Homework 12🔗

Last updated: Tue, 25 Nov 2025 12:39:01 -0500

Out: Mon Nov 24, 12:00pm EDT (noon) Due: Fri Dec 05, 12:00pm EDT (noon)

Note: Assignments are not officially "released" until—and are subject to change without notice up to—the indicated "Out" date and time. If an assignment is posted early, students may look ahead but are responsible for ensuring that they are always working with the most recent version of the homework.

This assignment looks at the topic of mapping reducibility.

Homework Problems

1. A Closed Operation for Undecidable Languages? (16 points)

2. E_TM is Undecidable, Again (24 points)

3. The Halting Problem is Universal (16 points)

4. Time Complexity Warmup Exercises (14 points)

5. README (2 point)

Total: 72 points

Submitting

Submit your solution to this assignment in Gradescope hw12. 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 A Closed Operation for Undecidable Languages?🔗

Prove that the set of undecidable language is closed under the COP operation from the Regular Languages Review Question problem in Homework 9.

Start by giving the IF-THEN version of the Statement above, and then prove it.

There is a slight wrinkle, as we’ve seen for these kinds of "not" problems, which is that you will have to use proof by contradiction, because the method of constructing / transforming machines won’t work when there are no machines to use. So start by negating the IF-THEN statement you came up with and then produce a contradiction, where contradiction means that the proof must eventually have two statements that are clearly negations of each other, e.g., "L is decidable" and "L is undecidable". It can conclude that the initial assumption is false, proving the Statement to Prove.

As usual, the proof must be in the form of a Statements and Justifications Table that consists of only valid Statements and approved Justifications from this course (a sentence or paragraph has never been a valid Statement or Justification), and make sure that any machine constructions have all the required components to Justify any Statements written about them.

The proof may not use any known closure properties of decidable / undecidable languages in this problem unless you prove it on your own.

As many have noticed, a correct answer in this course is usually short and precise, so answers that span multiple pages are almost always incorrect. Further, any extraneous info that is submitted will only serve to make the answer incorrect so don’t include such info.

2 E_TM is Undecidable, Again🔗

Prove that E_\textsf{TM} is undecidable again, but this time using mapping reducibility. The proof must reduce from A_\textsf{TM}.

The solution has already been presented in class, in a few steps, and is also available in the Sipser textbook, so your only job is to demonstrate understanding by filling in some details and writing out the proof in full. This proof does have a few more steps than usual, so attention to detail will be needed for a correct answer.

The answer must use the template from class and fill in all necessary details. Any other formatting or formulations will be not be accepted.

The answer must also use the result from the A Closed Operation for Undecidable Languages? problem above.

Also, the contrapositive must be used in the reverse direction of the mapping reducibility if-and-only-if.

3 The Halting Problem is Universal🔗

This problem will show that the famous "Halting Problem" is universal, in some sense.

Specifically, prove that any Turing-recognizable language is mapping reducible to HALT_\textsf{TM}.

Start by giving an IF-THEN version of the Statement above.

Then prove that IF-THEN statement in the usual way.

The contrapositive must be used in the reverse direction of the mapping reducibility if-and-only-if.

4 Time Complexity Warmup Exercises🔗

Answer true or false for each Big-O statement below.

You may give an extra explanation if you think it would help clarify your answer, but graders are not obligated to look at it nor consider it in grading.

(COMING SOON It’s here!)

1. 420 = O(n^2)

2. 420 = O(1)

3. 620^4 = O(n^3)

4. 620 n^4 = O(n^4)

5. 2025^n = 2^{O(n)}

6. 2025\log n = O(\log n)

7. n\log n = O(\log n)