Closure Properties of Regular Languages

Closure Properties of Regular Languages Lecture 13 Section 4.1 Robb T. Koether Hampden-Sydney College Wed, Sep 21, 2016 Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 1 / 28

Outline 1 Closure Properties of Regular Languages 2 Additional Closure Properties 3 Examples 4 Right Quotients 5 Example 6 Assignment Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 2 / 28

Closure Theorem (Closure Properties of Regular Languages) The class of regular languages is closed under the operations of complementation, union, concatenation, and Kleene star. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 4 / 28

Closure Proof for unions. A B Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 5 / 28

Closure Proof for unions. A B Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 6 / 28

Closure Proof for concatenations. A B Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 7 / 28

Closure Proof for concatenations. A B Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 8 / 28

Closure Proof for Kleene star. A Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 9 / 28

Closure Proof for Kleene star. A Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 10 / 28

Closure Example (Closure) A DFA for the language (A BC). A B C Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 11 / 28

Closure Example (Closure) A DFA for the language (A BC). A B C Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 12 / 28

Closure Example (Closure) A DFA for the language (A BC). A B C Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 13 / 28

Closure Example (Closure) A DFA for the language (A BC). ε A B C Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 14 / 28

Closure Example (Closure) If A = {a}, B = {b}, and C = {c}, then we have. A B C a c b Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 15 / 28

Closure Example (Closure) The equivalent DFA is. a a b a b c b Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 16 / 28

Closure Example (Closure) This can be minimized to. a a b c b Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 17 / 28

More Closure Properties Corollary The set of regular languages is closed under intersection and set difference. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 19 / 28

Example (Intersection) Let Σ = {a, b} and L 1 = {w w contains aba} L 2 = {w w contains bab} Design a DFA for L 1 L 2. Design a DFA for L 1 L 2. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 21 / 28

Example (Intersection) Let Σ = {a, b} and L 1 = {w w contains aba} L 2 = {w w contains bab} Design a DFA for L 1 L 2. Design a DFA for L 1 L 2. Design a DFA for L 1 L 2. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 21 / 28

Right Quotients Definition (Right Quotient) Let L 1 and L 2 be languages on an alphabet Σ. The right quotient of L 1 with L 2 is L 1 /L 2 = {x xy L 1 for some y L 2 }. Theorem If L 1 and L 2 are regular languages, then L 1 /L 2 is regular. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 23 / 28

Right Quotients Proof. Let L 1 = L(M) and M = (Q, Σ, δ, q 0, F ). Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 24 / 28

Right Quotients Proof. Let L 1 = L(M) and M = (Q, Σ, δ, q 0, F ). Define M = (Q, Σ, δ, q 0, F ) with F defined as follows. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 24 / 28

Right Quotients Proof. Let L 1 = L(M) and M = (Q, Σ, δ, q 0, F ). Define M = (Q, Σ, δ, q 0, F ) with F defined as follows. For each q i Q, let M i = (Q, Σ, δ, q i, F ). Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 24 / 28

Right Quotients Proof. Let L 1 = L(M) and M = (Q, Σ, δ, q 0, F ). Define M = (Q, Σ, δ, q 0, F ) with F defined as follows. For each q i Q, let M i = (Q, Σ, δ, q i, F ). Determine whether L(M i ) L 2 =. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 24 / 28

Right Quotients Proof. Let L 1 = L(M) and M = (Q, Σ, δ, q 0, F ). Define M = (Q, Σ, δ, q 0, F ) with F defined as follows. For each q i Q, let M i = (Q, Σ, δ, q i, F ). Determine whether L(M i ) L 2 =. If L(M i ) L 2, then q i F. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 24 / 28

Right Quotients Proof. Let L 1 = L(M) and M = (Q, Σ, δ, q 0, F ). Define M = (Q, Σ, δ, q 0, F ) with F defined as follows. For each q i Q, let M i = (Q, Σ, δ, q i, F ). Determine whether L(M i ) L 2 =. If L(M i ) L 2, then q i F. If L(M i ) L 2 =, then q i / F. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 24 / 28

Right Quotients Proof. Let L 1 = L(M) and M = (Q, Σ, δ, q 0, F ). Define M = (Q, Σ, δ, q 0, F ) with F defined as follows. For each q i Q, let M i = (Q, Σ, δ, q i, F ). Determine whether L(M i ) L 2 =. If L(M i ) L 2, then q i F. If L(M i ) L 2 =, then q i / F. The idea is that x goes from q 0 to q i for some q i Q. y goes from q i to q f for some q f F and y L 2. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 24 / 28

Right Quotients Example (Right Quotients) Let Σ = {a, b} and L 1 = L(aba ) L 2 = L(b a) That is, L 1 = {ab, aba, abaa, abaaa,...} L 2 = {a, ba, bba, bbba,...} Use the construction in the proof to create a DFA for L 1 /L 2. Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 26 / 28

Assignment Assignment Section 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 L 1, L 2, L 3,... are regular languages, is L 1 L 2 L 3 necessarily a regular language? What about infinite intersections of regular languages? Must L 1 L 2 L 3 be regular? Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages Wed, Sep 21, 2016 28 / 28