Costas Busch1 More Applications of The Pumping Lemma

Costas Busch2 The Pumping Lemma: there exists an integer such that for any string we can write For infinite context-free language with lengths and it must be:

Costas Busch3 Context-free languages Non-context free languages

Costas Busch4 Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages

Costas Busch5 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma

Costas Busch6 Pumping Lemma gives a magic number such that: Pick any string of with length at least we pick:

Costas Busch7 We can write: with lengths and Pumping Lemma says: for all

Costas Busch8 We examine all the possible locations of string in

Costas Busch9 Case 1: is within the first

Costas Busch10 Case 1: is within the first

Costas Busch11 Case 1: is within the first

Costas Busch12 Case 1: is within the first Contradiction!!! However, from Pumping Lemma:

Costas Busch13 is in the first Case 2:

Costas Busch14 is in the first Case 2:

Costas Busch15 is in the first Case 2:

Costas Busch16 is in the first Case 2: Contradiction!!! However, from Pumping Lemma:

Costas Busch17 is in the first overlaps the first Case 3:

Costas Busch18 is in the first overlaps the first Case 3:

Costas Busch19 is in the first overlaps the first Case 3:

Costas Busch20 is in the first overlaps the first Case 3: Contradiction!!! However, from Pumping Lemma:

Costas Busch21 Overlaps the first in the first Case 4: Analysis is similar to case 3

Costas Busch22 Other cases: is within or Analysis is similar to case 1:

Costas Busch23 More cases: overlaps or Analysis is similar to cases 2,3,4:

Costas Busch24 Since, it is impossible to overlap: There are no other cases to consider nor

Costas Busch25 In all cases we obtained a contradiction Therefore: The original assumption that is context-free must be wrong Conclusion:is not context-free

Costas Busch26 Context-free languages Non-context free languages

Costas Busch27 Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages

Costas Busch28 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma

Costas Busch29 Pumping Lemma gives a magic number such that: Pick any string of with length at least we pick:

Costas Busch30 We can write: with lengths and Pumping Lemma says: for all

Costas Busch31 We examine all the possible locations of string in There is only one case to consider

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Costas Busch36 Since, for we have:

Costas Busch37

Costas Busch38 Contradiction!!! However, from Pumping Lemma:

Costas Busch39 We obtained a contradiction Therefore: The original assumption that is context-free must be wrong Conclusion:is not context-free

Costas Busch40 Context-free languages Non-context free languages

Costas Busch41 Theorem: The language is not context free Proof: Use the Pumping Lemma for context-free languages

Costas Busch42 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma

Costas Busch43 Pumping Lemma gives a magic number such that: Pick any string of with length at least we pick:

Costas Busch44 We can write: with lengths and Pumping Lemma says: for all

Costas Busch45 We examine all the possible locations of string in

Costas Busch46 Most complicated case: is in

Costas Busch47

Costas Busch48 Most complicated sub-case:and

Costas Busch49 Most complicated sub-case:and

Costas Busch50 Most complicated sub-case:and

Costas Busch51 and

Costas Busch52

Costas Busch53 However, from Pumping Lemma: Contradiction!!!

Costas Busch54 When we examine the rest of the cases we also obtain a contradiction

Costas Busch55 In all cases we obtained a contradiction Therefore: The original assumption that is context-free must be wrong Conclusion:is not context-free