Advertisement

Partial Derivatives of Regular Expressions over Alphabet-Invariant and User-Defined Labels

  • Stavros KonstantinidisEmail author
  • Nelma Moreira
  • João Pires
  • Rogério Reis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11601)

Abstract

We are interested in regular expressions that represent word relations in an alphabet-invariant way—for example, the set of all word pairs u, v where v is a prefix of u independently of what the alphabet is. This is the second part of a recent paper on this topic which focused on labelled graphs (transducers and automata) with alphabet-invariant and user-defined labels. In this paper we study derivatives of regular expressions over labels (atomic objects) in some set B. These labels can be any strings as long as the strings represent subsets of a certain monoid. We show that one can define partial derivative labelled graphs of type B expressions, whose transition labels can be elements of another label set X as long as X and B refer to the same monoid. We also show how to use derivatives directly to decide whether a given word pair is in the relation of a regular expression over pairing specs. Set specs and pairing specs are useful label sets allowing one to express languages and relations over large alphabets in a natural and compact way.

Keywords

Alphabet-invariant expressions Regular expressions Partial derivatives Algorithms Monoids 

Notes

Acknowledgement

We are grateful to the reviewers of CIAA 2019 for their constructive suggestions for improvement. We have applied most of these suggestions, and we plan to apply the remaining ones in the journal version where more pages are allowed. The idea of using special labels on automata to denote sets is also explored in [13] with different objectives.

References

  1. 1.
    Antimirov, V.M.: Partial derivatives of regular expressions and finite automaton constructions. Theoret. Comput. Sci. 155(2), 291–319 (1996)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bastos, R., Broda, S., Machiavelo, A., Moreira, N., Reis, R.: On the average complexity of partial derivative automata for semi-extended expressions. J. Autom. Lang. Comb. 22(1–3), 5–28 (2017).  https://doi.org/10.25596/jalc-2017-005MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Broda, S., Machiavelo, A., Moreira, N., Reis, R.: On the average state complexity of partial derivative automata: an analytic combinatorics approach. Internat. J. Found. Comput. Sci. 22(7), 1593–1606 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brzozowski, J.: Derivatives of regular expressions. J. ACM 11, 481–494 (1964)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Caron, P., Champarnaud, J.-M., Mignot, L.: Partial derivatives of an extended regular expression. In: Dediu, A.-H., Inenaga, S., Martín-Vide, C. (eds.) LATA 2011. LNCS, vol. 6638, pp. 179–191. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-21254-3_13CrossRefzbMATHGoogle Scholar
  6. 6.
    Champarnaud, J.M., Ziadi, D.: From Mirkin’s prebases to Antimirov’s word partial derivatives. Fund. Inform. 45(3), 195–205 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Champarnaud, J.M., Ziadi, D.: Canonical derivatives, partial derivatives and finite automaton constructions. Theoret. Comput. Sci. 289, 137–163 (2002).  https://doi.org/10.1016/S0304-3975(01)00267-5MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Demaille, A.: Derived-term automata of multitape expressions with composition. Sci. Ann. Comput. Sci. 27(2), 137–176 (2017).  https://doi.org/10.7561/SACS.2017.2.137MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    FAdo: Tools for formal languages manipulation. http://fado.dcc.fc.up.pt/. Accessed Mar 2019
  10. 10.
    Konstantinidis, S., Moreira, N., Reis, R., Young, J.: Regular expressions and transducers over alphabet-invariant and user-defined labels. CoRR abs/1805.01829 (2018). http://arxiv.org/abs/1805.01829CrossRefGoogle Scholar
  11. 11.
    Lombardy, S., Sakarovitch, J.: Derivatives of rational expressions with multiplicity. Theoret. Comput. Sci. 332(1–3), 141–177 (2005).  https://doi.org/10.1016/j.tcs.2004.10.016MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mirkin, B.G.: An algorithm for constructing a base in a language of regular expressions. Eng. Cybern. 5, 51–57 (1966)Google Scholar
  13. 13.
    Newton, J.: Representing and computing with types in dynamically typed languages. Ph.D. thesis, Sorbonne Université, Paris, France, November 2018Google Scholar
  14. 14.
    Pires, J.: Transducers and 2D regular expressions. Master’s thesis, Departamento de Ciência de Computadores, Faculdade de Ciências da Universidade do Porto, Porto, Portugal (2018)Google Scholar
  15. 15.
    Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press, Berlin (2009)CrossRefGoogle Scholar
  16. 16.
    Sakarovitch, J.: Automata and rational expressions. CoRR abs/1502.03573 (2015). http://arxiv.org/abs/1502.03573

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Stavros Konstantinidis
    • 1
    Email author
  • Nelma Moreira
    • 2
  • João Pires
    • 2
  • Rogério Reis
    • 2
  1. 1.Saint Mary’s UniversityHalifaxCanada
  2. 2.CMUP and DCCFaculdade de Ciências da Universidade do PortoPortoPortugal

Personalised recommendations