# prof. busch - lsu1 more applications of the pumping lemma

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• Slide 1
• Prof. Busch - LSU1 More Applications of the Pumping Lemma
• Slide 2
• Prof. Busch - LSU2 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:
• Slide 3
• Prof. Busch - LSU3 Regular languages Non-regular languages
• Slide 4
• Prof. Busch - LSU4 Theorem: The language is not regular Proof: Use the Pumping Lemma
• Slide 5
• Prof. Busch - LSU5 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma
• Slide 6
• Prof. Busch - LSU6 We pick Let be the critical length for Pick a string such that: lengthand
• Slide 7
• Prof. Busch - LSU7 we can write: with lengths: From the Pumping Lemma: Thus:
• Slide 8
• Prof. Busch - LSU8 From the Pumping Lemma: Thus:
• Slide 9
• Prof. Busch - LSU9 From the Pumping Lemma: Thus:
• Slide 10
• Prof. Busch - LSU10 BUT: CONTRADICTION!!!
• Slide 11
• Prof. Busch - LSU11 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore: END OF PROOF
• Slide 12
• Prof. Busch - LSU12 Regular languages Non-regular languages
• Slide 13
• Prof. Busch - LSU13 Theorem: The language is not regular Proof: Use the Pumping Lemma
• Slide 14
• Prof. Busch - LSU14 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma
• Slide 15
• Prof. Busch - LSU15 We pick Let be the critical length of Pick a string such that: length and
• Slide 16
• Prof. Busch - LSU16 We can write With lengths From the Pumping Lemma: Thus:
• Slide 17
• Prof. Busch - LSU17 From the Pumping Lemma: Thus:
• Slide 18
• Prof. Busch - LSU18 From the Pumping Lemma: Thus:
• Slide 19
• Prof. Busch - LSU19 BUT: CONTRADICTION!!!
• Slide 20
• Prof. Busch - LSU20 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore: END OF PROOF
• Slide 21
• Prof. Busch - LSU21 Regular languages Non-regular languages
• Slide 22
• Prof. Busch - LSU22 Theorem: The language is not regular Proof: Use the Pumping Lemma
• Slide 23
• Prof. Busch - LSU23 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma
• Slide 24
• Prof. Busch - LSU24 We pick Let be the critical length of Pick a string such that: length
• Slide 25
• Prof. Busch - LSU25 We can write With lengths From the Pumping Lemma: Thus:
• Slide 26
• Prof. Busch - LSU26 From the Pumping Lemma: Thus:
• Slide 27
• Prof. Busch - LSU27 From the Pumping Lemma: Thus:
• Slide 28
• Prof. Busch - LSU28 Since: There must exist such that:
• Slide 29
• Prof. Busch - LSU29 However:for for any
• Slide 30
• Prof. Busch - LSU30 BUT: CONTRADICTION!!!
• Slide 31
• Prof. Busch - LSU31 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore: END OF PROOF