The Inclusion Problem for Regular Expressions

  • Dag Hovland
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6031)


This paper presents a new polynomial-time algorithm for the inclusion problem for certain pairs of regular expressions. The algorithm is not based on construction of finite automata, and can therefore be faster than the lower bound implied by the Myhill-Nerode theorem. The algorithm automatically discards unnecessary parts of the right-hand expression. In these cases the right-hand expression might even be 1-ambiguous. For example, if r is a regular expression such that any DFA recognizing r is very large, the algorithm can still, in time independent of r, decide that the language of ab is included in that of (a + r)b. The algorithm is based on a syntax-directed inference system. It takes arbitrary regular expressions as input, and if the 1-ambiguity of the right-hand expression becomes a problem, the algorithm will report this.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Book, R., Even, S., Greibach, S., Ott, G.: Ambiguity in graphs and expressions. IEEE Transactions on Computers c-20(2), 149–153 (1971)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brüggemann-Klein, A.: Regular expressions into finite automata. Theoretical Computer Science 120(2), 197–213 (1993)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brüggemann-Klein, A., Wood, D.: One-unambiguous regular languages. Information and Computation 140(2), 229–253 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brzozowski, J.A.: Derivatives of regular expressions. J. ACM 11(4), 481–494 (1964)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, H., Chen, L.: Inclusion test algorithms for one-unambiguous regular expressions. In: Fitzgerald, J.S., Haxthausen, A.E., Yenigün, H. (eds.) ICTAC 2008. LNCS, vol. 5160, pp. 96–110. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Glushkov, V.M.: The abstract theory of automata. Russian Mathematical Surveys 16(5), 1–53 (1961)CrossRefGoogle Scholar
  7. 7.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)MATHGoogle Scholar
  8. 8.
    Hosoya, H., Vouillon, J., Pierce, B.C.: Regular expression types for XML. ACM Trans. Program. Lang. Syst. 27(1), 46–90 (2005)CrossRefGoogle Scholar
  9. 9.
    McNaughton, R., Yamada, H.: Regular expressions and state graphs for automata. IRE Transactions on Electronic Computers 9, 39–47 (1960)CrossRefGoogle Scholar
  10. 10.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: Proceedings of FOCS, pp. 125–129. IEEE, Los Alamitos (1972)Google Scholar
  11. 11.
    Nerode, A.: Linear automaton transformations. Proceedings of the American Mathematical Society 9(4), 541–544 (1958)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Salomaa, A.: Two complete axiom systems for the algebra of regular events. J. ACM 13(1), 158–169 (1966)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Dag Hovland
    • 1
  1. 1.Institutt for InformatikkUniversitetet i BergenNorway

Personalised recommendations