FREYDENBERGER, D.D. and REIDENBACH, D., 2010. Existence and nonexistence of descriptive patterns. Theoretical Computer Science, 411 (34-36), pp. 3274-3286.
In the present paper, we study the existence of descriptive patterns, i. e. patterns that cover
all words in a given set through morphisms and that are optimal in terms of revealing
commonalities of these words. Our main result shows that if patterns may be mapped to
words by arbitrary morphisms, then there exist infinite sets of words that do not have a
descriptive pattern. This answers a question posed by Jiang et al. (Pattern languages with
and without erasing, International Journal of Computer Mathematics 50 (1994)). Since the
problem of whether a pattern is descriptive depends on the inclusion relation of so-called
pattern languages, our technical considerations lead to a number of deep insights into the
inclusion problem for and the topology of the class of terminal-free E-pattern languages.