CS517 Homework 3

Due May, 20, 2:00 P.M.

All the numbers are from Arora & Barak.

3.1.
Use diagonalization or reduce the halting problem to the current problem.

3.2.
Hint: Consider an operation Pad, where Pad(x)= x1^|x|^2, i.e., Pad appends a string x with a string of 1's, whose length is the square of the input string x. We define Pad(L) as {Pad(x)|x in L}. First show that for any L, if Pad(L) is in NP, then L is in NP.

Pick a language L from SPACE(n^2) - SPACE(n), which must exist by the space hierarchy theorem. Show that Pad(L) is in SPACE(n) using the fact that L is in SPACE(n^2). Assume that Space(n) = NP, and derive a contradiction. In particular, is L in NP or not?

3.3.
Hint: start from the language B constructed in Theorem 3.7 and make it deterministic. Show that it is in EXP.

4.1.

4.2.

4.3.

4.4.

4.5. Hint: Show 2-Sat is in Co-NL by reducing it to reachability in a graph using log-space reductions.

4.10.

4.11.