Abstract
We summarize results on the complexity of regular(-like) expressions and tour a fragment of the literature. In particular we focus on the descriptional complexity of the conversion of regular expressions to equivalent finite automata and vice versa, to the computational complexity of problems on regular-like expressions such as, e.g., membership, inequivalence, and non-emptiness of complement, and finally on the operation problem measuring the required size for transforming expressions with additional language operations (built-in or not) into equivalent ordinary regular expressions.
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References
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)
Aho, A.V., Sethi, R., Ullman, J.D.: Compilers: Principles, Techniques, and Tools. Addison-Wesley, Reading (1986)
Allauzen, C., Mohri, M.: A unified construction of the Glushkov, follow, and Antimirov automata. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 110–121. Springer, Heidelberg (2006)
Alon, N., Dewdney, A.K., Ott, T.J.: Efficient simulation of finite automta by neural nets. J. ACM 38, 495–514 (1991)
Antimirov, V.M.: Partial derivatives of regular expressions and finite automaton constructions. Theoret. Comput. Sci. 155, 291–319 (1996)
Berry, G., Sethi, R.: From regular expressions to deterministic automata. Theoret. Comput. Sci. 48, 117–126 (1986)
Bille, P., Thorup, M.: Regular expression matching with multi-strings and intervals. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 1297–1308. SIAM, Philadelphia (2010)
Brüggemann-Klein, A.: Regular expressions into finite automata. Theoret. Comput. Sci. 120, 197–213 (1993)
Brzozowski, J.A., McCluskey, E.J.: Signal flow graph techniques for sequential circuit state diagrams. IEEE Trans. Comput. C-12, 67–76 (1963)
Brzozowski, J.A.: Derivatives of regular expressions. J. ACM 11, 481–494 (1964)
Champarnaud, J.M., Ouardi, F., Ziadi, D.: Normalized expressions and finite automata. Int. J. Algebra Comput. 17, 141–154 (2007)
Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, Boca Raton (1971)
Eggan, L.C.: Transition graphs and the star height of regular events. Michigan Math. J. 10, 385–397 (1963)
Ehrenfeucht, A., Zeiger, H.P.: Complexity measures for regular expressions. J. Comput. System Sci. 12, 134–146 (1976)
Ellul, K., Krawetz, B., Shallit, J., Wang, M.W.: Regular expressions: New results and open problems. J. Autom., Lang. Comb. 10, 407–437 (2005)
Fürer, M.: The complexity of the inequivalence problem for regular expressions with intersection. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 234–245. Springer, Heidelberg (1980)
Geffert, V.: Translation of binary regular expressions into nondeterministic ε-free automata with O(nlogn) transitions. J. Comput. System Sci. 66, 451–472
Gelade, W.: Succinctness of regular expressions with interleaving, intersection and counting. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 363–374. Springer, Heidelberg (2008)
Gelade, W.: Foundations of XML: Regular Expressions Revisited. PhD thesis, School voor Informatietechnologie, University of Hasselt, Belgium, and University of Maastricht, the Netherlands (2009)
Gelade, W., Neven, F.: Succinctness of the complement and intersection of regular expressions. In: Theoretical Aspects of Computer Science (STACS 2008), Schloss Dagstuhl, Germany. Dagstuhl Seminar Proceedings, IBFI, vol. 08001, pp. 325–336 (2008)
Ginzburg, A.: A procedure for checking equality of regular expressions. J. ACM 14, 355–362 (1967)
Glushkov, V.M.: The abstract theory of automata. Russian Math. Surveys 16, 1–53 (1961)
Gruber, H.: On the Descriptional and Algorithmic Complexity of Regular Languages. PhD thesis, Institut für Informatik, Universität Giessen, Germany (2010)
Gruber, H., Gulan, S.: Simplifying regular expressions. A quantitative perspective. In: Language and Automata Theory and Applications (LATA 2010), LNCS. Springer, Heidelberg (to appear 2010)
Gruber, H., Holzer, M.: Finite automata, digraph connectivity, and regular expression size. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 39–50. Springer, Heidelberg (2008)
Gruber, H., Holzer, M.: Provably shorter regular expressions from deterministic finite automata. In: Ito, M., Toyama, M. (eds.) DLT 2008. LNCS, vol. 5257, pp. 383–395. Springer, Heidelberg (2008)
Gruber, H., Holzer, M.: Language operations with regular expressions of polynomial size. Theoret. Comput. Sci. 410, 3281–3289 (2009)
Gruber, H., Holzer, M.: Tight bounds on the descriptional complexity of regular expressions. In: DLT 2009. LNCS, vol. 5583, pp. 276–287. Springer, Heidelberg (2009)
Gruber, H., Johannsen, J.: Optimal lower bounds on regular expression size using communication complexity. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 273–286. Springer, Heidelberg (2008)
Gulan, S., Fernau, H.: An optimal construction of finite automata from regular expressions. In: Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2008), Dagstuhl Seminar Proceedings, IBFI, Schloss Dagstuhl, Germany, vol. 08002, pp. 211–222 (2008)
Hagenah, C., Muscholl, A.: Computing epsilon-free NFA from regular expressions in O(n log2(n)) time. RAIRO Inform. Théor. 34, 257–278 (2000)
Hashiguchi, K.: Algorithms for determining relative star height and star height. Inform. Comput. 78, 124–169 (1988)
Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)
Horne, B.G., Hush, D.R.: Bounds on the complexity of recurrent neural network implementations of finite state machines. Neural Networks 9, 243–252 (1996)
Hromkovič, J., Seibert, S., Wilke, T.: Translating regular expressions into small ε-free nondeterministic finite automata. J. Comput. System Sci. 62, 565–588 (2001)
Hromkovič, J.: Descriptional complexity of finite automata: Concepts and open problems. J. Autom., Lang. Comb. 7, 519–531 (2002)
Hunt III, H.B.: The equivalence problem for regular expressions with intersections is not polynomial in tape. Technical Report TR 73-161, Department of Computer Science, Cornell University, Ithaca, New York (1973)
Ilie, L., Yu, S.: Follow automata. Inform. Comput. 186, 140–162 (2003)
Indyk, P.: Optimal simulation of automata by neural nets. In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 337–348. Springer, Heidelberg (1995)
Jiang, T., Ravikumar, B.: A note on the space complexity of some decision problems for finite automata. Inform. Process. Lett. 40, 25–31 (1991)
Kilpeläinen, P., Tuhkanen, R.: Regular expressions with numerical occurrence indicators – Preliminary results. In: Symposium on Programming Languages and Software Tools, Department of Computer Science, pp. 163–173. University of Kuopio, Finland (2003)
Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Automata Studies, pp. 3–42. Princeton University Press, Princeton (1956)
Leiss, E.L.: The complexity of restricted regular expressions and the synthesis problem for finite automata. J. Comput. System Sci. 23, 348–354 (1981)
Lifshits, Y.: A lower bound on the size of ε-free NFA corresponding to a regular expression. Inform. Process. Lett. 85, 293–299 (2003)
McCulloch, W.S., Pitts, W.H.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophysics 5, 115–133 (1943)
McNaughton, R.: The loop complexity of pure-group events. Inform. Control 11, 167–176 (1967)
McNaughton, R.: The loop complexity of regular events. Inform. Sci. 1, 305–328 (1969)
McNaughton, R., Yamada, H.: Regular expressions and state graphs for automata. IRE Trans. Elect. Comput. EC-9, 39–47 (1960)
Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: Symposium on Switching and Automata Theory (SWAT 1972), pp. 125–129. IEEE, Los Alamitos (1972)
Minsky, M.L.: Computation: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs (1967)
Petersen, H.: Decision problems for generalized regular expressions. In: Descriptional Complexity of Automata, Grammars and Related Structures (DCAGRS 2000), London, Ontario, pp. 22–29 (2000)
Petersen, H.: The membership problem for regular expressions with intersection is complete in LOGCFL. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 513–522. Springer, Heidelberg (2002)
Rangel, J.L.: The equivalence problem for regular expressions over one letter is elementary. In: Symposium on Switching and Automata Theory (SWAT 1974), pp. 24–27. IEEE, Los Alamitos (1974)
Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)
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)
Stockmeyer, L.J.: The Complexity of Decision Problems in Automata Theory and Logic. PhD thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts (1974)
Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time. In: Symposium on Theory of Computing (STOC 1973), pp. 1–9. ACM Press, New York (1973)
Thompson, K.: Regular expression search algorithm. J. ACM 11, 419–422 (1968)
Watson, B.: A taxonomy of finite automata construction algorithms. Technical Report 93/43, Eindhoven University of Technology, Department of Mathematics and Computing Science (1995)
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Holzer, M., Kutrib, M. (2010). The Complexity of Regular(-Like) Expressions. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds) Developments in Language Theory. DLT 2010. Lecture Notes in Computer Science, vol 6224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14455-4_3
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DOI: https://doi.org/10.1007/978-3-642-14455-4_3
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