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Unformatted text preview: 1 Module 17 • Closure Properties of Language class LFSA – Remember ideas used in solvable languages unit – Set complement – Set intersection, union, difference, symmetric difference 2 LFSA is closed under set complement • If L is in LFSA, then L c is in LFSA • Proof – Let L be an arbitrary language in LFSA – Let M be the FSA such that L(M) = L • M exists by definition of L in LFSA – Construct FSA M’ from M – Argue L(M’) = L c – There exists an FSA M’ such that L(M’) = L c – L c is in LFSA 3 Visualization • Let L be an arbitrary language in LFSA • Let M be the FSA such that L(M) = L • M exists by definition of L in LFSA • Construct FSA M’ from M • Argue L(M’) = L c • L c is in LFSA L c L LFSA FSA’s M M’ 4 Construct FSA M’ from M • What did we do when we proved that REC, the set of solvable languages, is closed under set complement? • Construct program P’ from program P • Can we translate this to the FSA setting? 5 Construct FSA M’ from M • M = (Q, Σ , q , A, δ ) • M’ = (Q’, Σ ’, q’, A’, δ ’) – M’ should say yes when M says no – M’ should say no when M says yes – How? • Q’ = Q • Σ ’ = Σ • q’ = q • δ ’ = δ • A’ = QA 6 Example 1 2 3 a a a b b b FSA M 1 2 3 a a a b b b FSA M’ Q’ = Q Σ ’ = Σ q’ = q δ ’ = δ A’ = QA 7 Construction is an algorithm * • Set Complement Construction – Algorithm Specification • Input: FSA M • Output: FSA M’ such that L(M’) = L(M) c – Comments • This algorithm can be in any computational model....
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 Fall '07
 TORNG
 Set Theory, Naive set theory, Intersection, Symmetric difference

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