Closure Properties of Regular Languages
Transcript of Closure Properties of Regular Languages
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Closure Properties of Regular LanguagesLecture 13Section 4.1
Robb T. Koether
Hampden-Sydney College
Wed, Sep 21, 2016
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Outline
1 Closure Properties of Regular Languages
2 Additional Closure Properties
3 Examples
4 Right Quotients
5 Example
6 Assignment
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Outline
1 Closure Properties of Regular Languages
2 Additional Closure Properties
3 Examples
4 Right Quotients
5 Example
6 Assignment
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Closure
Theorem (Closure Properties of Regular Languages)The class of regular languages is closed under the operations ofcomplementation, union, concatenation, and Kleene star.
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Closure
Proof for unions.
A
B
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Closure
Proof for unions.
A
B
λ
λ λ
λ
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Closure
Proof for concatenations.
A B
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Closure
Proof for concatenations.
A Bλ λ λ
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Closure
Proof for Kleene star.
A
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Closure
Proof for Kleene star.
Aλ
λ
λ
λ
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Closure
Example (Closure)A DFA for the language (A ∪ BC)∗.
A
B C
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Closure
Example (Closure)A DFA for the language (A ∪ BC)∗.
A
B Cλλ λ
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Closure
Example (Closure)A DFA for the language (A ∪ BC)∗.
A
B Cλλ λ
λ
λ
λ
λ
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Closure
Example (Closure)A DFA for the language (A ∪ BC)∗.
A
B Cλλ λ
ε
λ
λ
λ
λ
λ
λ
λ
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Closure
Example (Closure)If A = {a}, B = {b}, and C = {c}, then we have.
A
B Cλλ λ
λ
λ
λ
λ
λ
λ
λ
λ
a
cb
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Closure
Example (Closure)The equivalent DFA is.
a
b
a
b
b
c
a
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Closure
Example (Closure)This can be minimized to.
a
b
b c
a
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Outline
1 Closure Properties of Regular Languages
2 Additional Closure Properties
3 Examples
4 Right Quotients
5 Example
6 Assignment
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More Closure Properties
CorollaryThe set of regular languages is closed under intersection and setdifference.
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Outline
1 Closure Properties of Regular Languages
2 Additional Closure Properties
3 Examples
4 Right Quotients
5 Example
6 Assignment
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Example (Intersection)Let Σ = {a,b} and
L1 = {w | w contains aba}L2 = {w | w contains bab}
Design a DFA for L1 ∩ L2.Design a DFA for L1 − L2.
Design a DFA for L1 ∪ L2.
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Example (Intersection)Let Σ = {a,b} and
L1 = {w | w contains aba}L2 = {w | w contains bab}
Design a DFA for L1 ∩ L2.Design a DFA for L1 − L2.Design a DFA for L1 ∪ L2.
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Outline
1 Closure Properties of Regular Languages
2 Additional Closure Properties
3 Examples
4 Right Quotients
5 Example
6 Assignment
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Right Quotients
Definition (Right Quotient)Let L1 and L2 be languages on an alphabet Σ. The right quotient of L1with L2 is
L1/L2 = {x | xy ∈ L1 for some y ∈ L2}.
TheoremIf L1 and L2 are regular languages, then L1/L2 is regular.
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Right Quotients
Proof.Let L1 = L(M) and M = (Q,Σ, δ,q0,F ).
Define M ′ = (Q,Σ, δ,q0,F ′) with F ′ defined as follows.
For each qi ∈ Q, let Mi = (Q,Σ, δ,qi ,F ).Determine whether L(Mi ) ∩ L2 = ∅.
If L(Mi) ∩ L2 6= ∅, then qi ∈ F ′.If L(Mi) ∩ L2 = ∅, then qi /∈ F ′.
The idea is thatx goes from q0 to qi for some qi ∈ Q.y goes from qi to qf for some qf ∈ F and y ∈ L2.
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Right Quotients
Proof.Let L1 = L(M) and M = (Q,Σ, δ,q0,F ).Define M ′ = (Q,Σ, δ,q0,F ′) with F ′ defined as follows.
For each qi ∈ Q, let Mi = (Q,Σ, δ,qi ,F ).Determine whether L(Mi ) ∩ L2 = ∅.
If L(Mi) ∩ L2 6= ∅, then qi ∈ F ′.If L(Mi) ∩ L2 = ∅, then qi /∈ F ′.
The idea is thatx goes from q0 to qi for some qi ∈ Q.y goes from qi to qf for some qf ∈ F and y ∈ L2.
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Right Quotients
Proof.Let L1 = L(M) and M = (Q,Σ, δ,q0,F ).Define M ′ = (Q,Σ, δ,q0,F ′) with F ′ defined as follows.
For each qi ∈ Q, let Mi = (Q,Σ, δ,qi ,F ).
Determine whether L(Mi ) ∩ L2 = ∅.
If L(Mi) ∩ L2 6= ∅, then qi ∈ F ′.If L(Mi) ∩ L2 = ∅, then qi /∈ F ′.
The idea is thatx goes from q0 to qi for some qi ∈ Q.y goes from qi to qf for some qf ∈ F and y ∈ L2.
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Right Quotients
Proof.Let L1 = L(M) and M = (Q,Σ, δ,q0,F ).Define M ′ = (Q,Σ, δ,q0,F ′) with F ′ defined as follows.
For each qi ∈ Q, let Mi = (Q,Σ, δ,qi ,F ).Determine whether L(Mi ) ∩ L2 = ∅.
If L(Mi) ∩ L2 6= ∅, then qi ∈ F ′.If L(Mi) ∩ L2 = ∅, then qi /∈ F ′.
The idea is thatx goes from q0 to qi for some qi ∈ Q.y goes from qi to qf for some qf ∈ F and y ∈ L2.
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Right Quotients
Proof.Let L1 = L(M) and M = (Q,Σ, δ,q0,F ).Define M ′ = (Q,Σ, δ,q0,F ′) with F ′ defined as follows.
For each qi ∈ Q, let Mi = (Q,Σ, δ,qi ,F ).Determine whether L(Mi ) ∩ L2 = ∅.
If L(Mi) ∩ L2 6= ∅, then qi ∈ F ′.
If L(Mi) ∩ L2 = ∅, then qi /∈ F ′.
The idea is thatx goes from q0 to qi for some qi ∈ Q.y goes from qi to qf for some qf ∈ F and y ∈ L2.
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Right Quotients
Proof.Let L1 = L(M) and M = (Q,Σ, δ,q0,F ).Define M ′ = (Q,Σ, δ,q0,F ′) with F ′ defined as follows.
For each qi ∈ Q, let Mi = (Q,Σ, δ,qi ,F ).Determine whether L(Mi ) ∩ L2 = ∅.
If L(Mi) ∩ L2 6= ∅, then qi ∈ F ′.If L(Mi) ∩ L2 = ∅, then qi /∈ F ′.
The idea is thatx goes from q0 to qi for some qi ∈ Q.y goes from qi to qf for some qf ∈ F and y ∈ L2.
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Right Quotients
Proof.Let L1 = L(M) and M = (Q,Σ, δ,q0,F ).Define M ′ = (Q,Σ, δ,q0,F ′) with F ′ defined as follows.
For each qi ∈ Q, let Mi = (Q,Σ, δ,qi ,F ).Determine whether L(Mi ) ∩ L2 = ∅.
If L(Mi) ∩ L2 6= ∅, then qi ∈ F ′.If L(Mi) ∩ L2 = ∅, then qi /∈ F ′.
The idea is thatx goes from q0 to qi for some qi ∈ Q.y goes from qi to qf for some qf ∈ F and y ∈ L2.
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Outline
1 Closure Properties of Regular Languages
2 Additional Closure Properties
3 Examples
4 Right Quotients
5 Example
6 Assignment
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Right Quotients
Example (Right Quotients)Let Σ = {a,b} and
L1 = L(aba∗)L2 = L(b∗a)
That is,
L1 = {ab,aba,abaa,abaaa, . . .}L2 = {a,ba,bba,bbba, . . .}
Use the construction in the proof to create a DFA for L1/L2.
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Outline
1 Closure Properties of Regular Languages
2 Additional Closure Properties
3 Examples
4 Right Quotients
5 Example
6 Assignment
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Assignment
AssignmentSection 4.1 Exercises 1a, 2, 4, 6b, 11, 12, 14, 16, 20.Is the family of regular languages closed under infinite union?That is, if L1,L2,L3, . . . are regular languages, is L1 ∪ L2 ∪ L3 ∪ · · ·necessarily a regular language?What about infinite intersections of regular languages? MustL1 ∩ L2 ∩ L3 ∩ · · · be regular?
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