COSC 341 Languages

1
COSC 341Languages

• We begin our investigation of formal languages.
• Lets ask some basic questions.
• Why do we care about formal languages?
• Answer Programming languages!
• What is a formal language?
• Answer A set of strings over some alphabet.
• What is an alphabet?
• Answer Any finite set, for example a, b . Or
  the English alphabet of 26 letters. Or 128 ASCII
  characters. Or the digits 0, 1, 2, , 9.
• What is a string over a, b ?
• Answer Any function f n ? a, b , where n
  0, 1, 2, , n-1.
• Example If g 3 ? a, b is given by g(0) b,
  g(1) a, and g(2) a, then g is the string
  usually written baa.

2
Examples of simple formal languages

• Definition If ? is any alphabet, then ? is the
  set of all strings over ?.
• Examples Suppose ? a, b .
• Simplest of all languages is ?, which has no
  strings in it at all.
• ? is a language with one string, namely the
  empty string (of length 0).
• a is a language with one string, namely the
  function f 1 ? a, b defined by f(0) a.
  (String of length 1.)
• ?, a, b is the finite language consisting of
  all strings of length 1.
• ?, a, b, aa, ab, ba, bb is the finite
  language consisting of all the strings of length
  2.
• ? ?, a, b, aa, ab, ba, bb, aaa, aab, is
  the infinite language consisting of all strings.
  Notice that ? includes the empty string ?.

3
Recursive definitions of languages

• We like to define languages by recursion because
  it shows how to generate strings of the language
• ? is the universal set, the atoms typically
  come from ?, and the operation used to build
  complex strings is usually concatenation
• Recall that we write the concatenation of two
  strings, e.g. abba and baab, by writing them side
  by side abbabaab
• L a, aa, aaa, is defined by
• a ? L
• if u ? L then ua ? L.
• L ?, a, aa, aaa,
• ? ? L
• if u ? L then ua ? L.
• L a, aaa, aaaaa, an n is odd
• ?
• ?

4
A more interesting language
Let Evenstring u u?? has even length,
where ? a, b . Maybe Evenstring can be
defined recursively ? ? L if u ? L, then uaa,
uab, uba, and ubb ? L. Is L Evenstring? How can
we be sure? We must show that L ? Evenstring and
that Evenstring ? L. That L ? Evenstring can be
proved by induction Let X ? L consist of the
strings of L that belong to Evenstring. Now ? ? X
since 0 is even. Suppose u ? X. Then so are uaa,
uab, uba, ubb. So XL, and strings in L all have
even length. Conversely, we show (again by
induction) that every string of even length
belongs to L. Let E 0, 2, 4, and let Y n
? E every string of length n is in L. Now 0 ?
Y and if n ?Y then n2?Y. Why? So?
5
Another way to define languages

• Suppose ? a, b .
• The simplest languages over a, b are
• ?, the set having no strings in it at all.
• ?, the set having just one string, namely the
  empty string (of length 0).
• a , the set having just one string, namely the
  function f 1 ? a, b defined by f(0) a.
  (String of length 1.)
• b , the set having just one string, namely the
  function g 1?a, b given by g(0) b.
• Idea
• Suppose we combine these languages with the help
  of set-theoretic operations like ? and ??
• Would still have a recursive way to define
  languages, but the basic building blocks are
  languages, not strings, and the operations are
  things we do to languages, not things we do to
  strings.

6
What operations may be used?
Union If L1 a, ab and L2 ba, bb
then L1?L2 a, ab, ba, bb . Concatenation of
languages If L1 a, ab and L2 ba, bb
then L1L2 aba, abb, abba, abbb . Note that
L1L2 ? L1?L2 and L1L2 ? L2L1 baa, baab, bba,
bbab . Kleene star If L1 a then L1 ?,
a, aa, aaa, . If L2 aa then L2 ?, aa,
aaaa, . (Why only these operations? Tradition.
We shall see in Ch 6 that we may also use ? and
relative complement if we choose.)
7
Regular expressions

• Suppose we want to define a language over
  alphabet ? a, b by giving a recursive
  definition on languages, where
• we say which of ?, ?, a , b are to be
  used as building blocks, and
• which of union, concatenation, and Kleene star
  are to be used as operations.
• Regular expressions are a notation for expressing
  the languages as building blocks option
  concisely.
• Simplest regular expressions (for ? a, b )
• ?, ?, a, b.
• More complex regular expressions are built up as
  follows
• if u and v are regular expressions, then so are
  u?v, uv, and u.

8
Using regular expressions
Let ? a, b . The regular expression ? defines
the empty language ? ? defines the language ? a
defines the language a a?b defines the
language a ?b a, b a defines the
language ?, a, aa, aaa, (ab) defines ?, ab,
abab, ababab, a?b defines a ?b ?,
a, aa, ??, b, bb, ?, a, b, aa, bb, aaa,
bbb, (a?b) defines a, b ?, a, b, aa,
ab, ba, bb, What regular expression defines
Evenstring?