Pushdown Automaton: CFL and the pumping lemma
Pushdown Automaton: CFL and the pumping lemma
Pushdown Automaton: CFL and the pumping lemma
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Complement<br />
Theorem: <strong>CFL</strong> are not closed under complement.<br />
Proof: by contradiction.<br />
• Take two <strong>CFL</strong>s L 1 <strong>and</strong> L 2 <strong>and</strong> assume that ~L 1<br />
<strong>and</strong> ~L 2 are <strong>CFL</strong>. (~L denotes complement)<br />
• <strong>CFL</strong> are closed under union operation so<br />
~L 1 ~L 2 is a <strong>CFL</strong>.<br />
• Using our assumption again we get that<br />
~(~L 1 ~L 2 ) is a <strong>CFL</strong> as well.<br />
• But ~(~L 1 ~L 2 ) =L 1 L 2 <strong>and</strong> we already know<br />
that <strong>CFL</strong>s are not closed under intersection.<br />
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