Linear size test sets for certain commutative languages

Issue		RAIRO-Theor. Inf. Appl. Volume 35, Number 5, September-October 2001


Page(s)		453 - 475
DOI		10.1051/ita:2001105

DOI: 10.1051/ita:2001105

Theoret. Informatics Appl. 35, 453-475 (2001)

Linear size test sets for certain commutative languages

Stepán Holub and Juha Kortelainen

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.

(Received April, 2001. Accepted December, 2001)

Abstract
We prove that for each positive integer n, the finite commutative language $E_n=c(a_1a_2\cdots a_n)$ possesses a test set of size at most 5n. Moreover, it is shown that each test set for E_n has at least n-1 elements. The result is then generalized to commutative languages L containing a word w such that (i) $\text{\rm alph}(w)=\text{\rm alph}(L);$ and (ii) each symbol $a\in\text{\rm alph}(L)$ 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 $n=\text{\rm Card}({\text{\rm alph}(L)})$ . The considerations rest on the analysis of some basic types of word equations.

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