Homework 10

Last updated: Sun, 28 Nov 2021 12:21:45 -0500

Out: Wed Dec 01, 00:00 EST Due: Tue Dec 07, 23:59 EST

This assignment explores space complexity.

Homework Problems

1. The Complement Operation in Space (8 points)

2. Space for Converting a Regular Expression to an NFA (10 points)

3. Equivalent or Not Equivalent NFAs? (10 points)

4. Equivalent or Not Equivalent Regular Expressions? (10 points)

5. README (2 point)

Total: 40 points

Submitting

Submit your solution to this assignment in Gradescope hw10. Please assign each page to the correct problem.

A submission must include a README containing the required information, in addition to the solution to the problems.

1 The Complement Operation in Space

Prove that PSPACE is closed under the complement operation.

2 Space for Converting a Regular Expression to an NFA

Recall the procedure to convert a regular expression into an NFA (from lecture 5, and also described in Sipser Lemma 1.55).

Prove that this procedure is a computable function that runs in polynomial space.

3 Equivalent or Not Equivalent NFAs?

Prove that the following language is in PSPACE:

NOTEQ_\textsf{NFA} = \left\{\left\langle N_1,N_2\right\rangle\mid N_1,N_2\textrm{ are NFAs where }L(N_1)\neq L(N_2)\right\}

4 Equivalent or Not Equivalent Regular Expressions?

Prove that the following language is in PSPACE:

EQ_\textsf{REX} = \left\{\left\langle R_1,R_2\right\rangle\mid R_1,R_2\textrm{ are regular expressions where }L(R_1)= L(R_2)\right\}