Lec 09 Pumping Lemma.ppt - Theory of Automata Pumping Lemma...

Theory of Automata
Pumping Lemma
Dr Aftab Maroof, Dr Waseem Shehzad, Dr Labiba
Fahad & Mehreen Alam
Wee 5

Theory of Automata 2
Contents
•
Definition of non-Regular Languages
•
Theorem 13
–
Proof & Example
•
Theorem 14
–
Proof & Example

Theory of Automata 3
Introduction
•
By using FAs and regular expressions, we have been able to define many
languages. Although these languages have many different structures, they
take only a few basic forms:
– Languages with required substrings
– Languages that forbid some substrings
– Languages that begin or end with certain substrings
– Languages with certain even (or odd) properties, and so on.
•
We now turn our attention to some new forms, such as the language
PALINDROME, or the language PRIME of all words a
p
, where p is a prime
number.
•
We shall see that neither of these is a regular language. We can describe them
in English, but they can not be defined by an FA. We need to build more
powerful machines to define them.

Theory of Automata 4
Definition of Non-Regular Languages
Definition:
•
A language that cannot be defined by a
regular expression is called a non-regular
language.
Notes:
– By Kleene’s theorem, a non-regular language can also
not be accepted by any FA or TG.
– All languages are either regular or non-regular; none
are both.

Theory of Automata 5
Case Study
•
Consider the language
L = {Λ; ab; aabb; aaabbb; aaaabbbb; aaaaabbbbb; …}
•
The language L can also be written as
L = {a
n
b
n
for n = 0; 1; 2; 3; …}
or for short
L = {a
n
b
n
}
•
Note that although L is a subset of many regular
languages, such as a*b*; the language defined by ab also
includes such strings as aab and bb that are not in L.

Theory of Automata 6
Example
•
We shall show that the language L = {a
n
b
n
} is non-regular.
•
Suppose on the contrary that L were regular. Then there must exist some FA that
accepts L.
•
Just for the sake of argument, let us assume that this FA has 95 states.
•
We know that this FA must accept the word a
96
b
96
.
•
The first 96 letter a’s of this input string trace a path through this machine.
•
The path cannot visit a new state when each input letter is read, because there
are only 95 states. Therefore, at some point the path returns to a state that it has
already visited.
•
In other words, the path must contain a circuit in it. (A circuit is a loop that can be
made of several edges).

Theory of Automata 7
•
So, the path first wanders up to the circuit and then starts to loop
around the circuit (maybe many times) until a b is read from the input.