Homework 9
Last updated: Mon, 21 Nov 2022 15:00:26 -0500
Out: Tue Nov 15, 00:00 EST Due: Mon Nov 21, 23:59 EST (Note the Mon deadline again)
This assignment explores undecidable languages.
Homework Problems
Countable or Not Countable? (6 + 6 = 12 points)
Trying To Decide About CS 420, Fall 2022 (10 points)
A Closed Operation For Decidable Languages (9 points)
Re-Proving that EQ_TM is Undecidable (10 points)
README (1 point)
Total: 42 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 Countable or Not Countable?
The Infinite Monkey Theorem states that:
"a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare"
Prove that the set of possible monkey outputs is at least countable. You may assume that a monkey output is a string drawn from the alphabet \Sigma = \left\{\texttt{a},\ldots,\texttt{z},\texttt{A},\ldots,\texttt{Z},\texttt{\_}\right\} where \texttt{\_} represents a space.
In class we saw that an infinite subset of natural numbers, e.g., the set of all even numbers, can be countable and have the same size as the set of natural numbers!
Now show that the set of all possible such sets, i.e., the set of all possible infinite subsets of the natural numbers, is not countable.
Use proof by contradiction and diagonalization.
2 Trying To Decide About CS 420, Fall 2022
Prove that the following language is undecidable:
\mathrm{\textit{CS420}}_{\textsf{F22}} = \left\{\left\langle M\right\rangle\mid M\textrm{ is a \textsf{TM} where }\texttt{CS420}\in L(M)\textrm { and }\texttt{Fall2022} \in L(M)\right\}
Your proof should reduce from A_\textsf{TM}.
3 A Closed Operation For Decidable Languages
Define the following operation, called \mathrm{OV}, on languages:
\mathrm{OV}(L_1,L_2)= \textrm{The largest set of all strings X such that }X\subseteq L_1 \textrm{ and }X\subseteq L_2
Prove that \mathrm{OV} is closed for decidable languages by answering the following:
Give the If-Then statement that must be proved.
Prove it by giving a table of Statements and Justifications, where the last statement is the If-Then statement to be proved from above.
4 Re-Proving that EQ_TM is Undecidable
Prove (again) that EQ_\textsf{TM} is undecidable.
This time, however, you must reduce from the \textit{CS420}_\textsf{F22} language from the Trying To Decide About CS 420, Fall 2022 problem.