Commutative images of rational languages and the Abelian kernel of a monoid

Manuel Delgado

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 35, Issue: 5, page 419-435
  • ISSN: 0988-3754

Abstract

top
Natural algorithms to compute rational expressions for recognizable languages, even those which work well in practice, may produce very long expressions. So, aiming towards the computation of the commutative image of a recognizable language, one should avoid passing through an expression produced this way. We modify here one of those algorithms in order to compute directly a semilinear expression for the commutative image of a recognizable language. We also give a second modification of the algorithm which allows the direct computation of the closure in the profinite topology of the commutative image. As an application, we give a modification of an algorithm for computing the Abelian kernel of a finite monoid obtained by the author in 1998 which is much more efficient in practice.

How to cite

top

Delgado, Manuel. "Commutative images of rational languages and the Abelian kernel of a monoid." RAIRO - Theoretical Informatics and Applications 35.5 (2010): 419-435. <http://eudml.org/doc/222037>.

@article{Delgado2010,
abstract = { Natural algorithms to compute rational expressions for recognizable languages, even those which work well in practice, may produce very long expressions. So, aiming towards the computation of the commutative image of a recognizable language, one should avoid passing through an expression produced this way. We modify here one of those algorithms in order to compute directly a semilinear expression for the commutative image of a recognizable language. We also give a second modification of the algorithm which allows the direct computation of the closure in the profinite topology of the commutative image. As an application, we give a modification of an algorithm for computing the Abelian kernel of a finite monoid obtained by the author in 1998 which is much more efficient in practice. },
author = {Delgado, Manuel},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Rational language; semilinear set; profinite topology; finite monoid.; algorithms; semilinear expressions; commutative images; rational languages; rational expressions; finite monoids},
language = {eng},
month = {3},
number = {5},
pages = {419-435},
publisher = {EDP Sciences},
title = {Commutative images of rational languages and the Abelian kernel of a monoid},
url = {http://eudml.org/doc/222037},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Delgado, Manuel
TI - Commutative images of rational languages and the Abelian kernel of a monoid
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 5
SP - 419
EP - 435
AB - Natural algorithms to compute rational expressions for recognizable languages, even those which work well in practice, may produce very long expressions. So, aiming towards the computation of the commutative image of a recognizable language, one should avoid passing through an expression produced this way. We modify here one of those algorithms in order to compute directly a semilinear expression for the commutative image of a recognizable language. We also give a second modification of the algorithm which allows the direct computation of the closure in the profinite topology of the commutative image. As an application, we give a modification of an algorithm for computing the Abelian kernel of a finite monoid obtained by the author in 1998 which is much more efficient in practice.
LA - eng
KW - Rational language; semilinear set; profinite topology; finite monoid.; algorithms; semilinear expressions; commutative images; rational languages; rational expressions; finite monoids
UR - http://eudml.org/doc/222037
ER -

References

top
  1. A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analisys of Computer Algorithms. Addison Wesley (1974).  
  2. C.J. Ash, Inevitable graphs: A proof of the type II conjecture and some related decision procedures. Internat. J. Algebra and Comput.1 (1991) 127-146.  
  3. L. Babai, On Lovász' lattice reduction and the nearest lattice point problem. Combinatorica6 (1986) 1-13.  
  4. J. Berstel, Transductions and Context-free Languages. Teubner, Stuttgart (1979).  
  5. J. Brzozowski and E. McCluskey, Signal flow graph techniques for sequential circuit state diagrams. IEEE Trans. Electronic Comput.12 (1963) 67-76.  
  6. T.J. Chou and G.E. Collins, Algorithms for the solution of systems of linear Diophantine equations. SIAM J. Comput.11 (1982) 687-708.  
  7. H. Cohen, A Course in Computational Algebraic Number Theory. GTM, Springer Verlag (1993).  
  8. M. Delgado, Abelian pointlikes of a monoid. Semigroup Forum56 (1998) 339-361.  
  9. M. Delgado and V.H. Fernandes, Abelian kernels of some monoids of injective partial transformations and an application. Semigroup Forum61 (2000) 435-452.  
  10. A. Ehrenfeucht and P. Zeiger, Complexity measures for regular expressions. J. Comput. System Sci.12 (1976) 134-146.  
  11. V. Froidure and J.-E. Pin, Algorithms for computing finite semigroups, edited by F. Cucker and M. Shub. Berlin, Lecture Notes in Comput. Sci. (1997) 112-126.  
  12. K. Henckell, S. Margolis, J.-E. Pin and J. Rhodes, Ash's type II theorem, profinite topology and Malcev products: Part I. Internat. J. Algebra and Comput.1 (1991) 411-436.  
  13. A.K. Lenstra, H.W. Lenstra Jr. and L. Lovász, Factoring polynomials with rational coefficients. Math. Ann.261 (1982) 515-534.  
  14. S. Linton, G. Pfeiffer, E. Robertson and N. Ruskuc, Monoid Version 2.0. GAPPackage (1997).  
  15. O. Matz, A. Miller, A. Pothoff, W. Thomas and E. Valkema, Report on the program AMoRE. Tech. Rep. 9507, Christian Albrechts Universität, Kiel (1995).  
  16. J.-E. Pin, Varieties of Formal Languages. Plenum, New-York (1986).  
  17. J.-E. Pin, A topological approach to a conjecture of Rhodes. Bull. Austral. Math. Soc.38 (1988) 421-431.  
  18. J.-E. Pin and C. Reutenauer, A conjecture on the Hall topology for the free group. Bull. London Math. Soc.23 (1991) 356-362.  
  19. M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory. Cambridge University Press (1989).  
  20. L. Ribes and P.A. Zalesski, On the profinite topology on a free group. Bull. London Math. Soc.25 (1993) 37-43.  
  21. M. Schönert et al., GAP- Groups, Algorithms, and Programming, Lehrstuhl D fur Mathematik, Rheinisch Westfalische Technische Hochschule. Aachen, Germany, fifth Edition (1995).  
  22. C.C. Sims, Computation with Finitely Presented Groups. Cambridge University Press (1994).  
  23. B. Steinberg, Finite state automata: A geometric approach. Trans. Amer. Math. Soc.353 (2001) 3409-3464.  
  24. A. Storjohann, Algorithms for matrix canonical forms, Ph.D. thesis. Department of Computer Science, Swiss Federal Institute of Technology (2000)  URIhttp://www.scg.uwaterloo.ca/ astorjoh/publications.html
  25. B. Tilson, Type II redux, edited by S.M. Goberstein and P.M. Higgins. Reidel, Dordrecht, Semigroups and their applications (1987) 201-205.  
  26. The GAPGroup, GAP- Groups, Algorithms, and Programming, Version 4.2. Aachen, St Andrews (1999),  URIhttp://www-gap.dcs.st-and.ac.uk/ gap
  27. S. Willard, General Topology. Addison Wesley (1970).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.