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Fall 2003 Costas Busch 1 More Applications of The Pumping Lemma

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how to design pumping lemma for the given string

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  • More Applicationsof The Pumping Lemma

    Costas Busch

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

    Costas Busch

  • Context-free languagesNon-context free languages

    Costas Busch

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

    Costas Busch

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

    Costas Busch

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

    Costas Busch

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

    Costas Busch

  • We examine all the possible locationsof string in

    Costas Busch

  • Case 1:is within the first

    Costas Busch

  • Case 1:is within the first

    Costas Busch

  • Case 1:is within the first

    Costas Busch

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

    Costas Busch

  • is in the firstis in the firstCase 2:

    Costas Busch

  • is in the firstis in the firstCase 2:

    Costas Busch

  • is in the firstis in the firstCase 2:

    Costas Busch

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

    Costas Busch

  • is in the firstoverlaps the firstCase 3:

    Costas Busch

  • is in the firstoverlaps the firstCase 3:

    Costas Busch

  • is in the firstoverlaps the firstCase 3:

    Costas Busch

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

    Costas Busch

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

    Costas Busch

  • Other cases:is withinororAnalysis is similar to case 1:

    Costas Busch

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

    Costas Busch

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

    Costas Busch

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

    Costas Busch

  • Context-free languagesNon-context free languages

    Costas Busch

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

    Costas Busch

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

    Costas Busch

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

    Costas Busch

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

    Costas Busch

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

    Costas Busch

  • Costas Busch

  • Costas Busch

  • Costas Busch

  • Costas Busch

  • Since , for we have:

    Costas Busch

  • Costas Busch

  • Contradiction!!!However, from Pumping Lemma:

    Costas Busch

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

    Costas Busch

  • Context-free languagesNon-context free languages

    Costas Busch

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

    Costas Busch

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

    Costas Busch

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

    Costas Busch

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

    Costas Busch

  • We examine all the possible locations

    of string in

    Costas Busch

  • Most complicated case:is inis in

    Costas Busch

  • Costas Busch

  • Most complicated sub-case:and

    Costas Busch

  • Most complicated sub-case:and

    Costas Busch

  • Most complicated sub-case:and

    Costas Busch

  • and

    Costas Busch

  • Costas Busch

  • However, from Pumping Lemma:Contradiction!!!

    Costas Busch

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

    Costas Busch

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

    Costas Busch


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