Abstract
Recently lower bounds on the minimum required size for the conversion of deterministic finite automata into regular expressions and on the required size of regular expressions resulting from applying some basic language operations on them, were given by Gelade and Neven [8]. We strengthen and extend these results, obtaining lower bounds that are in part optimal, and, notably, the presented examples are over a binary alphabet, which is best possible. To this end, we develop a different, more versatile lower bound technique that is based on the star height of regular languages. It is known that for a restricted class of regular languages, the star height can be determined from the digraph underlying the transition structure of the minimal finite automaton accepting that language. In this way, star height is tied to cycle rank, a structural complexity measure for digraphs proposed by Eggan and Büchi, which measures the degree of connectivity of directed graphs.
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
Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S.: Dag-width and parity games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 524–536. Springer, Heidelberg (2006)
Berwanger, D., Grädel, E.: Entanglement—A measure for the complexity of directed graphs with applications to logic and games. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 209–223. Springer, Heidelberg (2005)
Björklund, A., Husfeldt, T., Khanna, S.: Approximating longest directed paths and cycles. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 222–233. Springer, Heidelberg (2004)
Book, R.V., Chandra, A.K.: Inherently nonplanar automata. Acta Informatica 6, 89–94 (1976)
Eggan, L.C.: Transition graphs and the star height of regular events. Michigan Mathematical Journal 10, 385–397 (1963)
Ehrenfeucht, A., Zeiger, H.P.: Complexity measures for regular expressions. Journal of Computer and System Sciences 12(2), 134–146 (1976)
Ellul, K., Krawetz, B., Shallit, J., Wang, M.: Regular expressions: New results and open problems. Journal of Automata, Languages and Combinatorics 10(4), 407–437 (2005)
Gelade, W., Neven, F.: Succinctness of the complement and intersection of regular expressions. In: Albers, S., Weil, P. (eds.) Symposium on Theoretical Aspects of Computer Science. Dagstuhl Seminar Proceedings, vol. 08001, pp. 325–336. IBFI (2008)
Gruber, H., Holzer, M.: Finite automata, digraph connectivity and regular expression size. Technical report, Technische Universität München (December 2007)
Gruber, H., Johannsen, J.: Optimal lower bounds on regular expression size using communication complexity. In: Amadio, R. (ed.) Foundations of Software Science and Computation Structures. LNCS, vol. 4962, pp. 273–286. Springer, Heidelberg (2008)
Hashiguchi, K.: Algorithms for determining relative star height and star height. Information and Computation 78(2), 124–169 (1988)
Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)
Ilie, L., Yu, S.: Follow automata. Information and Computation 186(1), 140–162 (2003)
Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. Journal of Combinatorial Theory, Series B 82(1), 138–154 (2001)
Kirsten, D.: Distance desert automata and the star height problem. RAIRO – Theoretical Informatics and Applications 39(3), 455–509 (2005)
Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies. Annals of Mathematics Studies, pp. 3–42. Princeton University Press, Princeton (1956)
Mayer, A.J., Stockmeyer, L.J.: Word problems – This time with interleaving. Information and Computation 115(2), 293–311 (1994)
McNaughton, R.: The loop complexity of pure-group events. Information and Control 11(1/2), 167–176 (1967)
McNaughton, R.: The loop complexity of regular events. Information Sciences 1, 305–328 (1969)
Morris, P.H., Gray, R.A., Filman, R.E.: Goto removal based on regular expressions. Journal of Software Maintenance 9(1), 47–66 (1997)
Nešetřil, J., de Mendez, P.O.: Tree-depth, subgraph coloring and homomorphism bounds. European Journal of Combinatorics 27(6), 1022–1041 (2006)
Obdržálek, J.: Dag-width: Connectivity measure for directed graphs. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 814–821. ACM Press, New York (2006)
Pinsker, M.S.: On the complexity of a concentrator. In: Annual Teletraffic Conference, pp. 318/1–318/4 (1973)
Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms 7(3), 309–322 (1986)
Sakarovitch, J.: The language, the expression, and the (small) automaton. In: Farré, J., Litovsky, I., Schmitz, S. (eds.) CIAA 2005. LNCS, vol. 3845, pp. 15–30. Springer, Heidelberg (2006)
Schnitger, G.: Regular expressions and NFAs without ε-transitions. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 432–443. Springer, Heidelberg (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gruber, H., Holzer, M. (2008). Finite Automata, Digraph Connectivity, and Regular Expression Size. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70583-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-70583-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70582-6
Online ISBN: 978-3-540-70583-3
eBook Packages: Computer ScienceComputer Science (R0)