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1 More Applications of the Pumping Lemma

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

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3 Regular languages Non-regular languages

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4 Theorem: The language is not regular Proof: Use the Pumping Lemma

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5 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

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6 We pick Let be the integer in the Pumping Lemma Pick a string such that: length and

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7 Write it must be that length From the Pumping Lemma Thus:

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8 From the Pumping Lemma: Thus:

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9 From the Pumping Lemma: Thus:

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10 BUT: CONTRADICTION!!!

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11 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore:

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12 Regular languages Non-regular languages

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13 Theorem: The language is not regular Proof: Use the Pumping Lemma

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14 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

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15 We pick Let be the integer in the Pumping Lemma Pick a string such that: length and

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16 Write it must be that length From the Pumping Lemma Thus:

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17 From the Pumping Lemma: Thus:

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18 From the Pumping Lemma: Thus:

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19 BUT: CONTRADICTION!!!

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20 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore:

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21 Regular languages Non-regular languages

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22 Theorem: The language is not regular Proof: Use the Pumping Lemma

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23 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

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24 We pick Let be the integer in the Pumping Lemma Pick a string such that: length

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25 Write it must be that length From the Pumping Lemma Thus:

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26 From the Pumping Lemma: Thus:

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27 From the Pumping Lemma: Thus:

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28 Since: There must exist such that:

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29 However:for for any

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30 BUT: CONTRADICTION!!!

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31 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore:

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