The paper investigates knowledge discovery based on learning of
languages generated by patterns from positive examples.
A pattern $p$ is a finite string of constant symbols and variables,
and the language defined by $p$ is the set of constant strings
obtained from $p$ by substituting nonempty and constant strings
for variables.
We consider the class ${\PL_*}^k$ of unions of at most $k$
intersections of finitely many pattern languages.
It is well known that the class of unions of finitely many pattern
languages is not inferable from positive examples.
Every intersection of finitely many pattern languages can be
represented as a union of pattern languages, but it may be
a union of infinitely many ones.
The class $\PL_*$ of intersections, however, is shown to have
finite thickness.
Using the result, we show that the class ${\PL_*}^k$ is refutably
inferable from complete examples as well as inferable from
positive examples.
In order to study efficient learning algorithm, we introduce two
kind of syntactic ordered relations for finite sets of {\it regular}
patterns, and show that the semantic containment of unions of
intersections of regular pattern languages is equivalent to the
syntactic containment of some sets of regular patterns.
In terms of this result, the class of intersections of regular
pattern languages is polynomial time (refutably) inferable from
positive (complete) examples, under some assumption.