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Closure Properties of Decidability

Closure Properties of Decidability. Lecture 28 Section 3.2 Fri, Oct 26, 2007. Closure Properties of Decidable Languages. The class of decidable languages is closed under Union Intersection Concatenation Complementation Star. Closure Under Union.

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Closure Properties of Decidability

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  1. Closure Properties of Decidability Lecture 28 Section 3.2 Fri, Oct 26, 2007

  2. Closure Properties of Decidable Languages • The class of decidable languages is closed under • Union • Intersection • Concatenation • Complementation • Star

  3. Closure Under Union • Theorem: If L1 and L2 are decidable, then L1L2 is decidable. • Proof: • Let D1 be a decider for L1 and let D2 be a decider for L2. • Then build a decider D for L1L2 as in the following diagram.

  4. Closure of Intersection D w no no no D1 D2 yes yes yes

  5. Closure Under Union • Theorem: If L1 and L2 are decidable, then L1L2 is decidable. • Proof: • Let D1 be a decider for L1 and let D2 be a decider for L2. • Then build a decider D for L1L2 as in the following diagram.

  6. Closure of Intersection D w yes yes yes D1 D2 no no no

  7. Closure Properties of Recognizable Languages • The class of recognizable languages is closed under • Union • Intersection • Concatenation • Star

  8. Closure Under Union • Theorem: If L1 and L2 are recognizable, then L1L2 is recognizable. • Proof: • Let R1 be a recognizer for L1 and let R2 be a recognizer for L2. • Then build a recognizer R for L1L2 as in the following diagram.

  9. Closure of Intersection R w yes yes yes R1 R2

  10. Closure Under Union • Theorem: If L1 and L2 are recognizable, then L1L2 is recognizable. • Proof: • Let R1 be a recognizer for L1 and let R2 be a recognizer for L2. • Then build a recognizer R for L1L2 as in the following diagram.

  11. R yes R1 w yes yes R2 Closure of Union

  12. Closure of Union • In that diagram, we must be careful to alternate execution between R1 and R2.

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