EXPSPACE

Template:Short description In computational complexity theory, Template:Sans-serif 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 Template:Sans-serif. If we use a nondeterministic machine instead, we get the class Template:Sans-serif, which is equal to Template:Sans-serif by Savitch's theorem.

A decision problem is Template:Sans-serif if it is in Template:Sans-serif, and every problem in Template:Sans-serif 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. Template:Sans-serif problems might be thought of as the hardest problems in Template:Sans-serif.

Template:Sans-serif is a strict superset of Template:Sans-serif, Template:Sans-serif, and Template:Sans-serif. It contains Template:Sans-serif and is believed to strictly contain it, but this is unproven.

Formal definition

In terms of Template:Sans-serif and Template:Sans-serif,

$\mathsf{E}\mathsf{X}\mathsf{P}\mathsf{S}\mathsf{P}\mathsf{A}\mathsf{C}\mathsf{E}=\bigcup _{k\in \mathbb{\mathbb{N}}}\mathsf{D}\mathsf{S}\mathsf{P}\mathsf{A}\mathsf{C}\mathsf{E}\left(2^{n^k}\right)=\bigcup _{k\in \mathbb{\mathbb{N}}}\mathsf{N}\mathsf{S}\mathsf{P}\mathsf{A}\mathsf{C}\mathsf{E}\left(2^{n^k}\right)$

Examples of problems

Formal languages

An example of an Template:Sans-serif 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).

Logic

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

Reasoning in the first-order theory of the real numbers with +, ×, = is in EXPSPACE and was conjectured to be EXPSPACE-complete in 1986.

Petri nets

The coverability problem for Petri Nets is Template:Sans-serif-complete.

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

See also

• Game complexity

References

• Section 9.1.1: Exponential space completeness, pp. 313–317. Demonstrates that determining equivalence of regular expressions with exponentiation is EXPSPACE-complete.