On this page:
1 Computing Chomsky Normal Form
2 Countably or Uncountably Infinite?
3 CS420 is Undecidable?
4 One more Undecidable Problem

Homework 9

Last updated: Mon, 22 Apr 2024 11:53:28 -0400

Out: Mon Apr 17, 12:00pm EST (noon) Due: Wed Apr 24, 12:00pm EST (noon)

This assignment continues explore decidability, and also begins to look at undecidability.

Homework Problems

  1. Computing Chomsky Normal Form (12 points)

  2. Countably or Uncountably Infinite? (6 + 6 = 12 points)

  3. CS420 is Undecidable? (12 points)

  4. One more Undecidable Problem (12 points)

  5. README (2 points)

Total: 50 26 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 Computing Chomsky Normal Form

Recall that a CFG is in Chomsky Normal Form (CNF) if all rules have the form:

A \rightarrow BC

A \rightarrow \texttt{a}

where \texttt{a} must be a terminal, A, B, C are variables, and B and C are not the start variable.

In addition, a grammar in CNF may include a rule A\rightarrow \varepsilon if A is the start variable.

Convert the following context-free grammar to Chomsky Normal Form using the algorithm described in class.

The start variable of the CFG is S and the set of terminals is \{\texttt{0},\texttt{1},\texttt{\#}\}

2 Countably or Uncountably Infinite?

In class, we learned about how infinite sets can have different sizes. Specifically, they can be either countable or uncountable.

3 CS420 is Undecidable?

Note: This problem has been moved to Homework 10.

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 EQ_TM HALT_\textsf{TM}. It should also use the "modify the TM" technique from class.

4 One more Undecidable Problem

Note: This problem has been moved to Homework 10.

Now let’s go the other way. Re-prove that EQ_\textsf{TM} is undecidable.

Your proof must reduce from the undecidable \textit{CS420} language from the CS420 is Undecidable? problem. (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.