Abstract
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.
The first and third authors were partially supported by the MIUR grants ex-60% 2006 and 2007 Università degli Studi di Salerno.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alur, R., Madhusudan, P.: Visibly pushdown languages. In: STOC, pp. 202–211 (2004)
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)
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)
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)
Carayol, A., Meyer, A.: Context-sensitive languages, rational graphs and determinism. Logical Methods in Computer Science 2(2) (2006)
Carayol, A., Meyer, A.: Linearly bounded infinite graphs. Acta Inf. 43(4), 265–292 (2006)
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)
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)
Chandra, A.K., Kozen, D., Stockmeyer, L.J.: Alternation. J. ACM 28(1), 114–133 (1981)
Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Springer, Heidelberg (1995)
Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)
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)
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)
Meyer, A.: Traces of term-automatic graphs. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 489–500. Springer, Heidelberg (2007)
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)
Muller, D.E., Schupp, P.E.: The theory of ends, pushdown automata, and second-order logic. Theor. Comput. Sci. 37, 51–75 (1985)
Post, E.L.: Formal reductions of the general combinatorial decision problem. American Journal of Mathematics 65(2), 197–215 (1943)
Rispal, C.: The synchronized graphs trace the context-sensitive languages. Electr. Notes Theor. Comput. Sci. 68(6) (2002)
Slutzki, G.: Alternating Tree Automata. Theor. Comput. Sci. 41, 305–318 (1985)
Thomas, W.: Languages, automata, and logic. Handbook of formal languages 3, 389–455 (1997)
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)
Thue, A.: Probleme über veränderungen von zeichenreihen nach gegebener regeln. Kra. Vidensk. Selsk. Skrifter. 1. Mat. Nat. Kl. 10 (1914)
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)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
La Torre, S., Madhusudan, P., Parlato, G. (2008). An Infinite Automaton Characterization of Double Exponential Time. In: Kaminski, M., Martini, S. (eds) Computer Science Logic. CSL 2008. Lecture Notes in Computer Science, vol 5213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87531-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-87531-4_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87530-7
Online ISBN: 978-3-540-87531-4
eBook Packages: Computer ScienceComputer Science (R0)