# Discussion on Theory of Computation

Dec 2020Final ExaminationTheory of ComputationCOMP 330 SEC 0016:30pm-Dec 11 to 6:30pm-Dec 14, 2020EXAMINER: Hamed Hatami ASSOC. EXAMINER:STUDENT NAME: McGILL ID:INSTRUCTIONSCLOSED BOOK OPEN BOOK X• Submit a pdf file – either typed, or a clear and readable scan – of your solution.• You can use the textbook, and lecture notes, but you are not allowed to search the internet forsolutions to these questions.1 2 3 4 5 6 7 8 Total/10 /10 /10 /15 /10 /15 /10 /20 /100Course: COMP 330 SEC 001 Theory of Computation Page number: 1 / 210

True or False? (Prove that your answer is correct). If L is a regular language and A ⊆ L, then A isdecidable. 10
True or False? (Prove that your answer is correct). If L1 is decidable and L2 is Turing recognizable, thenL1 ∩ L2 is decidable. 10
True or False? (Prove that your answer is correct). If L1 and L2 are in NP, then L1 ∩ L2 is in NP. 15
Describe the language of the following TM over the alphabet {0, 1, 2}: There are three states qstart,qaccept, and qreject. Three arrows from qstart to itself with labels 1 → 0, R and 0 → 1, L, and 2 → 2, L.There is also one arrow from qstart to qaccept with label t → t, R.10
Rigorously establish the decidability or undecidability of the following language:L = {hM1, M2i | M1, M2 are TM’s and L(M1) ⊆ L(M2)}.15
Let us call a deterministic Turing Machine M super-fast if there exists a constant c (here c can depend onM but it does not depend on the input) such that the following holds: On every input w, the TM M haltsafter at most c steps. Rigorously establish the decidability or undecidability of the following languages:L = {hMi | M is a super-fast TM}.10
Let L be the set of all hpi such that p is a multivariate polynomial with integer coefficients that evaluatesto zero for some assignment of positive integers to its variables. For example hx21 + x22 − 5i ∈ L as itevaluates to 0 if we set x1 = 1 and x2 = 2. On the other hand hx21 − 5i is not in L. Prove that L canbe decided by an oracle Turing Machine that has access to an oracle forHaltTM = {hM, wi | M is a TM that halts on input w}.20
Consider the following language:L = {hM, wi|M is a TM and L(M) = {w}}.(a) Is L Turing recognizable? (Prove your answer)(b) Is the complement of L Turing recognizable?(Prove your answer)Course: COMP 330 SEC 001 Theory of Computation Page number: 2 / 2

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# Discussion on Theory of Computation

Dec 2020Final ExaminationTheory of ComputationCOMP 330 SEC 0016:30pm-Dec 11 to 6:30pm-Dec 14, 2020EXAMINER: Hamed Hatami ASSOC. EXAMINER:STUDENT NAME: McGILL ID:INSTRUCTIONSCLOSED BOOK OPEN BOOK X• Submit a pdf file – either typed, or a clear and readable scan – of your solution.• You can use the textbook, and lecture notes, but you are not allowed to search the internet forsolutions to these questions.1 2 3 4 5 6 7 8 Total/10 /10 /10 /15 /10 /15 /10 /20 /100Course: COMP 330 SEC 001 Theory of Computation Page number: 1 / 210

True or False? (Prove that your answer is correct). If L is a regular language and A ⊆ L, then A isdecidable. 10
True or False? (Prove that your answer is correct). If L1 is decidable and L2 is Turing recognizable, thenL1 ∩ L2 is decidable. 10
True or False? (Prove that your answer is correct). If L1 and L2 are in NP, then L1 ∩ L2 is in NP. 15
Describe the language of the following TM over the alphabet {0, 1, 2}: There are three states qstart,qaccept, and qreject. Three arrows from qstart to itself with labels 1 → 0, R and 0 → 1, L, and 2 → 2, L.There is also one arrow from qstart to qaccept with label t → t, R.10
Rigorously establish the decidability or undecidability of the following language:L = {hM1, M2i | M1, M2 are TM’s and L(M1) ⊆ L(M2)}.15
Let us call a deterministic Turing Machine M super-fast if there exists a constant c (here c can depend onM but it does not depend on the input) such that the following holds: On every input w, the TM M haltsafter at most c steps. Rigorously establish the decidability or undecidability of the following languages:L = {hMi | M is a super-fast TM}.10
Let L be the set of all hpi such that p is a multivariate polynomial with integer coefficients that evaluatesto zero for some assignment of positive integers to its variables. For example hx21 + x22 − 5i ∈ L as itevaluates to 0 if we set x1 = 1 and x2 = 2. On the other hand hx21 − 5i is not in L. Prove that L canbe decided by an oracle Turing Machine that has access to an oracle forHaltTM = {hM, wi | M is a TM that halts on input w}.20
Consider the following language:L = {hM, wi|M is a TM and L(M) = {w}}.(a) Is L Turing recognizable? (Prove your answer)(b) Is the complement of L Turing recognizable?(Prove your answer)Course: COMP 330 SEC 001 Theory of Computation Page number: 2 / 2

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Our essay writers are graduates with diplomas, bachelor’s, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college diploma. When assigning your order, we match the paper subject with the area of specialization of the writer.

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1.      Place an order

You fill all the paper instructions in the order form. Make sure you include all the helpful materials so that our academic writers can deliver the perfect paper. It will also help to eliminate unnecessary revisions.

2.      Pay for the order

Proceed to pay for the paper so that it can be assigned to one of our expert academic writers. The paper subject is matched with the writer’s area of specialization.

3.      Track the progress

You communicate with the writer and know about the progress of the paper. The client can ask the writer for drafts of the paper. The client can upload extra material and include additional instructions from the lecturer. Receive a paper.

The paper is sent to your email and uploaded to your personal account. You also get a plagiarism report attached to your paper.

Get Professional Assignment Help Cheaply
Are you busy and do not have time to handle your assignment? Are you scared that your paper will not make the grade? Do you have responsibilities that may hinder you from turning in your assignment on time? Are you tired and can barely handle your assignment? Are your grades inconsistent?
Whichever your reason may is, it is valid! You can get professional academic help from our service at affordable rates. We have a team of professional academic writers who can handle all your assignments.
Our essay writers are graduates with diplomas, bachelor’s, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college diploma. When assigning your order, we match the paper subject with the area of specialization of the writer.
Why Choose Our Academic Writing Service?

Plagiarism free papers
Timely delivery
Skilled, Experienced Native English Writers
Ability to tackle bulk assignments
Reasonable prices

How It Works
1.      Place an order
You fill all the paper instructions in the order form. Make sure you include all the helpful materials so that our academic writers can deliver the perfect paper. It will also help to eliminate unnecessary revisions.
2.      Pay for the order
Proceed to pay for the paper so that it can be assigned to one of our expert academic writers. The paper subject is matched with the writer’s area of specialization.
3.      Track the progress
You communicate with the writer and know about the progress of the paper. The client can ask the writer for drafts of the paper. The client can upload extra material and include additional instructions from the lecturer. Receive a paper.
The paper is sent to your email and uploaded to your personal account. You also get a plagiarism report attached to your paper.

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"Honestly, I was afraid to send my paper to you, but you proved you are a trustworthy service. My essay was done in less than a day, and I received a brilliant piece. I didn’t even believe it was my essay at first 🙂 Great job, thank you!"

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"This company is the best there is. They saved me so many times, I cannot even keep count. Now I recommend it to all my friends, and none of them have complained about it. The writers here are excellent."