Relational morphisms, transductions and operations on languages

  • Jean-Eric Pin
Mathematical Foundations Of The Theory Of Automata
Part of the Lecture Notes in Computer Science book series (LNCS, volume 386)


We have seen that, for the most part, natural operations on (varieties of) languages corrspond to natural operations on (varieties of) monoids: concatenation corresponds to aperiodic relational morphisms and to Schützenberger products, non ambiguous concatenation to locally trivial relational morphisms, length preserving morphisms to power monoids, sequential functions to wreath products, etc. In some cases, the corresponding operation is still unknown: for instance, it would be interesting to find an operation on languages corresponding to nilpotent relational morphisms,a nd an operation on monoids corresponding to shuffle. This correspondence between languages and monoids has motivated numerous research articles, and a number of problems are still open: on the semigroup side, the complete classification of the varieties of the form PV or the (many) decidability problems about varieties of the form V-1W or V*W; on the language side, the characterisation of varieties closed under shuffle or under pure star, and all problems related to the star operation and to the concatenation product (see the articles of P. Weil and K. Hashiguchi in this volume).


Wreath Product Concatenation Product Finite Semigroup Rational Subset Relational Morphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Berstel, Transductions and Context-free Languages, Teubner, Stuttgart, 1979.Google Scholar
  2. [2]
    S. Eilenberg, Automata, Languages and Machines, Academic Press, New York, Vol. A, 1974; Vol B, 1976.Google Scholar
  3. [3]
    K. Hashiguchi, A decision procedure for the order of regular events, Theoret. Comput. Sci. 8 (1979) 69–72.Google Scholar
  4. [4]
    K. Hashiguchi, Limitedness theorem on finite automata with distance functions, J. of Computer and System Sciences 24, (1982), 232–244.Google Scholar
  5. [5]
    K. Hashiguchi, Representation theorems on regular languages, J. Comput. System Sci. 27 (1983) 101–115.CrossRefGoogle Scholar
  6. [6]
    K. Hashiguchi, Regular languages of star-height one, Information and Control 53 (1982) 199–210.CrossRefGoogle Scholar
  7. [7]
    K. Hashiguchi, Improved limitedness theorems on finite automata with distance functions, Rapport LITP 86–72 (1986).Google Scholar
  8. [8]
    J.E. Pin, Langages reconnaissables et codage préfixe pur. 8th ICALP, Lecture Notes in Computer Science 115 (1981) 184–192.Google Scholar
  9. [9]
    J.E. Pin, Varieties of formal languages, Masson, Paris (1984), North Oxford Academic, London and Plenum, New-York (1986).Google Scholar
  10. [10]
    J.E. Pin, Power semigroups and related varieties of finite semigroups, Semigroups and Their Applications, édité par S.M. Goberstein et P.M. Higgins, D. Reidel (1986) 139–152.Google Scholar
  11. [11]
    J.E. Pin, H. Straubing and D. Thérien, Locally trivial categories and unambiguous concatenation, Journal of Pure and Applied Algebra 52 (1988) 297–311.Google Scholar
  12. [12]
    J.E. Pin and J. Sakarovitch, Operations and transductions that preserve rationality, 6ème GI Conference, Lecture Notes in Computer Science 145 (1983) 617–628.Google Scholar
  13. [13]
    J.E. Pin and J. Sakarovitch, Une application de la représentation matricielle des transductions, Theoretical Computer Science 35 (1985) 271–293.CrossRefGoogle Scholar
  14. [14]
    Ch. Reutenauer, Sur les variétés de langages et de monoïdes, Lecture Notes in Computer Science 67, Springer, Berlin, 1979, 260–265.Google Scholar
  15. [15]
    H. Straubing, Recognizable sets and power sets of finite semigroups, Semigroup Forum, 18 (1979) 331–340.Google Scholar
  16. [16]
    H. Straubing, Aperiodic homomorphisms and the concatenation product of recognizable sets, Journal of Pure and Applied Algebra 15 (1979) 319–327.CrossRefGoogle Scholar
  17. [17]
    H. Straubing, Relational morphisms and operations on recognizable sets, RAIRO Inf. Théor., 15 (1981) 149–159.Google Scholar
  18. [18]
    B. Tilson, Chapters XI and XII of reference [2].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Jean-Eric Pin
    • 1
  1. 1.LITP, Université Paris VI et CNRSFrance

Personalised recommendations