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 log²n is DSPACE-constructible as defined in Lecture 17. (Note that log² 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