PSPACE-hard

In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space.

If we denote by SPACE(t(n)), the set of all problems that can be solved by Turing machines using O(t(n)) space for some function t of the input size n, then we can define PSPACE formally as

PSPACE is a strict superset of the set of context-sensitive languages.

It turns out that allowing the Turing machine to be nondeterministic does not add any extra power. Because of Savitch's theorem, NPSPACE is equivalent to PSPACE, essentially because a deterministic Turing machine can simulate a non-deterministic Turing machine without needing much more space (even though it may use much more time). Also, the complements of all problems in PSPACE are also in PSPACE, meaning that co-PSPACE = PSPACE.

The following relations are known between PSPACE and the complexity classes NL, P, NP, PH, EXPTIME and EXPSPACE (note that ⊊ is not the same as ⊈):

It is known that in the first and second line, at least one of the set containments must be strict, but it is not known which. It is widely suspected that all are strict.

The containments in the third line are both known to be strict. The first follows from direct diagonalization (the space hierarchy theorem, NL ⊊ NPSPACE) and the fact that PSPACE = NPSPACE via Savitch's theorem. The second follows simply from the space hierarchy theorem.

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