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Linear size test sets for certain commutative languages

Published online by Cambridge University Press:  15 August 2002

Štěpán Holub  and
Juha Kortelainen
Štěpán Holub
Affiliation:
Turku Centre for Computer Science & Charles University, Prague, Czech Republic Department of Information Processing Science, University of Oulu, P.O. Box 3000, 90014 Oulun Yliopisto, Finland
Juha Kortelainen
Affiliation:
Turku Centre for Computer Science & Charles University, Prague, Czech Republic Department of Information Processing Science, University of Oulu, P.O. Box 3000, 90014 Oulun Yliopisto, Finland
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Abstract

We prove that for each positive integer n, the finite commutative language En = c(a1a2...an) possesses a test set of size at most 5n. Moreover, it is shown that each test set for En has at least n-1 elements. The result is then generalized to commutative languages L containing a word w such that (i) alph(w) = alph}(L); and (ii) each symbol a ∈ alph}(L) occurs at least twice in w if it occurs at least twice in some word of L: each such L possesses a test set of size 11n, where n = Card(alph(L)). The considerations rest on the analysis of some basic types of word equations.

Keywords

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 35 , Issue 5 , September 2001 , pp. 453 - 475
Copyright
© EDP Sciences, 2001

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References

J. Albert, On test sets, checking sets, maximal extensions and their effective constructions. Habilitationsschirft, Fakultät für Wirtschaftswissenschaften der Universität Karlsruhe (1968).
Albert, M.H. and Lawrence, J., A proof of Ehrenfeucht`s Conjecture. Theoret. Comput. Sci. 41 (1985) 121-123. CrossRef
Albert, J. and Wood, D., Checking sets, test sets, rich languages and commutatively closed languages. J. Comput. System Sci. 26 (1983) 82-91. CrossRef
Ch. Choffrut and J. Karhumäki, Combinatorics on words, in Handbook of Formal Languages, Vol. I, edited by G. Rosenberg and A. Salomaa. Springer-Verlag, Berlin (1997) 329-438.
Ehrenfeucht, A., Karhumäki, J. and Rosenberg, G., On binary equality sets and a solution to the test set conjecture in the binary case. J. Algebra 85 (1983) 76-85. CrossRef
Erdös, P. and Szekeres, G., A combinatorial problem in geometry. Compositio Math. 2 (1935) 464-470.
I. Hakala, On word equations and the morphism equivqalence problem for loop languages. Academic dissertation, Faculty of Science, University of Oulu (1997).
Hakala, I. and Kortelainen, J., On the system of word equations x 0 u 1 i x 1 u 2 i x 2 u 3 i x 3 = y 0 v 1 i y 1 i v 2 i y 2 v 3 i y 3(i = 0,1,2,...) in a free monoid. Theoret. Comput. Sci. 225 (1999) 149-161. CrossRef
Hakala, I. and Kortelainen, J., Linear size test sets for commutative languages. RAIRO: Theoret. Informatics Appl. 31 (1997) 291-304.
T. Harju and J. Karhumäki, Morphisms, in Handbook of Formal Languages, Vol. I, edited by G. Rosenberg and A. Salomaa. Springer-Verlag, Berlin (1997) 439-510.
M.A. Harrison, Introduction to Formal Language Theory. Addison-Wesley, Reading, Reading Massachusetts (1978).
Holub, S., Local and global cyclicity in free monoids. Theoret. Comput. Sci. 262 (2001) 25-36. CrossRef
Karhumäki, J. and Plandowski, W., On the size of independent systems of equations in semigroups. Theoret. Comput. Sci. 168 (1996) 105-119. CrossRef
Kortelainen, J., On the system of word equations x 0 u 1 i x 1 u 2 i x 2...u m i x m=y 0 v 1 i y 1 v 2 i y 2...v n i y n(i=1,2,...) in a free monoid. J. Autom. Lang. Comb. 3 (1998) 43-57.
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading Massachusetts (1983).
Salomaa, A., The Ehrenfeucht conjecture: A proof for language theorists. Bull. EATCS 27 (1985) 71-82.