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6. Prove or disprove the following conjecture. If M = (Q, Σ, δ, q0, F) is...

6. Prove or disprove the following conjecture. If M = (Q, Σ, δ, q0, F) is a minimal dfa for a regular language L, then M̂ = (Q, Σ, δ, q0, Q − F) is a minimal dfa for L¯.

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Answer #1

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  • Lets say that is not a minimal DFA of .
  • Then there must be another DFA P which accepts .
  • Since is not minimal , P will have lesser number of states than .
  • We can swap the final states to non-final and non-final to final for P so that the resultant DFA accepts which is equivalent to L.
  • So now recognizes L.
  • we know that and P have the same number of states.
  • And and M also have the same number of states.
  • We assumed that P will have the lesser number of states than .
  • Then we can say that will have lesser states than M.
  • But recognizes L , which is a contradiction to our assumption. So M is a minimal DFA for L and is minimal DFA for .
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