Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T17:52:01.235Z Has data issue: false hasContentIssue false
  • English
  • Français

An aperiodicity problem for multiwords

Published online by Cambridge University Press:  23 November 2011

Véronique Bruyère ,
Olivier Carton ,
Alexandre Decan ,
Olivier Gauwin  and
Jef Wijsen
Véronique Bruyère
Affiliation:
Universitéde Mons – UMONS, Institut d’Informatique, Belgium. [email protected]
Olivier Carton
Affiliation:
Université Paris Diderot, LIAFA, France; [email protected]
Alexandre Decan
Affiliation:
Universitéde Mons – UMONS, Institut d’Informatique, Belgium. [email protected]
Olivier Gauwin
Affiliation:
Universitéde Mons – UMONS, Institut d’Informatique, Belgium. [email protected]
Jef Wijsen
Affiliation:
Universitéde Mons – UMONS, Institut d’Informatique, Belgium. [email protected]
Get access

Abstract

Multiwords are words in which a single symbol can be replaced by a nonempty set of symbols. They extend the notion of partial words. A word w is certain in a multiword M if it occurs in every word that can be obtained by selecting one single symbol among the symbols provided in each position of M. Motivated by a problem on incomplete databases, we investigate a variant of the pattern matching problem which is to decide whether a word w is certain in a multiword M. We study the language CERTAIN(w) of multiwords in which w is certain. We show that this regular language is aperiodic for three large families of words. We also show its aperiodicity in the case of partial words over an alphabet with at least three symbols.

Keywords

Pattern matchingaperiodicitypartial words
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 46 , Issue 1: Special issue dedicated to the 13th "Journées Montoises d'Informatique Théorique" , January 2012 , pp. 33 - 50
Copyright
© EDP Sciences 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aho, A.V. and Corasick, M.J., Efficient string matching: An aid to bibliographic search. Commun. ACM 18 (1975) 333–340. Google Scholar
A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms. Addison-Wesley (1974).
Berstel, J. and Boasson, L., Partial words and a theorem of Fine and Wilf. Theoret. Comput. Sci. 218 (1999) 135–141. Google Scholar
F. Blanchet-Sadri, Algorithmic Combinatorics on Partial Words (Discrete Mathematics and Its Applications). Chapman & Hall/CRC (2007).
Boyer, R.S. and Mooren, J.S., A fast string searching algorithm. Commun. ACM 20 (1977) 762–772. Google Scholar
V. Bruyère, A. Decan and J. Wijsen, On first-order query rewriting for incomplete database histories, in Proc. of the 16th International Symposium on Temporal Representation and Reasoning (TIME) (2009) 54–61.
M. Crochemore and W. Rytter, Text Algorithms. Oxford University Press (1994).
M. Crochemore, C. Hancart and T. Lecroq, Algorithms on Strings. Cambridge University Press (2007) 392.
Fine, N.J. and Wilf, H.S., Uniqueness theorems for periodic functions. Proc. of Amer. Math. Soc. 16 (1965) 109–114. Google Scholar
M.J. Fischer and M.S. Paterson, String matching and other products. SIAM-AMS Proceedings, Complexity of Computation 7 (1974) 113–125.
Halava, V., Harju, T. and Kärki, T., Relational codes of words. Theoret. Comput. Sci. 389 (2007) 237–249. Google Scholar
Holub, J., Smyth, W.F. and Wang, S., Fast pattern-matching on indeterminate strings. J. Discrete Algorithms 6 (2008) 37–50. Google Scholar
Knuth, D.E., Morris, J.H. and Pratt, V.R., Fast pattern matching in strings. SIAM J. Comput. 6 (1977) 323–350. Google Scholar
G. Kucherov, L. Noé and M.A. Roytberg, Subset seed automaton, in Proc. of the 12th International Conference on Implementation and Application of Automata (CIAA). Springer (2007) 180–191.
M. Lothaire, Combinatorics on words. Cambridge University Press (1997).
R. McNaughton and S. Papert, Counter-free Automata. MIT Press, Cambridge, MA (1971).
J.-É. Pin, Varieties of Formal Languages. North Oxford, London and Plenum, New-York (1986).
M.S. Rahman, C.S. Iliopoulos and L. Mouchard, Pattern matching in degenerate DNA/RNA sequences, in Workshop on Algorithms and Computation (WALCOM), edited by M. Kaykobad and M.S. Rahman. Bangladesh Academy of Sciences (BAS) (2007) 109–120.
Schützenberger, M.P., On finite monoids having only trivial subgroups. Inform. Control 8 (1965) 190–194. Google Scholar