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.