Q1. Let  ∑ = {0,1}. Define the following language:

L = {x | x contains an equal number of occurrences of 01 and 10}

Either prove L is regular (by constructing a DFA/NFA or a regex) or prove that it is not regular using the Pumping Lemma for regular languages.

Q2. Given languages L, L' (over ∑ ), define the shuffle of L, L' to be

Shuffle(L, L') = {x₁y₁x₂y₂......x_ny_n | x_i ∈ L and y_i ∈ L' for i = 1,......, n}
Suppose that M = ( ∑, Q, q₀,δ, F) is a DFA for L and M0 = (∑ ,Q', q₀',δ', F') is a DFA for L'. Construct an NFA that accepts Shuffle(L).
[This proves that Shu e is a closed regular operator, i.e., this proves that if L; L0 are regular, then Shu e(L; L0) is also regular.]

Q3. Let   ∑={a, b, c}. Prove that the following language (defined over)

L = {uvuvu | u ∈ {a, b}* ; v ∈ {b; c}*} is not regular.
[HINT: Clean up and then use the pumping lemma for regular languages.]

Q4. Prove that the following languages are context-free by constructing the a context-free grammar for L₁ and a push-down automata for L₂.

(a) L1 = {a^mb^mcⁿ | m  ≥ 0,  n ≥ 0}

(b) L2 = {a^mbⁿcⁿ |m  ≥ 0,  n  ≥ 0}

Finally:

(c) What is L1 ∩ L2? [Use set notation.]

(d) TRUE or FALSE: The intersection of any two context-free languages gives a context- free language.

Q5. Prove that

L = {a^mbⁿ |m ≠ n and 2m ≠ n}

is a context-free language.

Q6. Suppose L is a context-free language and L' is regular. Show that L ∩ L' is a context- free language. Specifically, if you're given a PDA diagram of L and a DFA diagram for L',describe informally how to draw a PDA diagram for L ∩ L'.

[... it's even better if you can describe the construction of L∩L' formally but you don't have
to.]
Q7. (a) A Turing machine M is considered a DFA1 if the read/prepare head always moves to the right and it halts (i.e., enters its q_accept or q_reject) when it reads a space (or blank) character on the input tape. Note that a DFA1 is like a DFA except that it must have exactly 1 accept state. (A DFA can have any number of accept states.) Is the following language:

Accept_DFA1 = {#| M is a DFA1 and accepts w}

a Turing{decidable language? If it is, describe clearly how to construct a Turing decider (i.e.,a Turing machine that always halts) that accepts the language. Otherwise prove that it's not decidable.

(b) Consider the following language:

NonEmpty_TM = {#| M is a Turing machine and L(M) ≠0}

Is NonEmpty_TM a Turing-decidable language?

[This is the only TM problem but it actually requires very little TM knowledge. You have everything you need to solve this problem if have a general understanding of TMs and you have paid attention to the Wednesday and Friday classes during the last week.]

Q8. This is the problem for the second version of the final exam. You only need to attempt one of the three parts. Do not turn in any work for Q1-Q7 if you plan to turn in work for Q8, or you will get an immediate 0.

1. Given two words x,y ∈ ∑*  , a DFA M separates x, y if L accepts either x or y but not both. What is the smallest number of states you need to construct a DFA that can separate any pair of distinct words of length ≤ n?

2. Consider the following problem: Given a PDA M and an integer n, is it true that ∑ⁿ ⊆ L(M)? The corresponding language is L = {#| M is a PDA and ∑ⁿ ⊆ L(M)} Is L regular, context-free, Turing{decidable, Turing{recognizable, not Turing{recognizable?

3. Either prove P = NP or P ≠ NP.