Derivatives and Finite Automata of Expressions in Star Normal Form

  • Haiming ChenEmail author
  • Ping Lu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)


This paper studies derivatives and automata for expressions in star normal form as defined by Brüggemann-Klein. For an expression in star normal form, the paper shows that the derivatives are either \(\emptyset \) or unique, while in general Berry and Sethi’s result shows the derivatives are either \(\emptyset \) or similar. It is known that the partial derivative automaton and the follow automaton are two small automata, each of which is a quotient of the position automaton. For the relation between the partial derivative and follow automata, however, Ilie and Yu stated that a rigorous analysis is necessary but difficult. The paper tackles the issue, and presents several results. Our work shows that there are different conditions under which the relation of the two automata can be different.


Regular expressions Finite automata Derivatives Partial derivatives Star normal form 


  1. 1.
    Antimirov, V.: Partial derivatives of regular expressions and finite automaton constructions. Theoret. Comput. Sci. 155, 291–319 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berry, G., Sethi, R.: From regular expressions to deterministic automata. Theoret. Comput. Sci. 48, 117–126 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brüggemann-Klein, A.: Regular expressions into finite automata. Theoret. Comput. Sci. 120, 197–213 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brüggemann-Klein, A., Wood, D.: One-unambiguous regular languages. Inf. Comput. 142(2), 182–206 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brzozowski, J.A.: Derivatives of regular expressions. J. ACM (JACM) 11(4), 481–494 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Champarnaud, J.-M., Ouardi, F., Ziadi, D.: Normalized expressions and finite automata. Int. J. Algebra Comput. 17(01), 141–154 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Champarnaud, J.-M., Ziadi, D.: Canonical derivatives, partial derivatives and finite automaton constructions. Theoret. Comput. Sci. 289(1), 137–163 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Champarnaud, J.-M., Nicart, F., Ziadi, D.: Computing the follow automaton of an expression. In: Domaratzki, M., Okhotin, A., Salomaa, K., Yu, S. (eds.) CIAA 2004. LNCS, vol. 3317, pp. 90–101. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-30500-2_9 CrossRefGoogle Scholar
  9. 9.
    Chia-Hsiang, C., Paige, R.: From regular expressions to DFA’s using compressed NFA’s. Theoret. Comput. Sci. 178(1), 1–36 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Glushkov, V.M.: The abstract theory of automata. Russ. Math. Surv. 16(5), 1 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ilie, L., Yu, S.: Follow automata. Inf. Comput. 186, 146–162 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    McNaughton, R., Yamada, H.: Regular expressions and state graphs for automata. IEEE Trans. Electron. Comput. 1(EC–9), 39–47 (1960)CrossRefzbMATHGoogle Scholar
  13. 13.
    Mirkin, B.G.: An algorithm for constructing a base in a language of regular expressions. Eng. Cybern. 5, 110–116 (1966)Google Scholar
  14. 14.
    Ponty, J.-L., Ziadi, D., Champarnaud, J.-M.: A new quadratic algorithm to convert a regular expression into an automaton. In: Raymond, D., Wood, D., Yu, S. (eds.) WIA 1996. LNCS, vol. 1260, pp. 109–119. Springer, Heidelberg (1997). doi: 10.1007/3-540-63174-7_9 CrossRefGoogle Scholar
  15. 15.
    Lombardy, S., Sakarovitch, J.: Derivatives of rational expressions with multiplicity. Theoret. Comput. Sci. 332(1–3), 141–177 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 41–110. Springer, Berlin (1997)CrossRefGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.State Key Laboratory of Computer ScienceInstitute of Software Chinese Academy of SciencesBeijingChina
  2. 2.BDBCBeihang UniversityBeijingChina

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