09.pumpinglemma.ink.pdf - CS 301 SEPTEMBER 21 ST – PUMPING...

CS 301
SEPTEMBER 21 ST – PUMPING LEMMA

Regular Languages
DFA = NFA = Regex
Regular languages are good, simple languages to build from
Very common in string search and data mining
What is next?

Regular Languages
DFA = NFA = Regex
Regular languages are good, simple languages to build from
Very common in string search and data mining
What is next?
We need a good understanding of non-regular languages

Regular Languages
How do we know if a language is irregular?

Regular Languages
How do we know if a language is irregular?
Much more difficult to prove than showing it is regular
Regular(x) = ∃
M
(Decides(M,x)

Regular Languages
How do we know if a language is irregular?
Much more difficult to prove than showing it is regular
Regular(x) = ∃
M
(Decides(M,x)
!Regular(x) = ∀
M
(!Decides(M,x)

Regular Languages
How do we know if a language is irregular?
What is the biggest constraint of the DFA?

Regular Languages
How do we know if a language is irregular?
What is the biggest constraint of the DFA?

Non-regular languages
A language is non-regular if there cannot exist a DFA which decides it

Non-regular languages
A language is non-regular if there cannot exist a DFA which decides it
◦ To prove this, we utilize the pumping lemma
◦ The DFA can only have finite states, and this is the primary constraint on the
language

Pumping Lemma
For any DFA, there exists a number p, called the pumping length , for which
any string with length > p, the string must pass through the same state
twice.

Pumping Lemma
For any DFA, there exists a number p, called the pumping length , for which
any string with length > p, the string must pass through the same state
twice.
What is p?

Pumping Lemma
For any DFA, there exists a number p, called the pumping length , for which
any string with length > p, the string must pass through the same state
twice.