Prof. Busch - LSU1 More Applications of the Pumping Lemma

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:

Prof. Busch - LSU3 Regular languages Non-regular languages

Prof. Busch - LSU4 Theorem: The language is not regular Proof: Use the Pumping Lemma

Prof. Busch - LSU5 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

Prof. Busch - LSU6 We pick Let be the critical length for Pick a string such that: lengthand

Prof. Busch - LSU7 we can write: with lengths: From the Pumping Lemma: Thus:

Prof. Busch - LSU8 From the Pumping Lemma: Thus:

Prof. Busch - LSU9 From the Pumping Lemma: Thus:

Prof. Busch - LSU10 BUT: CONTRADICTION!!!

Prof. Busch - LSU11 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore: END OF PROOF

Prof. Busch - LSU12 Regular languages Non-regular languages

Prof. Busch - LSU13 Theorem: The language is not regular Proof: Use the Pumping Lemma

Prof. Busch - LSU14 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

Prof. Busch - LSU15 We pick Let be the critical length of Pick a string such that: length and

Prof. Busch - LSU16 We can write With lengths From the Pumping Lemma: Thus:

Prof. Busch - LSU17 From the Pumping Lemma: Thus:

Prof. Busch - LSU18 From the Pumping Lemma: Thus:

Prof. Busch - LSU19 BUT: CONTRADICTION!!!

Prof. Busch - LSU20 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore: END OF PROOF

Prof. Busch - LSU21 Regular languages Non-regular languages

Prof. Busch - LSU22 Theorem: The language is not regular Proof: Use the Pumping Lemma

Prof. Busch - LSU23 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

Prof. Busch - LSU24 We pick Let be the critical length of Pick a string such that: length

Prof. Busch - LSU25 We can write With lengths From the Pumping Lemma: Thus:

Prof. Busch - LSU26 From the Pumping Lemma: Thus:

Prof. Busch - LSU27 From the Pumping Lemma: Thus:

Prof. Busch - LSU28 Since: There must exist such that:

Prof. Busch - LSU29 However:for for any

Prof. Busch - LSU30 BUT: CONTRADICTION!!!

Prof. Busch - LSU31 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore: END OF PROOF