Words over a Partially Commutative Alphabet

  • Dominique Perrin
Part of the NATO ASI Series book series (volume 12)


Many interesting combinatorial problems on words deal with rearrangements of words. One of the goals of such rearrangements is to provide bijective mappings between sets of words satisfying certain properties and therefore give some enumeration results on words. The interested reader may consult the Chapter by D. Foata in Lothaire’s book [11] where some examples of rearrangements are developped. The algorithms involved in such rearrangements are, by the way, close to the more popular ones since many sorting problems can usefully be formulated in terms of rearrangements. The study of rearrangements has lead D. Foata to consider words over an alphabet in which some of the letters are allowed to commute. And this, in turn, could have raised the interest for studying “in abstracto” problems concerning these words and the structure of the commutation monoids which is their habitat. Nonetheless, it happened on the contrary that words on partially commutative alphabets became of interest to computer scientists studying problems of concurrency control. Roughly speaking, the alphabet considered in this framework is made of functions and the commutation between these functions corresponds to the commutation of mappings under composition. A typical problem is then to decide wether, up to the commutation rule, a given word is equivalent to one in a special form (see [13], chapter 10 for an exposition of this problem). My own interest in such questions was motivated by the work of M.P. Flé and G. Roucairol [9] who proved a surprising result on finite automata in commutation monoids motivated by problems of concurrency control. The aim of this paper is to present a survey of results obtained recently on commutation monoids including a generalization of the above mentionned. It does not intend to be a comprehensive exposition and many facets of the question have been left in the dark. The first section introduces some terminology and definitions. The second section contains the discussion of two normal form theorems in commutation monoids. The third section contains some results on the structure of commutation monoids. Finally, in the last section, I will discuss the problem of finite automata and commutation monoids.


Normal Form Direct Product Dependency Graph Finite Automaton Concurrency Control 
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]
    Anisimov, A.V., Knuth D., Inhomogeneous sorting, unpublishedGoogle Scholar
  2. [2]
    Manuscript Berstel, J. Transductions and Context Free Languages,Teubner, 1979.Google Scholar
  3. [3]
    Cartier, P., Foata, D., Problèmes Combinatoires de Commutation et Réarrangements, Lecture Notes in Math., 85, Springer Verlag, 1969.Google Scholar
  4. [4]
    Clerbout, M., Latteux, M., Partial Commutations and faithful rational transductions, research report, Univ. Lille.Google Scholar
  5. [5]
    Cori, R., Métivier, Y. Rational subsets of some partially commutative monoids, Theoret. Comput. Sci.,to appear.Google Scholar
  6. [6]
    Cori, R., Perrin, D., Sur la reconnaissabilité dans les monoides partiellement commutatifs libres, Rairo In format. Theor., to appearGoogle Scholar
  7. [7]
    Duboc, C., Some properties of commutations in free partially commutative monoids Inform. Processing Letters,to appear.Google Scholar
  8. [8]
    Dulucq, S., Viennot, G., Bijective proof and generalizations of McMahon Master Theorem, unpublished manuscript.Google Scholar
  9. [9]
    Flé, M.P., Roucairol, G., Maximal serializability of iterated transactions, Theoret. Comput. Sci. to appear (see also ACM SIGACT SIGOPS, 1982, 194–200 ).Google Scholar
  10. [10]
    Lallement, G., Semigroups and Combinatorial Applications,Wiley, 1979.Google Scholar
  11. [11]
    Lothaire, M., Combinatorics on Words,Addison Wesley, 1983.Google Scholar
  12. [12]
    Métivier, Y., Une condition suffisante de reconnaissabilité dans les monoides partiellement commutatifs, Rairo In format. Theor.,to appear.Google Scholar
  13. [13]
    Ullman, J., Principles of Database Systems,Computer Science Press, 1980.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Dominique Perrin
    • 1
  1. 1.L.I.T.P.Université Paris 7France

Personalised recommendations