The paper deals with the problems related to finding a pattern common
to all words in a given set. We restrict our attention to patterns exp
ressible by the use of variables ranging over words. Two essentially d
ifferent cases result, depending on whether or not the empty word belo
ngs to the range. We investigate equivalence and inclusion problems, p
atterns descriptive for a set, as well as some complexity issues. The
inclusion problem between two pattern languages turns out to be of fun
damental theoretical importance because many problems in the classical
combinatorics of words can be reduced to it.