NP not equal to SPACE(n) - MathOverflow

NP not equal to SPACE(n)

2011-02-22T10:30:52Z

Exercise 3.2 of Computational Complexity, a Modern Approach states:

Prove: NP != SPACE(n) [Hint: we don't know if either is a subset of the other.]

I don't know how to solve this problem. It's in the diagonalization chapter.

I've looked around google a bit, but it basically ends up linking back to the Arora/Barak book.

Anyone know how to attack this?

Thanks!

More generally: to prove a language to be uncommputable, I can use diagonalization -- but to prove that two sets of languages (Space(N) and NP) are different, when it's not known that either is contained in the other -- what techniques are there for these proofs?

Thanks again!

Answer by wood for NP not equal to SPACE(n)

2011-02-22T10:48:33Z

I think that a common technique for proving such statements is for example the following type:

One class shares a closure property, while the other cannot because of a hierarchy theorem. Thus they cannot be equal.

In this particular case a proof could proceed along these lines: Since NP is closed under polynomial time reductions, so would SPACE(n), if they were equal. Then deduce that polynomial time reductions would imply that SPACE(n^2) is contained in SPACE(n), which is impossible by the space hierarchy theorem.

Answer by Thierry Zell for NP not equal to SPACE(n)

2011-02-22T11:34:39Z

Scott Aaronson has a blog post, Sidesplitting Proofs, which is a highly recommended read for a variety of reasons. The first proof in the list, which is said to be folklore, is that E, the class of problems solvable in $2^{O(n)}$ time, is not equal to PSPACE. The key is again padding: if the two are equal, then E=EXP, and we derive a contradiction. Like in your case, which is bigger (if one is even contained in the other) is not known.

Answer by fathollah for NP not equal to SPACE(n)

2012-07-06T23:35:11Z

this prob is hard we dont know nL!=np and dont know np!=pspace if u have l in np and not in space(n) you proved np!=nl if u have l in space(n) and not in np you proved np!=pspace