# CMSC 651 Homepage

## Homework Assignments

From Introduction to the Theory of Computation by Michael Sipser, PWS Publishing unless otherwise noted.

HU = Introduction to Automata Theory, Languages and Computation, Hopcroft & Ullman, 1979.

1. (Due 02/08) 1.4 (f, h & i), 1.17, 1.24. Optional: 1.32.

Note: for people with the first printing of the textbook, Ex 1.17(b) should read "{ www | ...}" (3 w's instead of 2).

2. (Due 02/15) 1.23, 1.31, 1.42

Note: for people with the first printing of the textbook, in Prob 1.23(d), the string w is in {0,1}* (the * is missing).

Optional: [HU Ex 3.18, p. 73] Show that if L is regular, then so is the following language:

LOG(L) = { x | for some y with |y| = 2|x|, xy is in L }

3. (Due 02/22) 2.6 (b & d), 2.18 (c), 2.17. Optional: 2.27.

4. (Due 03/14) 3.10, 4.19, 5.12.

5. (Due 03/28) 5.20 and the following two problems:

• Show that K17 defined below is many-one complete for the Turing recognizable languages.

K17 = { <M> | <M> is an encoding of a Turing machine and |L(M)| >= 17 }

• Show that ALLTM and INFTM defined below are equivalent under many-one reductions.

ALLTM = { <M> | <M> is an encoding of a Turing machine and L(M) = Sigma* }

INFTM = { <M> | <M> is an encoding of a Turing machine and L(M) contains an infinite number of strings}

6. (Due 04/04) 4.18, 5.21 and the following problem:

A set is cofinite if it is the complement of a finite set. Let

COF = { < M > | M is a TM and L(M) is cofinite}
FIN = { < M > | M is a TM and L(M) is finite}

Prove that FIN reduces to COF via a many-one reduction.

7. (Due 04/25) 7.21, 7.27, 7.29

8. (Due 05/02) 8.19, 8.20, 9.3

9. (Due 05/09) 9.19, 9.20, 9.21