Finite Automata, Digraph Connectivity, and Regular Expression Size
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 . 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.
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- 2.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)Google Scholar
- 3.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)Google Scholar
- 8.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)Google Scholar
- 9.Gruber, H., Holzer, M.: Finite automata, digraph connectivity and regular expression size. Technical report, Technische Universität München (December 2007)Google Scholar
- 10.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)Google Scholar
- 16.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)Google Scholar
- 23.Pinsker, M.S.: On the complexity of a concentrator. In: Annual Teletraffic Conference, pp. 318/1–318/4 (1973)Google Scholar