An Infinite Automaton Characterization of Double Exponential Time

  • Salvatore La Torre
  • P. Madhusudan
  • Gennaro Parlato
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5213)


Infinite-state automata are a new invention: they are automata that have an infinite number of states represented by words, transitions defined using rewriting, and with sets of initial and final states. Infinite-state automata have gained recent interest due to a remarkable result by Morvan and Stirling, which shows that automata with transitions defined using rational rewriting precisely capture context-sensitive (NLinSpace) languages. In this paper, we show that infinite automata defined using a form of multi-stack rewriting precisely defines double exponential time (more precisely, 2ETime, the class of problems solvable in \(2^{2^{O(n)}}\) time). The salient aspect of this characterization is that the automata have no ostensible limits on time nor space, and neither direction of containment with respect to 2ETime is obvious. In this sense, the result captures the complexity class qualitatively, by restricting the power of rewriting.


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  1. 1.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: STOC, pp. 202–211 (2004)Google Scholar
  2. 2.
    Ball, T., Rajamani, S.K.: Bebop: A symbolic model checker for boolean programs. In: Havelund, K., Penix, J., Visser, W. (eds.) SPIN 2000. LNCS, vol. 1885, pp. 113–130. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Bouajjani, A., Habermehl, P., Vojnar, T.: Abstract regular model checking. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 372–386. Springer, Heidelberg (2004)Google Scholar
  4. 4.
    Carayol, A., Wöhrle, S.: The Caucal hierarchy of infinite graphs in terms of logic and higher-order pushdown automata. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 112–123. Springer, Heidelberg (2003)Google Scholar
  5. 5.
    Carayol, A., Meyer, A.: Context-sensitive languages, rational graphs and determinism. Logical Methods in Computer Science 2(2) (2006)Google Scholar
  6. 6.
    Carayol, A., Meyer, A.: Linearly bounded infinite graphs. Acta Inf. 43(4), 265–292 (2006)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Caucal, D.: On infinite transition graphs having a decidable monadic theory. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 194–205. Springer, Heidelberg (1996)Google Scholar
  8. 8.
    Caucal, D., Knapik, T.: A Chomsky-like hierarchy of infinite graphs. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 177–187. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Chandra, A.K., Kozen, D., Stockmeyer, L.J.: Alternation. J. ACM 28(1), 114–133 (1981)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Springer, Heidelberg (1995)MATHGoogle Scholar
  11. 11.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)MATHGoogle Scholar
  12. 12.
    La Torre, S., Madhusudan, P., Parlato, G.: A robust class of context-sensitive languages. In: LICS, pp. 161–170. IEEE Computer Society, Los Alamitos (2007)Google Scholar
  13. 13.
    La Torre, S., Madhusudan, P., Parlato, G.: Context-bounded analysis of queue systems. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 299–314. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Meyer, A.: Traces of term-automatic graphs. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 489–500. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Morvan, C., Stirling, C.: Rational graphs trace context-sensitive languages. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 548–559. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Muller, D.E., Schupp, P.E.: The theory of ends, pushdown automata, and second-order logic. Theor. Comput. Sci. 37, 51–75 (1985)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Post, E.L.: Formal reductions of the general combinatorial decision problem. American Journal of Mathematics 65(2), 197–215 (1943)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Rispal, C.: The synchronized graphs trace the context-sensitive languages. Electr. Notes Theor. Comput. Sci. 68(6) (2002)Google Scholar
  19. 19.
    Slutzki, G.: Alternating Tree Automata. Theor. Comput. Sci. 41, 305–318 (1985)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Thomas, W.: Languages, automata, and logic. Handbook of formal languages 3, 389–455 (1997)Google Scholar
  21. 21.
    Thomas, W.: A short introduction to infinite automata. In: Kuich, W., Rozenberg, G., Salomaa, A. (eds.) DLT 2001. LNCS, vol. 2295, pp. 130–144. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  22. 22.
    Thue, A.: Probleme über veränderungen von zeichenreihen nach gegebener regeln. Kra. Vidensk. Selsk. Skrifter. 1. Mat. Nat. Kl. 10 (1914)Google Scholar
  23. 23.
    Vardi, M.: Reasoning about The Past with Two-Way Automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Salvatore La Torre
    • 1
  • P. Madhusudan
    • 2
  • Gennaro Parlato
    • 1
    • 2
  1. 1.Università di SalernoItaly
  2. 2.University of IllinoisUrbana-ChampaignUSA

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