EXPSPACE

In computational complexity theory,  is the set of all decision problems solvable by a deterministic Turing machine in exponential space, i.e., in $$O(2^{p(n)})$$ space, where $$p(n)$$ is a polynomial function of $$n$$. Some authors restrict $$p(n)$$ to be a linear function, but most authors instead call the resulting class. If we use a nondeterministic machine instead, we get the class, which is equal to by Savitch's theorem.

A decision problem is if it is in, and every problem in  has a polynomial-time many-one reduction to it. In other words, there is a polynomial-time algorithm that transforms instances of one to instances of the other with the same answer. problems might be thought of as the hardest problems in.

is a strict superset of, , and and is believed to be a strict superset of.

Formal definition
In terms of and ,


 * $$\mathsf{EXPSPACE} = \bigcup_{k\in\mathbb{N}} \mathsf{DSPACE}\left(2^{n^k}\right) = \bigcup_{k\in\mathbb{N}} \mathsf{NSPACE}\left(2^{n^k}\right)$$

Examples of problems
An example of an problem is the problem of recognizing whether two regular expressions represent different languages, where the expressions are limited to four operators: union, concatenation, the Kleene star (zero or more copies of an expression), and squaring (two copies of an expression).

Alur and Henzinger extended linear temporal logic with times (integer) and prove that the validity problem of their logic is EXPSPACE-complete.

The coverability problem for Petri Nets is -complete.

The reachability problem for Petri nets was known to be -hard for a long time, but shown to be nonelementary, so probably not in. In 2022 it was shown to be Ackermann-complete.

Relationship to other classes
is known to be a strict superset of, , and. It is further suspected to be a strict superset of, however this is not known.