CSC433_Lecture 3_07.04.21.pdf - Theory Of Computation CSC...

Theory Of
Computation
CSC 433 – SPRING 2021
LECTURE 3 : PUMPING LEMMA
Dr. Basma Hassan
bhassan[email protected]

Non-Regular Languages
4/6/2021 CSC 433 – Spring 2021
• Finite Automata have limitations, they have limited
number of states.
• Certain languages cannot be recognized by FA.
• Example, languages over the  = {0, 1}:
• A = { 0 n 1 n | n  0 }
• B = { w | w has an equal number of 0s and 1s}
• How to prove that certain languages are not regular,
i.e. cannot be recognized by FA?

Pumping Lemma for Regular
Languages
4/6/2021 CSC 433 – Spring 2021
• Pumping Lemma is a theorem states that all regular
languages have a special property.
• The property states that:
• All strings in the language that are at least as long as a pumping
length , contains a section that can be repeated any number of times
with the resulting string remaining in the language.
• Any language does not have this property is
guaranteed to be not regular.

Pumping Lemma Theorem
4/6/2021 CSC 433 – Spring 2021
• If A is a regular language, then there is a number p
( the pumping length ) where, if s is any string in A of
length at least p , then s may be divided into three
pieces, s = xyz , satisfying the following conditions:
1. for each i  0, xy
i
z  A, → ( xz, xyz, xyyz, xyyyz , …..  A)
2. |y| > 0, and
3. |xy|  p.

Pumping Lemma Theorem
4/6/2021 CSC 433 – Spring 2021
• Example, string s with length n ≥ p

Prove Non-regularity of a Language
4/6/2021 CSC 433 – Spring 2021
• Pumping Lemma is used to prove non-regularity of a
language B (Proof by contradiction).