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## PowerPoint Slideshow about ' Examples' - kaden-bradshaw

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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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Summary of Proof Techniques Employed Impossibility Proofs Reduction Proofs : Closure Properties

- Counter Examples
- Constructions/Simulations
- Induction Proofs

- Proofs by Contradiction

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