Introduction to Computational Linguistics

10. Characters, Strings and Regular Expressions 29
(N is the set of natural numbers. It contains all integers starting from 0.)· is often
omitted. Hence, st is the same as s · t. Hence, cat = c · a · t. It is easily seen
that every set containing exactly one string is a regular language. Hence, every
set containing a finite set of strings is also regular. With more effort one can show
that a set containing all but a finite set of strings is regular, too.
Notice that whereas s is a term, L(s) is a language, the language of terms that
fall under the term s. In ordinary usage, these two are not distinguished, though.
We write a both for the string a and the regular term whose language is {a}. It
follows that
(85)
(86)
L(a · (b ∪ c)) = {ab, ac}
L((cb) ∗ a) = {a, cba, cbcba, . . . }
A couple of abbreviations are also used:
(87)
(88)
(89)
s? := ε ∪ s
s + := s · s ∗
s n := s · s · · · · · s
(n–times)
We say that s ⊆ t if L(s) ⊆ L(t). This is the same as saying that L(s ∪ t) = L(t).
Regular expressions are used in a lot of applications. For example, if you are
searching for a particular string, say department in a long document, it may
actually appear in two shapes, department or Department. Also, if you are
searching for two words in a sequence, you may face the fact that they appear on
different lines. This means that they are separated by any number of blanks and
an optional carriage return. In order not to loose any occurrence of that sort you
will want to write a regular expression that matches any of these occurrences. The
Unix command egrep allows you to search for strings that match a regular term.
The particular constructors look a little bit different, but the underlying concepts
are the same.
Notice that there are regular expressions which are different but denote the
same language. We have in general the following laws. (A note of caution. These
laws are not identities between the terms; the terms are distinct. These are identities
concerning the languages that these terms denote.)
(90a)
(90b)
s · (t · u) = (s · t) · u
s · ε = s
ε · s = s