952)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gruber, H., Holzer, M., Kutrib, M.: More on the size of Higman-Haines sets: effective constructions. Fundamenta Informaticae 91(1), 105–121 (2009).  https://doi.org/10.3233/FI-2009-0035MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Karandikar, P., Niewerth, M., Schnoebelen, Ph.: On the state complexity of closures and interiors of regular languages with subwords and superwords. Theor. Comput. Sci. 610(A), 91–107 (2016).  https://doi.org/10.1016/j.tcs.2015.09.028MathSciNetCrossRefGoogle Scholar
  14. 14.
    van Leeuven, J.: Effective construction in well-partially-ordered free monoids. Discrete Math. 21(3), 237–252 (1978).  https://doi.org/10.1016/0012-365X(78)90156-5MathSciNetCrossRefGoogle Scholar
  15. 15.
    Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Math. Dokl. 11, 1373–1375 (1970)zbMATHGoogle Scholar
  16. 16.
    Mehlhorn, K.: Pebbling mountain ranges and its application to DCFL-recognition. In: de Bakker, J., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 422–435. Springer, Heidelberg (1980).  https://doi.org/10.1007/3-540-10003-2_89CrossRefGoogle Scholar
  17. 17.
    Okhotin, A.: On the state complexity of scattered substrings and superstrings. Fundamenta Informaticae 99(3), 325–338 (2010).  https://doi.org/10.3233/FI-2010-252MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Okhotin, A.: Input-driven languages are linear conjunctive. Theor. Comput. Sci. 618, 52–71 (2016).  https://doi.org/10.1016/j.tcs.2016.01.007MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Okhotin, A., Salomaa, K.: Complexity of input-driven pushdown automata. SIGACT News 45(2), 47–67 (2014).  https://doi.org/10.1145/2636805.2636821MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Okhotin, A., Salomaa, K.: Descriptional complexity of unambiguous input-driven pushdown automata. Theor. Comput. Sci. 566, 1–11 (2015).  https://doi.org/10.1016/j.tcs.2014.11.015MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Okhotin, A., Salomaa, K.: State complexity of operations on input-driven pushdown automata. J. Comput. Syst. Sci. 86, 207–228 (2017).  https://doi.org/10.1016/j.jcss.2017.02.001MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Okhotin, A., Salomaa, K.: Edit distance neighbourhoods of input-driven pushdown automata. In: Weil, P. (ed.) CSR 2017. LNCS, vol. 10304, pp. 260–272. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-58747-9_23CrossRefGoogle Scholar
  23. 23.
    Okhotin, A., Salomaa, K.: The quotient operation on input-driven pushdown automata. In: Pighizzini, G., Câmpeanu, C. (eds.) DCFS 2017. LNCS, vol. 10316, pp. 299–310. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-60252-3_24CrossRefGoogle Scholar
  24. 24.
    Piao, X., Salomaa, K.: Operational state complexity of nested word automata. Theor. Comput. Sci. 410, 3290–3302 (2009).  https://doi.org/10.1016/j.tcs.2009.05.002MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Salomaa, K.: Limitations of lower bound methods for deterministic nested word automata. Inf. Comput. 209, 580–589 (2011).  https://doi.org/10.1016/j.ic.2010.11.021MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Seiferas, J.I., McNaughton, R.: Regularity-preserving relations. Theor. Comput. Sci. 2(2), 147–154 (1976).  https://doi.org/10.1016/0304-3975(76)90030-XMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2018

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySaint PetersburgRussia
  2. 2.School of ComputingQueen’s UniversityKingstonCanada

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