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# Examples - PowerPoint PPT Presentation

Examples. Applying Pumping Lemma. Proof by contradiction: Let be accepted by a k -state DFA. Choose For all prefixes of length show there exists such that i.e.,. Choose

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### Examples

Applying Pumping Lemma

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• Let be accepted by a k-state DFA.

• Choose

• For all prefixes of length

• show there exists such that

• i.e.,

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• Choose

(For this specific problem happens to be independent of j, but that need not always be the case.)

• is non-regular because it violates the necessary condition.

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Proof :(For this example, choice of initial string is crucial.)

• For this choice of s, the pumping lemma cannot generate a contradiction!

• However, let instead.

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• For

• Thus, by pumping the substring containing a’s 0 times (effectively deleting it), the number of a’s can be made smaller than the number of b’s.

• So, by pumping lemma, L is non-regular.

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• Proof by contradiction:

• If is regular, then so is , the complement of

• But which is known to be non-regular.

• So, cannot be regular.

• Proving to be non-regular using pumping lemma may be difficult/impossible.

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Source of the problem?

Regular (ultimately periodic)

Prime (sparse)

Composite(dense)

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• Counter Examples

• Constructions/Simulations

• Induction Proofs

• Impossibility Proofs

• Proofs by Contradiction

• Reduction Proofs : Closure Properties

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