More Applications of the Pumping Lemma

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# More Applications of the Pumping Lemma - PowerPoint PPT Presentation

More Applications of the Pumping Lemma. The Pumping Lemma:. Given a infinite regular language . there exists an integer (critical length). for any string with length . we can write. with and. such that:. Non-regular languages.

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## More Applications of the Pumping Lemma

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### More Applications ofthe Pumping Lemma

Prof. Busch - LSU

The Pumping Lemma:
• Given a infinite regular language
• there exists an integer (critical length)
• for any string with length
• we can write
• with and
• such that:

Prof. Busch - LSU

Non-regular languages

Regular languages

Prof. Busch - LSU

Theorem:

The language

is not regular

Proof:

Use the Pumping Lemma

Prof. Busch - LSU

that is a regular language

Since is infinite

we can apply the Pumping Lemma

Prof. Busch - LSU

Let be the critical length for

Pick a string such that:

and

length

We pick

Prof. Busch - LSU

From the Pumping Lemma:

we can write:

with lengths:

Thus:

Prof. Busch - LSU

From the Pumping Lemma:

Thus:

Prof. Busch - LSU

From the Pumping Lemma:

Thus:

Prof. Busch - LSU

BUT:

Prof. Busch - LSU

Therefore:

Our assumption that

is a regular language is not true

Conclusion:

is not a regular language

END OF PROOF

Prof. Busch - LSU

Non-regular languages

Regular languages

Prof. Busch - LSU

Theorem:

The language

is not regular

Proof:

Use the Pumping Lemma

Prof. Busch - LSU

that is a regular language

Since is infinite

we can apply the Pumping Lemma

Prof. Busch - LSU

Let be the critical length of

Pick a string such that:

and

length

We pick

Prof. Busch - LSU

From the Pumping Lemma:

We can write

With lengths

Thus:

Prof. Busch - LSU

From the Pumping Lemma:

Thus:

Prof. Busch - LSU

From the Pumping Lemma:

Thus:

Prof. Busch - LSU

BUT:

Prof. Busch - LSU

Therefore:

Our assumption that

is a regular language is not true

Conclusion:

is not a regular language

END OF PROOF

Prof. Busch - LSU

Non-regular languages

Regular languages

Prof. Busch - LSU

Theorem:

The language

is not regular

Proof:

Use the Pumping Lemma

Prof. Busch - LSU

that is a regular language

Since is infinite

we can apply the Pumping Lemma

Prof. Busch - LSU

Let be the critical length of

Pick a string such that:

length

We pick

Prof. Busch - LSU

From the Pumping Lemma:

We can write

With lengths

Thus:

Prof. Busch - LSU

From the Pumping Lemma:

Thus:

Prof. Busch - LSU

From the Pumping Lemma:

Thus:

Prof. Busch - LSU

Since:

There must exist such that:

Prof. Busch - LSU

However:

for

for any

Prof. Busch - LSU

BUT:

Prof. Busch - LSU

Therefore:

Our assumption that

is a regular language is not true

Conclusion:

is not a regular language

END OF PROOF

Prof. Busch - LSU