# costas busch1 more applications of the pumping lemma

Post on 15-Jan-2016

220 views

Category:

## Documents

Tags:

• #### possible locationsof

Embed Size (px)

TRANSCRIPT

• 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

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