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Examples

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

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  1. Examples Applying Pumping Lemma L11PLEG

  2. Proof by contradiction: • Let be accepted by a k-state DFA. • Choose • For all prefixes of length • show there exists such that • i.e., L11PLEG

  3. 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. L11PLEG

  4. 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. L11PLEG

  5. 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. L11PLEG

  6. 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. L11PLEG

  7. … Source of the problem? Regular (ultimately periodic) Prime (sparse) … Composite(dense) … L11PLEG

  8. Summary of Proof Techniques Employed • Counter Examples • Constructions/Simulations • Induction Proofs • Impossibility Proofs • Proofs by Contradiction • Reduction Proofs : Closure Properties L11PLEG

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