The quotient of two languages L_1/L_2 = { x | ∃y ∈ L_2 (xy ∈ L_1) } That is, the set of strings from which a string in L_1 could be created by appending a string in L_2. So, {aaa, aab, aba, abb} / {a, bb} = {aa, ab, a} Are regular languages closed with respect to quotient? What if L_1 is not regular but L_2 is? Is L_1/L_2 guaranteed to be regular? What if L_2 is not regular but L_1 is? Is L_1/L_2 guaranteed to be regular? Exhibit a nonregular language L1 and regular language L2 for which L1/L2 is nonregular.

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The quotient of two languages L_1/L_2 = { x | ∃y ∈ L_2 (xy ∈ L_1) }

That is, the set of strings from which a string in L_1 could be created by appending a string in L_2. So, {aaa, aab, aba, abb} / {a, bb} = {aa, ab, a}

Are regular languages closed with respect to quotient?

What if L_1 is not regular but L_2 is? Is L_1/L_2 guaranteed to be regular?

What if L_2 is not regular but L_1 is? Is L_1/L_2 guaranteed to be regular?

Exhibit a nonregular language L1 and regular language L2 for which L1/L2 is nonregular.

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