Examples

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Examples

• Applying Pumping Lemma

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• Proof by contradiction
• 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 as 0
  times (effectively deleting it), the number of
  as can be made smaller than the number of bs.
• 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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Summary of Proof Techniques Employed

• Counter Examples
• Constructions/Simulations
• Induction Proofs
• Impossibility Proofs
• Proofs by Contradiction
• Reduction Proofs Closure Properties