Reference no: EM131058101

COSC1105/1107 - Computing Theory Assignment

EXERCISE 1: TURING MACHINES               

1. Consider the following Turing Machine M = (Q, Σ, Γ, #, δ, q₀, F), where Σ = {1, 2, 3} is the input alphabet and Γ = {1, 2, 3, a, b, #} is the tape alphabet (# is the blank symbol):

A transition of the form 1/a, R means that when the head is reading 1, M can legally write a and move right.

(a) Trace the computation of M on input string 11222233 by completing the following derivation (location of head is denoted by underlining the respective symbol):

q₀#11222233# |- q₁#11222233# |- q₂#a1222233# |- ...

Is the input string accepted?

(b) Explain informally, the working of the machine M.

(c) Give the language L(M ) in set notation.

(d) Provide a grammar having 12 or less rules that accepts the same language as M. What type of grammar is it?

(e) What properties of formal languages would you use (and how) to formally prove that there is no NFA that can accept the language L(M). Explain briefly (note: you do not need to actually do any proof).

(f) Does there exist a PDA that accepts L(M )? Briefly explain your answer in one short sentence.            

(g) We obtain Turing Machine M' by modifying M as follows: (a) replace transition 3/3, R from state q₁  to q₅  with transition b/b, R; and (b) delete loop transition 3/3, R in state q₅. Give the language L(Mt) in set notation.

2. Show formally, by resorting to problem reduction, that the problem, of deciding whether a Turing machine halts on an input w where |w| is a multiple of 5, is undecidable.

EXERCISE 2: AUTOMATA

3. Let M₁ = (Q, Σ, δ₁, q₀, F₁) be the following finite state automaton defined over the alphabet Σ = {a, b, c, d}:  

Here, λ denotes the empty transition.

(a) You are told that the following DFA M₂ = (Q₂, Σ, δ₂, q₀, F₂), built using the subset construction technique, is a deterministic equivalent version of automaton M₁:

After careful analysis, you realise that M₂ is incorrect: it is not a deterministic equivalent version of M₁.

Construct a DFA M₃ = (Q₂, Σ, δ₃, q₀, F₃), from M₂, by only adding or removing transitions and toggling whether a state is final or not (but without changing the set of states and the initial state), such that L(M₁) = L(M₃). In addition, you are asked ensure that the transition function δ₃ is total. You do not     need to provide a diagram of M₃; just answer the following questions by interpreting transition functions as relations (e.g., (q₀, c, q₁) ∈ δ₂, since δ₂(q₀, c) = q₁):

i. What is δ₃\δ₂?

ii. What is δ₂\δ₃?

iii. What is F₃\F₂?

iv. What is F₂\F₃?

(b) What modifications are required in M₁ in order to make it λ-free without altering its language?  Your resulting machine M₄ = (Q, Σ, δ₄, q₀, F₄) will still be non-deterministic. You can only remove λ transitions, add transitions over Σ, or toggle whether a state is final or not, you must not change the set of states and you cannot change the initial state. You do not need to provide a diagram of M₄; just answer the following questions:

i. What is δ₄\δ₁?

ii. What is δ₁\δ₄?

iii. What is F₄\F₁?

iv. What is F₁\F₄?

4. Emma's Pumping Lemma Dilemma: For her job application to tutor CT, Emma has been asked to prove that L = {aⁿb^ma^2m+3n | m, n ≥ 1} is not regular. The night before it was due, she finally finished the proof, wrote it all down neatly, and went to bed. During the night, burglars came into her house with their dog, and the dog chewed up her proof. In an attempt to cover their tracks, the burglars tried to recreate the proof. They gathered    all the shreds they could find, but on some of them the ink had washed out from the dog's saliva, and so they couldn't read some parts of it.  Can you help them put the proof back together?  For each shred they found, they rewrote it on a piece of paper. If they couldn't rewrite it, they wrote different interpretations of what they thought it read onto different pieces of paper. Now they have many pieces of paper, but they don't know which    of them are correct and in which order they need to go. Help Emma's burglars.

There is one big piece that reads:

Reminder: the Pumping lemma states, that for any regular language L, there exists a pumping length k ∈ N, and for all w ∈ L, where |w| ≥ k, there exist substrings x, y, z ∈ Σ*, such that:

w = xyz

|y| ≥ 1

|xy| ≤ k

xy'z ∈ L, for all i ≥ 0.       

Here are the other pieces they've transcribed:

(a) Now consider w = a^kb^ka^5k.

(b) Now consider w = abaaaaa.

(c) Thus w = a^kb^ka^5k = xyz such that |xy| ≤ k and |y| ≥ 1.

(d) Then, it follows that w ∈ L (where m = n = 1) and |w| = 7 ≥ 1.

(e) Then, it follows that w ∈ L (where m = n = k) and |w| = 7k ≥ k.

(f) Consider now i = 1. Then, xy'z = xyz = a^qa^ra^sb^ka^5k = a^q+r+sb^ka^5k.

(g) Consider now i = 0. Then, xy'z = xy⁰z = xz = a^qa^sb^ka^5k = a^q+sb^ka^5k.

(h) So, it follows that x = a, y = b, and z = aaaaa. Clearly w = abaaaaa = xyz.

(i) Assume that L is regular. Then, there exists a k ∈ N as per the Pumping Lemma.

(j) Consider now i = 2. Then, xy'z = xyyz = abbaaaaa. It is then clear that xy'z ∉ L.

(k) Now, because q + r + s = k and r ≥ 1, we know that q + s ≠ k, and therefore, xy'z ∉ L.

(l) Thus we have a contradiction: our assumption (that L is regular) must be blue and L is true.

(m) Now, because q + r + s = k and r ≥ 1, we know that a^qa^ra^s = a^k, and therefore, xy'z ∈ L.

(n) So, according to the Pumping lemma, there exist substrings x, y, z ∈ Σ* such that (1)-(4) hold.

(o) Thus we have a contradiction: our assumption (that L is regular) must be false and L is not regular.

(p) So, it follows that x = a^q, y = a^r, z = a^sb^ka^5k, where q + r ≤ k, r ≥ 1, s ≥ 0, and q + r + s = k.

You need not write out the entire proof, just provide the corresponding letters in the correct order as a word. For example, the word badf denotes the proof built from statements (b), (a), (d), and (f) in that exact order. And remember:  the burglars wrote down multiple different interpretations of the same piece of the proof, so not all of these will be necessary or correct. However, all necessary pieces are listed above.

5. Prove that L = {a^mb^2maⁿ | m, n ≥ 0} is not regular.         

6. Show that the language L₂ = {a, b}*\{a^mb^2maⁿ|m, n ≥ 0} is not regular. For this part, you can rely on the   fact that L, from the previous question, is not regular.

7. Show that the language L₃ = {aⁿ | n ≥ 0, n ≠ 3 and n ≠ 6} is regular.        

EXERCISE 3-       

8. Prove mathematically that f (n) = 10n³ + 20n² + 30 ∈ O(n⁴), for n ≥ 0.  

9. Suppose you are studying the computational complexity of a given problem X. It is known that the boolean satisfiability  problem  can  be  reduced  to  X  in  polynomial  time.  In  addition,  it  is  known  that  problem  X  is  in class PSPACE, as somebody has developed an algorithm for X that is guaranteed to consume no more than  polynomial amount of space. How would you prove that:

(a) X is NP-COMPLETE.

(b) X is PSPACE-Complete.

Observe that you do not need to develop any proof, but just explain briefly what task would you do.

10. Let X₁ and X₂ be two decision problems.  Suppose it is known that X₁ is NP-COMPLETE. Suppose further that there exists an exponential reduction of problem X₁ into problem X₂, that is, there exists a function f: X₁ → X₂, whose implementation runs in exponential time, such that for any given instance x₁ of X₁ the answer to x₁ is "Yes" for problem X₁ if and only if the answer to f (x₁) is "Yes" for problem X₂. What can be said about the computation complexity of the problem X₂. Is X₂ NP-HARD? Explain in one or two paragraphs.

EXERCISE 4-       

11. Explain the phase transition phenomenon observed in 3-SAT.

You're expected to submit between 400 and 500 words (excluding references).

Think of it, as if you were hired to write a textbook for Computer Science undergraduates who are taking Computing Theory.

Besides providing enough technical information, your text needs to be enjoyable to read (or else the book will    not sell well!). Make sure you use multiple paragraphs, correct English grammar and spelling, and language of appropriate formality with an adequate flow.

You must not copy and paste a single sentence from the web (this includes Wikipedia).  Note that copied sentences are trivial to identify, so you should re-phrase in your own words whatever you read. You should also reference the material you used for your research.

Finally, we strongly recommend you proof-read your text, reflect on whether the reader (e.g., a fellow student) would be satisfied with what you wrote and revise accordingly.