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

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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

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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

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  9. D. Reidenbach, A non-learnable class of E-pattern languages. Theoret. Comput. Sci.350 (2006) 91–102.  
  10. D. Reidenbach, Discontinuities in pattern inference. Theoret. Comput. Sci.397 (2008) 166–193.  
  11. D. Reidenbach and J.C. Schneider, Morphically primitive words, in Proc. 6th International Conference on Words, WORDS 2007 (2007) 262–272.  
  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.  

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