▼CS420:   Intro to Theory of Computation
►Homework 10
On this page:
1 Practice with Mapping Reducibility
2 A Closed Operation for Decidable Languages
3 Mapping Reducibility and Unrecognizable Languages
4 Turing-Recognizable Languages and the Halting Problem
    ← prev  up  next → 

Homework 10

Last updated: Fri, 21 Apr 2023 12:56:50 -0400

Out: Mon Apr 24, 00:00 EST Due: Sun Apr 30, 23:59 EST

This assignment explores decidability and deciders.

Homework Problems

  1. Practice with Mapping Reducibility (8 + 8 = 16 points)

  2. A Closed Operation for Decidable Languages (10 points)

  3. Mapping Reducibility and Unrecognizable Languages (11 points)

  4. Turing-Recognizable Languages and the Halting Problem (12 points)

  5. README (1 point)

Total: 50 points

Submitting

Submit your solution to this assignment in Gradescope hw10. 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 Practice with Mapping Reducibility

In lecture, we learned that showing mapping reducibility between two languages requires doing two things:

  • creating a computable function between the two languages, and

  • showing that the appropriate if-and-only-if statement (which itself has two parts) holds for that computable function.

We then started to show mapping reducibility between some languages we’ve previously seen. For this problem, you’ll complete what we started.

  1. Use the computable function TM from lecture mapping strings in E_\textsf{TM} to strings in EQ_\textsf{TM} to show mapping reducibility between these two languages.

    Specifically,

    1. give the if-and-only-if statement that must be proved, and

    2. show that the computable function satisfies this if-and-only-if requirement. Don’t forget that proving an if-and-only-if statement requires proving two separate statements. Further, the contrapositive must be used for the reverse direction.

  2. Now do the same thing, but for A_\textsf{TM} and \overline{E}_\textsf{TM}:

    1. give the if-and-only-if statement that must be proved, for there to be a mapping reducibility between these two languages;

    2. and show that the computable function from class satisfies this if-and-only-if requirement. Don’t forget that proving an if-and-only-if statement requires proving two separate statements. Further, the contrapositive must be used for the reverse direction.

2 A Closed Operation for Decidable Languages

The Are Turing-Recognizable Languages Are Closed under the OPPO operation? problem from Homework 7 explored whether Turing-recognizable languages were closed under the \mathrm{OPPO} operation (originally seen in Homework 3).

Now prove that Turing decidable languages are closed under \mathrm{OPPO}.

As usual, first write the if-then statement to be proved, and then prove it.

Remember that deciders require termination arguments. Also, an example table as shown in class, with at least one accepted string and one rejected string, is required to "prove" that a machine recognizes some language.

3 Mapping Reducibility and Unrecognizable Languages

In class we showed that both EQ_\textsf{TM} and EQ_\textsf{CFG} are Turing-unrecognizable languages.

Explain why this means that there cannot exist a mapping reduction from EQ_\textsf{TM} to EQ_\textsf{CFG}?

Structure your proof as a proof by contradiction. In other words, start with an assumption of the opposite.

4 Turing-Recognizable Languages and the Halting Problem

Show that every Turing-recognizable language is mapping reducible to HALT_\textsf{TM}.

    ← prev  up  next →