Document Open Access Logo

Optimal Regular Expressions for Permutations (Track B: Automata, Logic, Semantics, and Theory of Programming)

Authors Antonio Molina Lovett , Jeffrey Shallit

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2019.121.pdf
  • Filesize: 435 kB
  • 12 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • regular expressions
  • lower bounds
  • divide-and-conquer

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

The permutation language P_n consists of all words that are permutations of a fixed alphabet of size n. Using divide-and-conquer, we construct a regular expression R_n that specifies P_n. We then give explicit bounds for the length of R_n, which we find to be 4^{n}n^{-(lg n)/4+Theta(1)}, and use these bounds to show that R_n has minimum size over all regular expressions specifying P_n.

Cite As Get BibTex

Antonio Molina Lovett and Jeffrey Shallit. Optimal Regular Expressions for Permutations (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 121:1-121:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.121

Author Details

Antonio Molina Lovett
  • University of Waterloo, Canada
Jeffrey Shallit
  • University of Waterloo, Canada

References

  1. Rakesh Agrawal, Giuseppe Psaila, Edward L. Wimmers, and Mohamed Zait. Querying Shapes of Histories. Technical Report RJ 9962 (87921), IBM Almaden Research Center, June 26 1995. Google Scholar
  2. P. R. J. Asveld. Generating all permutations by context-free grammars in Chomsky normal form. Theoret. Comput. Sci., 354:118-130, 2006. Google Scholar
  3. P. R. J. Asveld. Generating all permutations by context-free grammars in Greibach normal form. Theoret. Comput. Sci., 409:565-577, 2008. Google Scholar
  4. D. Chistikov, S. Ivan, A. Lubiw, and J. Shallit. Fractional Coverings, Greedy Coverings, and Rectifier Networks. In Heribert Vollmer and Brigitte Vallée, editors, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), volume 66 of LIPIcs, pages 23:1-23:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  5. Da Jung Cho, Daniel Goč, Yo-Sub Han, Sang Ki Ko, Alexandros Palioudakis, and Kai Salomaa. State complexity of permutation on finite languages over a binary alphabet. Theoretical Computer Science, 682:67-78, June 2017. URL: http://dx.doi.org/10.1016/j.tcs.2017.03.007.
  6. A. Ehrenfeucht and P. Zeiger. Complexity measures for regular expressions. J. Comput. System Sci., 12:134-146, 1976. Google Scholar
  7. K. Ellul, B. Krawetz, J. Shallit, and M.-w. Wang. Regular expressions: new results and open problems. J. Autom. Lang. Combin., 10:407-437, 2005. Google Scholar
  8. Y. Filmus. Lower bounds for context-free grammars. Info. Proc. Letters, 111:895-898, 2011. Google Scholar
  9. G. Gramlich and G. Schnitger. Minimizing nfa’s and regular expressions. J. Comput. System Sci., 73:908-923, 2007. Google Scholar
  10. J. L. W. V. Jensen. Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math., 30:175-193, 1906. URL: http://dx.doi.org/10.1007/BF02418571.
  11. T. Jiang and B. Ravikumar. Minimal NFA problems are hard. SIAM J. Comput., 22:1117-1141, 1993. Google Scholar
  12. A. R. Meyer and L. J. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. In Proc. 13th Ann. IEEE Symp. on Switching and Automata Theory, pages 125-129. IEEE, 1972. Google Scholar
  13. H. Mousavi. Lower bounds on regular expression size. Preprint available at https://arxiv.org/abs/1712.00811, 2017.
  14. H. Robbins. A Remark on Stirling’s Formula. Amer. Math. Monthly, 62:26-29, 1955. Google Scholar
  15. N. J. A. Sloane et al. The On-Line Encyclopedia of Integer Sequences. Available at https://oeis.org, 2018.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact