Context Free Pumping Examples

More Applicationsof The Pumping Lemma

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The Pumping Lemma:there exists an integer such that for any string we can writeFor infinite context-free language with lengthsand it must be:

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Context-free languagesNon-context free languages

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Theorem:The languageis not context freeProof:Use the Pumping Lemmafor context-free languages

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Assume for contradiction thatis context-freeSince is context-free and infinitewe can apply the pumping lemma

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Pumping Lemma gives a magic numbersuch that: Pick any string of with length at least we pick:

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We can write:with lengths andPumping Lemma says:for all

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We examine all the possible locationsof string in

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Case 1:is within the first

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Case 1:is within the first

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Case 1:is within the first

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Case 1:is within the firstContradiction!!!However, from Pumping Lemma:

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is in the firstis in the firstCase 2:

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is in the firstis in the firstCase 2:

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is in the firstis in the firstCase 2:

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is in the firstis in the firstCase 2:Contradiction!!!However, from Pumping Lemma:

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is in the firstoverlaps the firstCase 3:

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is in the firstoverlaps the firstCase 3:

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is in the firstoverlaps the firstCase 3:

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is in the firstoverlaps the firstCase 3:Contradiction!!!However, from Pumping Lemma:

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Overlaps the firstin the firstCase 4:Analysis is similar to case 3

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Other cases:is withinororAnalysis is similar to case 1:

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More cases:overlapsorAnalysis is similar to cases 2,3,4:

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Since , it is impossible to overlap:There are no other cases to considernornor

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In all cases we obtained a contradictionTherefore:The original assumption that is context-free must be wrongConclusion:is not context-free

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Context-free languagesNon-context free languages

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Theorem:The languageis not context freeProof:Use the Pumping Lemmafor context-free languages

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Assume for contradiction thatis context-freeSince is context-free and infinitewe can apply the pumping lemma

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Pumping Lemma gives a magic numbersuch that: Pick any string of with length at least we pick:

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We can write:with lengths andPumping Lemma says:for all

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We examine all the possible locationsof string in There is only one case to consider

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Since , for we have:

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Contradiction!!!However, from Pumping Lemma:

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We obtained a contradictionTherefore:The original assumption that is context-free must be wrongConclusion:is not context-free

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Context-free languagesNon-context free languages

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Theorem:The languageis not context freeProof:Use the Pumping Lemmafor context-free languages

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Assume for contradiction thatis context-freeSince is context-free and infinitewe can apply the pumping lemma

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Pumping Lemma gives a magic numbersuch that: Pick any string of with length at least we pick:

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We can write:with lengths andPumping Lemma says:for all

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We examine all the possible locations

of string in

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Most complicated case:is inis in

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Most complicated sub-case:and

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Most complicated sub-case:and

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Most complicated sub-case:and

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and

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However, from Pumping Lemma:Contradiction!!!

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When we examine the rest of the caseswe also obtain a contradiction

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In all cases we obtained a contradictionTherefore:The original assumption that is context-free must be wrongConclusion:is not context-free

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