Unambiguous erasing morphisms in free monoids

Johannes C. Schneider

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 44, Issue: 2, page 193-208
  • ISSN: 0988-3754

Abstract

top
This paper discusses the fundamental combinatorial question of whether or not, for a given string α, there exists a morphism σ such that σ is unambiguous with respect to α, i.e. there exists no other morphism τ satisfying τ(α) = σ(α). While Freydenberger et al. [Int. J. Found. Comput. Sci. 17 (2006) 601–628] characterise those strings for which there exists an unambiguous nonerasing morphism σ, little is known about the unambiguity of erasing morphisms, i.e. morphisms that map symbols onto the empty string. The present paper demonstrates that, in contrast to the main result by Freydenberger et al., the existence of an unambiguous erasing morphism for a given string can essentially depend on the size of the target alphabet of the morphism. In addition to this, those strings for which there exists an erasing morphism over an infinite target alphabet are characterised, complexity issues are discussed and some sufficient conditions for the (non-)existence of unambiguous erasing morphisms are given.

How to cite

top

Schneider, Johannes C.. "Unambiguous erasing morphisms in free monoids." RAIRO - Theoretical Informatics and Applications 44.2 (2010): 193-208. <http://eudml.org/doc/250782>.

@article{Schneider2010,
abstract = { This paper discusses the fundamental combinatorial question of whether or not, for a given string α, there exists a morphism σ such that σ is unambiguous with respect to α, i.e. there exists no other morphism τ satisfying τ(α) = σ(α). While Freydenberger et al. [Int. J. Found. Comput. Sci. 17 (2006) 601–628] characterise those strings for which there exists an unambiguous nonerasing morphism σ, little is known about the unambiguity of erasing morphisms, i.e. morphisms that map symbols onto the empty string. The present paper demonstrates that, in contrast to the main result by Freydenberger et al., the existence of an unambiguous erasing morphism for a given string can essentially depend on the size of the target alphabet of the morphism. In addition to this, those strings for which there exists an erasing morphism over an infinite target alphabet are characterised, complexity issues are discussed and some sufficient conditions for the (non-)existence of unambiguous erasing morphisms are given. },
author = {Schneider, Johannes C.},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Combinatorics on words; morphisms in free monoids; unambiguity; complexity; combinatorics on words; unambiguity},
language = {eng},
month = {5},
number = {2},
pages = {193-208},
publisher = {EDP Sciences},
title = {Unambiguous erasing morphisms in free monoids},
url = {http://eudml.org/doc/250782},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Schneider, Johannes C.
TI - Unambiguous erasing morphisms in free monoids
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/5//
PB - EDP Sciences
VL - 44
IS - 2
SP - 193
EP - 208
AB - This paper discusses the fundamental combinatorial question of whether or not, for a given string α, there exists a morphism σ such that σ is unambiguous with respect to α, i.e. there exists no other morphism τ satisfying τ(α) = σ(α). While Freydenberger et al. [Int. J. Found. Comput. Sci. 17 (2006) 601–628] characterise those strings for which there exists an unambiguous nonerasing morphism σ, little is known about the unambiguity of erasing morphisms, i.e. morphisms that map symbols onto the empty string. The present paper demonstrates that, in contrast to the main result by Freydenberger et al., the existence of an unambiguous erasing morphism for a given string can essentially depend on the size of the target alphabet of the morphism. In addition to this, those strings for which there exists an erasing morphism over an infinite target alphabet are characterised, complexity issues are discussed and some sufficient conditions for the (non-)existence of unambiguous erasing morphisms are given.
LA - eng
KW - Combinatorics on words; morphisms in free monoids; unambiguity; complexity; combinatorics on words; unambiguity
UR - http://eudml.org/doc/250782
ER -

References

top
  1. C. Choffrut and J. Karhumäki, Combinatorics of words, edited by G. Rozenberg and A. Salomaa, Handbook of Formal Languages1, Chap. 6. Springer (1997) 329–438.  
  2. A. Ehrenfeucht and G. Rozenberg, Finding a homomorphism between two words is NP-complete. Inform. Process. Lett.9 (1979) 86–88.  Zbl0414.68022
  3. D.D. Freydenberger and D. Reidenbach, The unambiguity of segmented morphisms. In Proc. 11th International Conference on Developments in Language Theory, DLT 2007. Lect. Notes Comput. Sci. (2007) 181–192.  Zbl1202.68225
  4. D.D. Freydenberger, D. Reidenbach and J.C. Schneider, Unambiguous morphic images of strings. Int. J. Found. Comput. Sci.17 (2006) 601–628.  Zbl1110.68119
  5. M.R. Garey and D.S. Johnson, Computers and Intractability – A Guide to the Theory of NP-Completeness. W.H. Freeman and Co., New York (1979).  Zbl0411.68039
  6. T. Head, Fixed languages and the adult languages of 0L schemes. Int. J. Comput. Math.10 (1981) 103–107.  Zbl0472.68034
  7. T. Jiang, A. Salomaa, K. Salomaa and S. Yu, Decision problems for patterns. J. Comput. System Sci.50 (1995) 53–63.  Zbl0827.68066
  8. A. Mateescu and A. Salomaa, Patterns, edited by G. Rozenberg and A. Salomaa, Handbook of Formal Languages1, Chap. 4.6. Springer (1997) 230–242.  
  9. D. Reidenbach, A non-learnable class of E-pattern languages. Theoret. Comput. Sci.350 (2006) 91–102.  Zbl1086.68116
  10. D. Reidenbach, Discontinuities in pattern inference. Theoret. Comput. Sci.397 (2008) 166–193.  Zbl1145.68027
  11. D. Reidenbach and J.C. Schneider, Morphically primitive words, in Proc. 6th International Conference on Words, WORDS 2007 (2007) 262–272.  Zbl1166.68036
  12. J.C. Schneider, Unambiguous erasing morphisms in free monoids, in Proc. SOFSEM 2009: Theorie and Practice of Computer Science. Lect. Notes Comput. Sci.5404 (2009) 473–484.  Zbl1206.68198

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.