# CMPSCI 601: Theory of Computation

### David Mix Barrington

### Spring, 2003

### Homework Assignment #1

### Due Monday 1 March 2004 in class

**(Wed 10 March 2:00 pm EST for off-campus students)
**

Most of the problems are from [BE] and are to be submitted
to the Grade Grinder *if* they are marked for electronic
submission. The last two are written here.
Question 3 was replaced on 21 February 2004.

**Question 1 (80):**
Do the following problems in [BE]: 3.3, 3.15, 3.21, 4.24, 6.2, 6.14, 6.32,
8.4, 8.46. 8.42 (10), 17.4, 17.13 (10), 17.16 (10). The problems not
marked "(10)" are five points each.
**Question 2 (10):**
Describe a one-tape Turing machine that when started on input "w", a string
in {0,1}^{*}, halts with "ww" on the tape. You may define the tape
alphabet as needed, and omit transitions that will not be used in a successful
computation.
**Question 3 (10+10):**
(Question modified Sat 21 Feb 2004)
Explain informally why the function Times (which inputs two $n$-bit binary
strings interpreted as numbers x and y, and outputs their product xy)
is in the class F(DTIME(n^{2})). For up to ten points **extra
credit**,
explain informally why Times is in the class
is in the class F(DSPACE(log n)). Recall that the definition of the latter
class refers to a Turing machine
with a read-only input tape, one or more read/write work tapes, and a
write-only output tape.

Last modified 4 March 2004