Homework 10

Last updated: Tue, 26 Apr 2022 17:08:35 -0400

Out: Wed April 20, 00:00 EST Due: Tue April 26, 23:59 EST

This assignment begins to explore time complexity.

Homework Problems

1. Big-O Exercises (6 points)

2. Regular Languages are in P (8 points)

3. More Closure Operations for Decidable Languages (3 + 3 + 3 = 9 points)

4. Closure Operations for Poly Time Languages (6 + 6 + 6 = 18 points)

5. README (1 point)

Total: 42 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 Big-O Exercises

Answer true or false for each statement below.

You may give an extra explanation if you think it would help clarify your answer.

1. 1 = O(n)

2. n = O(n)

3. n^3 = O(n)

4. 10^n = 2^{O(n)}

5. n\log^2n = O(n^2)

6. n\log^2n = 2^{O(n)}

2 Regular Languages are in P

Show that every regular language is in P.

3 More Closure Operations for Decidable Languages

Prove that each of the following operations are closed for decidable languages:

1. \textrm{EITHER}(A,B) = \left\{w\mid w\in A\textrm{ or }w\in B\right\}

2. \textrm{COMBINE}(A,B) = \left\{w\mid w = ab \textrm{ where } a\in A \textrm{ and } b\in B\right\}

3. \textrm{REPEAT}(A) = \left\{w\mid w=x_1\ldots x_n, n\geq0\textrm{ and each }x_i\in A\right\}

4 Closure Operations for Poly Time Languages