Homework 9
Last updated: Thu, 20 Apr 2023 11:08:02 -0400
Out: Wed Apr 12, 00:00 EST Due: Sun Apr 23, 23:59 EST (extended)
This assignment explores decidability and deciders.
Homework Problems
Countably or Uncountably Infinite? (6 + 6 = 12 points)
CS420 is Undecidable? (13 points)
CS420 is Undecidable? (Again) (12 points)
One more Undecidable Problem (12 points)
README (1 point)
Total: 50 points
Submitting
Submit your solution to this assignment in Gradescope hw9. Please assign each page to the correct problem and make sure your solutions are legible.
A submission must also include a README containing the required information.
1 Countably or Uncountably Infinite?
In class, we learned about how different infinite sets can have different sizes. Specifically, they can be either countable or uncountable.
One example we showed in class is that the set of all natural numbers and the set of all even (natural numbers) have the same size. Prove that even if we add in negative even numbers, the resulting set will still have the same size as the natural numbers.
An example of a set that is uncountable is the set of all languages. Prove that this is true. In this proof, assume that the strings in each language (even though it is a set) have a definitive ordering, e.g., alphabetical order. UPDATE: The proof must be a proof by contradiction and use diagonalization.
2 CS420 is Undecidable?
Prove that the following language is undecidable:
\textit{CS420} = \left\{\left\langle M\right\rangle\mid M\textrm{ is a \textsf{TM} where }\texttt{CS420}\in L(M)\right\}
Your proof should be a proof by contradiction and should reduce from A_\textsf{TM}. It should also use the "modify the TM" technique from class.
3 CS420 is Undecidable? (Again)
Now prove that \textit{CS420} (from the above CS420 is Undecidable? problem) is undecidable, again.
This time, your proof should reduce from E_\textsf{TM}. It must be a proof by contradiction.
4 One more Undecidable Problem
Re-prove that E_\textsf{TM} is undecidable.
Your proof must reduce from the undecidable \textit{CS420} language from the CS420 is Undecidable? and CS420 is Undecidable? (Again) problems. (You may assume that the \textit{CS420} language is undecidable, even if you were unable to answer the earlier questions.) The proof must be a proof by contradiction.