In complexity theory, 'EXPSPACE' is the set of all decision problems solvable by a deterministic Turing machine in space, where is a polynomial function of . (Some authors restrict to be a linear function, but most authors instead call the resulting class ESPACE.) If we use a nondeterministic machine instead, we get the class NEXPSPACE, which is equal to EXPSPACE by Savitch's theorem.
A decision problem is EXPSPACE-complete if it is in EXPSPACE, and every problem in EXPSPACE 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. EXPSPACE-complete problems might be thought of as the hardest problems in EXPSPACE.
Examples of problems
An example of an EXPSPACE-complete 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).
If the Kleene star is left out, then that problem becomes NEXPTIME-complete, which is like EXPTIME-complete, except it is defined in terms of non-deterministic Turing machines rather than deterministic.
It has also been shown by L. Berman in 1980 that the problem of verifying/falsifying any first-order statement about real numbers that involves only addition and comparison (but no multiplication) is in EXPSPACE.
The reachability problem for Petri Nets is EXPSPACE-hard.
- Meyer, A.R. and L. Stockmeyer. The equivalence problem for regular expressions with squaring requires exponential space. 13th IEEE Symposium on Switching and Automata Theory, Oct 1972, pp.125–129.
- Alur, Rajeev; Henzinger, Thomas A. (1994-01-01). "A Really Temporal Logic". J. ACM. 41 (1): 181–203. doi:10.1145/174644.174651. ISSN 0004-5411.
- Lipton, R. (1976). "The Reachability Problem Requires Exponential Space". Technical Report 62. Yale University.
- L. Berman The complexity of logical theories, Theoretical Computer Science 11:71-78, 1980.
- Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Section 9.1.1: Exponential space completeness, pp. 313–317. Demonstrates that determining equivalence of regular expressions with exponentiation is EXPSPACE-complete.