comp2022 Assignment 3 (70 marks) s2 2024
This assignment is due in Week 10 and should be submitted to Gradescope.
All work must be done individually without consulting anyone else’s solutions in accordance
with the University’s “Academic Dishonesty and Plagiarism” policies.
Go to the last page of this document and read the Submission Instructions. For clariffcations
and updates, monitor “Assignment FAQ”.
Problem 1. (10 marks) Consider the following deterministic Turing Machine M
over input alphabet Σ = {a, b}:
0 _ _ L 1
0 * * R 0
1 b _ L 2
2 a _ L 3
1 _ _ * halt_accept
3 _ _ R 0
3 * * L 3
1. (5 marks) State ffve strings that are in L(M), and ffve that are not. The
strings should be over Σ.
2. (5 marks) Provide a low level description in Morphett notation of a (**tape
deterministic) Turing Machine for the language that has time complexity at
most 5n + 5.
Problem 2. (10 marks) Consider the following nondeterministic Turing Machine
N over input alphabet Σ = {a, b}:
0 _ _ * halt-reject
0 a a r 0
0 b b r 0
0 b x l 1
1 x x l 1
1 a x r 2
1 b x r 2
1 _ _ r 4
1comp2022 Assignment 3 (70 marks) s2 2024
2 x x r 2
2 a x r 3
2 b x r 3
2 _ _ * halt-reject
3 x x r 3
3 a x l 1
3 b x l 1
3 _ _ * halt-reject
4 x x r 4
4 a a * halt-reject
4 b b * halt-reject
4 _ _ * halt-accept
1. (5 marks) State ffve strings that are in L(N), and ffve that are not. The
strings should be over Σ.
2. (5 marks) Provide a low level description in Morphett notation of a (**tape
deterministic) Turing Machine for the language.
Note: Morphett’s simulator of nondeterministic TMs uses randomness to resolve
nondeterminism. This is not the semantics of NTMs.
Problem 3. (30 marks) For each of the following languages over the input alphabet
Σ = {a, b, c}, provide a low level description in Morphett notation of a
(**tape deterministic) TM for the language.
1. The language of non-empty strings where the ffnal character appears at
most 3 times in the string (including the ffnal character).
E.g., abccaba is in the language, while abcbcbab is not.
2. The language of strings of the form a
E.g., aabbccaa is in the language, while abc is not.
3. The language of strings that can be turned into a palindrome by replacing
at most two characters by other characters.
E.g., aba is in the language because it is a palindrome, abb is in the language
because we can change one character to get a palindrome (e.g., aba),
and aabc is in the language because we can change two characters to get a
palindrome (e.g., aaaa); however aabbccc is not in the language.
4. The language of strings for which the longest substring that matches a
∗
is
longer than the longest substring that matches b
∗
.
E.g., caaaccbbaabaaac, baaacbbcaaabb and aaaa are in the language, while
aabbbcacacacaca is not.
2comp2022 Assignment 3 (70 marks) s2 2024
5. The language of strings of the form uvcvu where u, v ∈ {a, b}
∗
.
E.g., aabbacbaaab is in the language (take u = aab, v = ba), while aabbcabab
is not.
6. The language of strings of the form uvw where v is a non-empty string with
the same number of as, bs, and cs. E.g., bbaabbbccaccbc is in the language,
while bbaabbbcc is not.
Problem 4. (5 marks + 5 bonus marks)
Your robot buddy GNPT-4 has come up with a revolutionary new strategy to
prove that it is in fact equal in computational power to its more well-known
cousin. It has a simple yet brilliant proof strategy: it will start by proving that
P in fact equals the set of Turing-decidable languages, by showing that every
decider runs in polynomial time. Once it has done this, it will obtain as a corollary
that NP is also equal to this set, and the result will follow. GNPT-4 would
like you to check its generated proof, and has generously offered you half of the
million dollar bounty for doing so.
Unfortunately, you’re starting to have some concerns about the claim that every
decider runs in polynomial time. GNPT-4’s proof of this claim is 2123 pages
long, so you don’t really feel like checking it in detail for a ffaw. Instead, you
have a much better idea: you’ll provide an explicit counterexample of a machine
that does not run in polynomial time.
1. (5 marks) Provide a low level description in Morphett notation of a (**tape
deterministic) TM over input alphabet Σ = {a} that accepts every string, has
at most 20 states, and has time complexity f(n) such that 2
n ≤ f(n) ≤ 2
2n+1
for all n.
2. (5 bonus marks) Provide a low level description in Morphett notation of a
(**tape deterministic) TM over input alphabet Σ = {a} that accepts every
string, has at most 40 states, and has time complexity exactly 2
n
.
Problem 5. (15 marks)
You’re a budding cartoonist, trying to create the next great TV animation. You’ve
come up with the perfect idea, but now you need to pitch it to the executives.
You know from your experience in the industry how the process works: you
make a proposal with a string over Σ = {a, b} and the network runs a Turing
machine Q on it. If Q accepts, your show will be ready for broadcast, but if
it doesn’t, you will be shown the door, fflled with eternal regret at what could
have been. Of course, as Q is a Turing machine, there is also the possibility that
Q will diverge. (For example, this is what happened after season 7 of Futurama.)
One of your shady contacts (apparently they’re a secret agent who uses ffnite automata,
or something?) has managed to obtain a copy of the network’s machine
Q for you. You now want to analyse Q to ffgure out how to pitch your show
3comp2022 Assignment 3 (70 marks) s2 2024
so it will be accepted. Furthermore, you’ve heard that it’s considered especially
fortuitous if Q runs in a number of steps that is a multiple of 77, and such shows
will be given air during the network’s prime timeslots. So you’d like a machine
that will analyse Q and your proposal to see if that will be the case.
1. (5 marks) Prove that the language {M, x: M halts on x in exactly 77n steps
for some integer n > 0} is undecidable.
Okay, so that was a bust. You’ve set your sights lower: at this point you just want
any description that will be accepted, and you’re willing to retool your proposal
to make it work. Rather than focusing on your speciffc string, you’d like a
machine that will analyse just Q, and ffnd some string, any string, that it will
accept. There is, however, the possibility that Q doesn’t accept any string. (That
would explain why there are no decent new shows these days.) In this event,
your endeavour is doomed and you don’t care about the output, but you’d like
the analysing machine to at least halt, so you’re not stuck waiting forever.
2. (10 marks) Consider the following speciffcation. The inputs are Turing machines
over input alphabet Σ = {a, b}.
(a) If the input is a Turing machine M that accepts some input, the output
should be any string x that M accepts.
(b) If the input is a Turing machine M that does not accept any input, the
output should be any string x. (There still must be an output, ie. the
machine satisfying this speciffcation must halt.)
Prove or disprove whether there exists a Turing Machine that halts on every
input and satisffes this speciffcation.
4comp2022 Assignment 3 (70 marks) s2 2024
Submission Instructions
You will submit answers to all the problems on Gradescope.
Problems 1, 2, 3 and 4 are autograded.
It is essential that you ensure that your submission is formatted so that the autograder can
understand it. Upon submitting your responses, you should wait for the autograder to provide
feedback on whether your submission format was correct. An incorrectly formatted submission
for a question will receive zero marks for that question. A scaffold will be provided on Ed
with the ffle names the autograder expects.
Problem 1.1, 2.1 format:
The ffrst line of each answer should contain a comma separated sequence of ffve strings that are
in the language, and the second line should contain a comma separated sequence of ffve strings
that are not in the language. For example, if the language consists of all strings that only contain
b’s, an example of a correct text ffle would be:
epsilon, b, bb, bbb, bbbb
a, aa, aaa, aaaa, aaaaa
Problem 1.2, 2.2, 3, 4 format (TMs):
All TMs that you are required to provide in this assignment are deterministic and have a single
tape, and that tape is doubly-inffnite. When asked to give a low-level description use Morphett’s
format. The initial state must be 0
Note that your machine should use an explicit transition to halt-reject when rejecting a string. If
the machine has no transition on a (state, input) pair, this will be treated as an error, and will not
be treated as rejecting the string. You may wish to include the following line in your machines,
to treat all undeffned transitions as rejects: * * * * halt-reject
Problem 5 format:
Problem 5 is handgraded. You will submit a single typed pdf (no pdf containing text as images,
no handwriting). Start by typing your student ID at the top of the ffrst page of each pdf. Do not
type your name. Do not include a cover page. Submit only your answers to the questions. Do
not copy the questions. Your pdf must be readable by Turnitin.
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