# regular pumping examples

Post on 24-Dec-2015

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• More Applications

of

the Pumping Lemma

Costas Busch - RPI

• 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:

Costas Busch - RPI

• Regular languagesNon-regular languages

Costas Busch - RPI

• Theorem:The languageis not regularProof:Use the Pumping Lemma

Costas Busch - RPI

• Assume for contradictionthat is a regular languageSince is infinitewe can apply the Pumping Lemma

Costas Busch - RPI

• We pickLet be the critical length forPick a string such that: lengthand

Costas Busch - RPI

• we can write:with lengths:From the Pumping Lemma: Thus:

Costas Busch - RPI

• From the Pumping Lemma:Thus:

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• From the Pumping Lemma:Thus:

Costas Busch - RPI

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• Our assumption thatis a regular language is not trueConclusion:is not a regular languageTherefore:END OF PROOF

Costas Busch - RPI

• Regular languagesNon-regular languages

Costas Busch - RPI

• Theorem:The languageis not regularProof:Use the Pumping Lemma

Costas Busch - RPI

• Assume for contradictionthat is a regular languageSince is infinitewe can apply the Pumping Lemma

Costas Busch - RPI

• We pickLet be the critical length of Pick a string such that: lengthand

Costas Busch - RPI

• We can writeWith lengthsFrom the Pumping Lemma:

Thus:

Costas Busch - RPI

• From the Pumping Lemma:Thus:

Costas Busch - RPI

• From the Pumping Lemma: Thus:

Costas Busch - RPI

Costas Busch - RPI

• Our assumption thatis a regular language is not trueConclusion:is not a regular languageTherefore:END OF PROOF

Costas Busch - RPI

• Regular languagesNon-regular languages

Costas Busch - RPI

• Theorem:The languageis not regularProof:Use the Pumping Lemma

Costas Busch - RPI

• Assume for contradictionthat is a regular languageSince is infinitewe can apply the Pumping Lemma

Costas Busch - RPI

• We pickLet be the critical length ofPick a string such that: length

Costas Busch - RPI

• We can writeWith lengthsFrom the Pumping Lemma:

Thus:

Costas Busch - RPI

• From the Pumping Lemma:Thus:

Costas Busch - RPI

• From the Pumping Lemma: Thus:

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• Since:There must exist such that:

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• However:forfor any

Costas Busch - RPI