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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

Lecture 28

Section 3.2

Fri, Oct 26, 2007

• The class of decidable languages is closed under

• Union

• Intersection

• Concatenation

• Complementation

• Star

• 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.

D

w

no

no

no

D1

D2

yes

yes

yes

• 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.

D

w

yes

yes

yes

D1

D2

no

no

no

• The class of recognizable languages is closed under

• Union

• Intersection

• Concatenation

• Star

• 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.

R

w

yes

yes

yes

R1

R2

• 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.

yes

R1

w

yes

yes

R2

Closure of Union

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