Skip to main content

The Shuffle Product: New Research Directions

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8977))

Abstract

In this paper we survey some recent researches concerning the shuffle operation that arise both in Formal Languages and in Combinatorics on Words.

Antonio Restivo—Partially supported by Italian MIUR Project PRIN 2010LYA9RH, “Automi e Linguaggi Formali: Aspetti Matematici e Applicativi”.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baeten, J., Weijland, W.: Process Algebra. Cambridge University Press (1990)

    Google Scholar 

  2. ter Beek, M.H., Martn-Vide, C., Mitrana, V.: Synchronized shuffles. Theoretical Computer Science 341(13), 263–275 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berstel, J., Boasson, L., Carton, O., Pin, J.É., Restivo, A.: The expressive power of the shuffle product. Inf. Comput. 208(11), 1258–1272 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Buss, S., Soltys, M.: Unshuffling a square is np-hard. J. Comput. Syst. Sci. 80(4), 766–776 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cano, A., Guaiana, G., Pin, J.É.: Regular languages and partial commutations. Information and Computation 230, 76–96 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Castiglione, G., Restivo, A.: On the shuffle of star-free languages. Fundam. Inform. 116(1–4), 35–44 (2012)

    MATH  MathSciNet  Google Scholar 

  7. Currie, J.: Shuffle squares are avoidable. Manuscript

    Google Scholar 

  8. Currie, J.D., Saari, K.: Square-free words with square-free self-shuffles. Electr. J. Comb., 1–9 (2014)

    Google Scholar 

  9. Diekert, V., Rozenberg, G.: The Book of Traces. World Scientific, River Edge (1995)

    Book  Google Scholar 

  10. Eilenberg, S.: Automata, Languages, and Machines. Academic Press Inc., Orlando (1976)

    MATH  Google Scholar 

  11. Ginsburg, S., Spanier, E.H.: Mappings of languages by two-tape devices. J. ACM 12(3), 423–434 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gómez, A.C., Pin, J.É.: Shuffle on positive varieties of languages. Theoretical Computer Science 312(23), 433–461 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gómez, A.C., Pin, J.É.: A robust class of regular languages. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 36–51. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  14. Gonçalves, D., Montassier, M., Pinlou, A.: Entropy compression method applied to graph colorings. CoRR abs/1406.4380 (2014). http://arxiv.org/abs/1406.4380

  15. Guaiana, G., Restivo, A., Salemi, S.: Star-free trace languages. Theor. Comput. Sci. 97(2), 301–311 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Guegan, G., Ochem, P.: Avoiding shuffle squares using entropy compression. 15e Journées Montoises d’Informatique Théorique (2014)

    Google Scholar 

  17. Harju, T.: A note on square-free shuffles of words. In: Karhumäki, J., Lepistö, A., Zamboni, L. (eds.) WORDS 2013. LNCS, vol. 8079, pp. 154–160. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  18. Harju, T., Müller, M.: Square-free shuffles of words. CoRR abs/1309.2137 (2013). http://arxiv.org/abs/1309.2137

  19. Henshall, D., Rampersad, N., Shallit, J.: Shuffling and unshuffling. Bulletin of the EATCS 107, 131–142 (2012)

    Google Scholar 

  20. Aoki, H., Uehara, R., Yamazaki, K.: Expected length of longest common subsequences of two biased random strings and its application. In: LA Symposium (2001)

    Google Scholar 

  21. Kececioglu, J., Gusfield, D.: Reconstructing a history of recombinations from a set of sequences. Discrete Applied Mathematics 88(13), 239–260 (1998). computational Molecular Biology DAM - CMB Series

    Article  MATH  MathSciNet  Google Scholar 

  22. Kimura, T.: An algebraic system for process structuring and interprocess communication. In: STOC 1976, pp. 92–100 (1976)

    Google Scholar 

  23. van Leeuwen, J., Nivat, M.: Efficient recognition of rational relations. Information Processing Letters 14(1), 34–38 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press (2002)

    Google Scholar 

  25. Lyndon, R.C., Schützenberger, M.P.: The equation \(a^m=b^nc^p\) in a free group. Michigan Math. J. 9(4), 289–298 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mansfield, A.: An algorithm for a merge recognition problem. Discrete Applied Mathematics 4(3), 193–197 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mansfield, A.: On the computational complexity of a merge recognition problem. Discrete Applied Mathematics 5(1), 119–122 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  28. McNaughton, R., Papert, S.: Counter-Free Automata. MIT Press, Cambridge (1971)

    MATH  Google Scholar 

  29. Perrot, J.F.: Varietes de langages et operations. Theor. Comput. Sci. 7, 197–210 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  30. Pin, J.É.: Varieties of formal languages. North Oxford, London and Plenum (1986). (Traduction de Variétés de langages formels)

    Book  MATH  Google Scholar 

  31. Pin, J.-É.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, chap. 10, vol. 1, pp. 679–746. Springer (1997)

    Google Scholar 

  32. Pin, J.É.: Theme and variations on the concatenation product. In: Winkler, F. (ed.) CAI 2011. LNCS, vol. 6742, pp. 44–64. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  33. Rizzi, R., Vialette, S.: On recognizing words that are squares for the shuffle product. In: Bulatov, A.A., Shur, A.M. (eds.) CSR 2013. LNCS, vol. 7913, pp. 235–245. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  34. Shaw, A.: Software descriptions with flow expressions. IEEE Transactions on Software Engineering SE–4(3), 242–254 (1978)

    Article  Google Scholar 

  35. Shyr, H.J., Yu, S.S.: Non-primitive words in the language \(p^+q^+\). Soochow J. Math. 20(4), 535–546 (1994)

    MATH  MathSciNet  Google Scholar 

  36. Tesson, P., Therien, D.: Diamonds are forever: The variety da. In: Semigroups, Algorithms, Automata and Languages, Coimbra (Portugal) 2001, pp. 475–500. World Scientific (2002)

    Google Scholar 

  37. Warmuth, M.K., Haussler, D.: On the complexity of iterated shuffle. Journal of Computer and System Sciences 28(3), 345–358 (1984)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Antonio Restivo .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Restivo, A. (2015). The Shuffle Product: New Research Directions. In: Dediu, AH., Formenti, E., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2015. Lecture Notes in Computer Science(), vol 8977. Springer, Cham. https://doi.org/10.1007/978-3-319-15579-1_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-15579-1_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15578-4

  • Online ISBN: 978-3-319-15579-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics