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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**Assume for contradiction**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:**CONTRADICTION!!! 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**Assume for contradiction**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:**CONTRADICTION!!! 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**Assume for contradiction**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:**CONTRADICTION!!! 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