COMP 4141 Harvard University Math Recursively Enumerable Questions

COMP 4141

Harvard University


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Assignment 5 COMP4141 2020 Term 1 Due: Friday, 3rd April, 17:00 Submission is through WebCMS/give and should be a single pdf file, maximum size 4Mb. Prose should be typed, not handwritten. Use of LATEX is encouraged, but not required. Discussion of assignment material with one other person is permitted, but the work submitted must be your own in line with the University’s plagiarism policy. Problem 1 Recall that a language L is verifiable if there is a Turing Machine that accepts the language: {hw, ci : w ∈ L for some string c}. (a) Show that if L is recursive, then it is verifiable. (4 marks) (b) Show that if L is verifiable, then it is recursive. (6 marks) Problem 2 Consider the following computational decision problem: Return to start Input: A Turing Machine M Question: Is there an input which will make M return to its start state? (a) Show that this problem is Recursively Enumerable. (4 marks) (b) Show that this problem is Undecidable. (6 marks) Problem 3 Consider the following decision problems: Hamiltonian Path Input: An undirected graph G Question: Is there a path in G that visits every vertex exactly once? Hamiltonian Cycle Input: An undirected graph G Question: Is there a cycle in G that contains every vertex exactly once? 1 (a) Show that Hamiltonian Path ≤ P Hamiltonian Cycle. (6 marks) (b) Show that both problems are in NP. (4 marks) 2 ...
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Final Answer

I'm still working on question 1

Recursively enumerable
A turning machine M, on an input 𝑠 either accept or reject thus halt or loop.
Denoting the language of a turing machine as 𝐿(𝑀), the set of all strings 𝑤 is what Turing
machine M accepts
If there is a Turing machine M that accepts a language L that is recursively enumerable will be
𝐿(𝑀) = 𝐿
When, 𝐿𝑑 = {𝑀 | 𝑀 ∉ 𝐿(𝑀)
Inputs strings over {0,1}
The respresentation are in 𝑖th binary string in the order (1,2,3, … … . 𝑖)
The turning machine accepts corresponding to 𝑖th a...

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