Homework 9

Last updated: Wed, 18 Nov 2020 10:01:45 -0500

Out: Wed Nov 18, 00:00 EST Due: Tues Nov 25, 23:59 EST

This homework covers material from Chapter 7 of the textbook.

Homework Problems

1. README (2 points)

2. Rice’s Theorem Again (6 points)

3. Brute Force Computing Languages (4 points)

4. Big-O (6 points)

5. Running time of E_DFA (6 points)

6. Closure of Polynomial Time Languages (8 points)

Total: 32 points

Submitting

Submit this assignment at Gradescope hw9.

1 README

Create a README file containing the required information, and submit it along with the rest of the homework.

2 Rice’s Theorem Again

Prove Rice’s Theorem again, this time using the Recursion Theorem from chapter 6.1 of the textbook.

Like in Homework 8, you may assume that you have both a TM M_{inP} that is in any given P, and a TM M_{notinP} that is not in P.

3 Brute Force Computing Languages

In earlier homework problems (e.g., DFAs and the Language They Recognize or NFAs and the Language They Recognize), you were asked to print out the language of a finite automaton by trying all strings with your "run" function.

1. Give an upper bound for the number of times "run" must be called. Express your answer as a function on n, where n is the length of the longest string in the language. You may assume an alphabet of alphanumeric ascii characters.

2. Express the answer to the first part in big-O notation.

4 Big-O

Give a big-O upper bound for the following functions on n:

1. 5n^3+3n^2+80n+100

2. 22n(4\log_2 n +30)

3. 100^{11n}+5

5 Running time of E_DFA

Give an upper bound, i.e., use big-O notation, on the running time for the decider for E_{DFA} from Theorem 4.4.

State this bound in terms of q and t, where for any given DFA, q is the number of states and t is the maximum transitions coming out of any one state.

6 Closure of Polynomial Time Languages

You showed in the Turing Machine and Closure Operations problem of Homework 5 that decidable languages are closed under union and concatenation. Closure under intersection can be proved in a similar way. Also, we showed in class that decidable languages are closed under complement. Now we look at the these operations in the context of time complexity.

Define P to be the set of languages that can be decided by a single-tape TM that runs in TIME(n^k), for some constant k.