Nonregular languages

1
Section 1.4
Nonregular languages

• Which of these languages are regular ?
• B 0n1n n 0
• C w 2 0,1 w has equal number of 0s and
  1s
• D w 2 0,1 w has equal number of 01s
  and 10s as substrings

2
Section 1.4
Nonregular languages

• Which of these languages are regular ?
• B 0n1n n 0
• C w 2 0,1 w has equal number of 0s and
  1s
• D w 2 0,1 w has equal number of 01s
  and 10s as substrings
• Proof by closure properties

not in the book
3
Section 1.4
Pumping lemma for regular lang.
Suppose we have a DFA with p states. Suppose
there is a string of length gt p that is accepted.
Are there other strings that are accepted?
4
Section 1.4
Pumping lemma for regular lang.
Thm 1.70 pumping lemma Let A be a regular
language. Then there exists a number p s.t. for
every string s 2 A of length p there exist
strings x, y, and z s.t. 0. s xyz, 1. For each
i 0, xyiz 2 A, 2. y gt 0, and 3. xy p.
5
Section 1.4
Pumping lemma for regular lang.
Example B 0n1n n 0
6
Section 1.4
Pumping lemma for regular lang.
Example C w w has equal number of 0s and
1s
7
Section 1.4
Pumping lemma for regular lang.
Example F ww w 2 0,1
8
Section 1.4
Pumping lemma for regular lang.
Example D 1k k 0 is a square
9
Section 1.4
Pumping lemma for regular lang.
Example E 0i1j i gt j