Consider the following languages over the alphabet Σ = {a, b}: 1. L1 = {xwx : x, w ∈ Σ∗} 2. L2 = {xwx : x, w ∈ Σ∗, |w| = 1} 3. L3 = {xwx : x, w ∈ Σ∗, |x| = 1} For each language, decide whether it is regular or not regular. Prove each of your claims. If you construct a FA or a regular expression, you do not need to prove the correctness of your construction.

Let B(x) mean x is a bird, let W(x) mean x is a worm, and let E(x,y) mean x eats y. Find an English sentence to describe the following expression: Vx Vy (E(x, y) –→ B(x) ^ W (y))

Given N = (Q, {a, b}, δ, q0, F) is an NFA that recognizes language L. Define a new language concatenate(L) that is formed by concatenating symbol a to the end of every string in L. That is, concatenatea(L) = {wa | w ∈ L(N)}. Please construct an NFA N′ with exactly one accepting state that recognizes language concatenate(L). All you need to do is fill in the missing parts in the definition below. N′ = (Q′ , {a, b}, δ′ , q′ 0 , F′ ) where (a) Q′ = (b) q ′ 0 = (c) F ′ = (d) δ ′ is defined as follows:

Please DRAW non-deterministic finite automata accepting the following languages over the alphabet Σ = {a,b}. Note: For w ∈ Σ∗,σ ∈ Σ, we use nσ(w) to mean the number of letters σ in w. You must use non-determinism.  1. L2 = {w ∈ Σ∗ :na(w) mod 2 = 1 or nb(w) mod 3 = 2} 2. L3 = {w ∈ Σ∗ : w starts and ends with the same letter}

For any string w 6=/ E, the principal suffix of w is the string resulting by removing the first symbol from w. We will denote this string by ps(w). For any language L, define ps(L) := {ps(w) : w ∈ L ∧ w 6=/ E}. Show that if L is regular, then ps(L) is regular. (The underlying alphabet is arbitrary.) Show by giving rules to transform a regex for L into a regex for ps(L).

For a Turing machine M, (M) refers to the binary representation of M. For a Turing machine M, L(M) contains the set of all strings accepted by M. For a Turing machine M and an input x = {0,1}*, Steps(M, x) refers to the number of steps taken by M to execute on x before it halts. Here, one step of execution of M on x = one movement (left or right) of the tape head. For a Turing machine M and an input x = {0,1}*, we define the following: ReachCells(M,x) = {i : M reaches ith tape cell when M is executed on x} Informally, it contains all locations on the tape that are visited when M is ecuted on x. The leftmost location on the tape is the first tape cell, the location next to it is the second tape cell, and so on. A string w₁ is an anagram of w2 if w₁ can be obtained by rearranging the alphabets of w2. Formally, if w₁ is an n length string, wê is called an anagram of w₁ if there exists a permutation à on n elements such that π(w₁) = W2.

Prove the language L = {1n | m ∈ N ∧n = 2m} is non-regular. Explain each step of your reasoning with precision.

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*.

For any string w = w0w1w2 ...wn, the reverse of w, written wR, is the string in the reverse order, wnwn−1 ...w1w0. For any language L ⊆ Σ∗, we can define its reverse as LR = {wR | w ∈ L}. Show that if L is regular, then so is LR.

Let regular language L1 recognized by DFA (Q1,Σ,δ1,s1,F1) and regular language L2 recognized by DFA (Q2,Σ,δ2,s2,F2). We will construct a product DFA as the quintuple (Q,Σ,δ,s,F) where: •Q=Q1×Q2•For allx∈Σ and (q1,q2)∈Q1×Q2,δ((q1,q2),x) = (δ1(q1,x),δ2(q2,x))•s= (s1,s2) DetermineFso that the product DFA recognizes: (i)L1∪L2(ii)L1∩L2(iii)L1\L2Based on your choice for F, for each case, provide a detailed proof why the product DFA recognizes each of the mentioned languages. Conclude that in-tersection of two regular languages is a regular language and set-theoretic dif-ference of two regular languages is regular.