### Pumping Lemma

```Pumping Lemma
• The pumping lemma describes something that the
regular languages satisfy
If regular then the properties of pumping lemma satisfied
So
If pumping lemma not satisfied then …..
NOT REGULAR
Remember contrapositive?
Regular languages
• Not all languages are regular!
• How to show a language is regular?
– Write a regular expression for it
– Break it down via unions, intersections,
concatenations, stars, complements, reversals
• {a^k | k is a multiple of 6} is regular because {a^k | k is
a multiple of 2} is regular and {a^k | k is a multiple of 3}
is regular
– Make a DFA/NFA
Central idea of the pumping lemma
• If the language contains only a finite number
of strings, we have a regular language. Why?
– Make an NFA for each string
– Union them all
• The finiteness of states is v important and is
the main concept the pumping lemma uses
Central idea of the pumping lemma
• Assume the language is accepted by a DFA
that has p states. Now consider what happens
when you have a string that is p in length.
What states are you going to hit?
• As you go from state to state while traversing
the string ….
You will repeat a state!
Central idea of pumping lemma
• How do states repeat themselves? That means
the NFA/DFA/graph contains ….
A loop!
• Using a loop we can pump!
Pumping just means using the loop over and
over again
Statement of pumping lemma
If A is a regular language then A has a ‘pumping
length’ p such that any string s that is longer
than p can be broken into 3 pieces s = xyz
so that
• xyiz is in A. Basically saying the string y is the
string you can read on the loop
• Length of y is non-zero
• Length of xy portion <= p
```