1. Let Σ be an alphabet. Prove or disprove that for every language L over alphabet Σ, if L2 = L, then L+ = L.

A language L over an alphabet Σ is co-finite, if Σ∗ \L is empty or finite. LetCOFNFA = {〈N 〉|N is a NFA accepting a co-finite language}.Show that COFNFA is decidable.

Question 1 Let LC * be a regular language over some alphabet Σ. The Kleene-closure L* of L is given by {V1V2 • Vn | n ≥ 0, V1, V2, . . . , Vn € L}. (a) Suppose DFA D = (Q, Σ, 8, qo, F) satisfies L(D) = L. Construct an NFA N = (Q', Σ, 8', %, F') such that L(N) = L*. L* (b) Prove by induction on n that each word v₁v2 Un with n ≥ 0, V₁, V2, ..., Un E L is accepted by the NFA N proposed in item (a). (c) Prove that each word accepted/recognized by the NFA N proposed in item (a) is a word of the language L*.

Let L be a language over the alphabet Σ = {a, b, c}. Prove that L = {aªbman for any positive integers n ‡m} is not regular. Proof: Assume that L is regular, so that the pumping lemma applies. Therefore L has a pumping length p. Let Since s belongs to L and S = This string is a member of L because has length longer than p, there exist strings x, y, and z such that s = xyz, where xz is a member of L, xyz is a member of L for all positive integers i, − [y] >0, and − |xy| ≤ p. Now consider the following argument: (You finish the rest of the proof.)