# CMPSCI 601: Theory of Computation

### David Mix Barrington

### Spring, 2004

### Homework Assignment #5

### Due Friday 16 April 2004 2:00 p.m. EST to CMPSCI office drop box

**(fax by
Wed 28 April 2:00 p.m. EST for off-campus students)
**

**Question 1 (35):**
Do the following seven problems in [BE] -- none are electronically
graded: 19.3, 19.4, 19.5, 19.15, 19.25, 19.26, and 19.27.
**Question 2 (15):**
Prove that a function is Floop-computable (or general recursive, or partial
recursive) iff it is representable by a ∀-bounded function as in
Lecture 16.
**Question 3 (10):**
Prove Theorem 16.11 on slide 21 of Lecture 16.
**Question 4 (10):**
Prove that the function log^{2}n
is DSPACE-constructible as defined in Lecture 17.
(Note that log^{2} means (log n)^{2}, not log (log n).)
**Question 5 (10):** Define "≤_{lin}" to be reducibility
where the function f must be in the class F(DTIME(n)). Prove that if B is
in DTIME(n^{k}) and A ≤_{lin} B, then A is in
DTIME(n^{k}).
**Question 6 (20):** Use the result of Question 5 and the Time
Hierarchy Theorem (though we did not prove it) to prove that *no*
language is complete for the class P under ≤_{lin} reductions.

Last modified 6 April 2004