EXPSPACE
Template:Short description In computational complexity theory, EXPSPACE is the set of all decision problems solvable by a deterministic Turing machine in exponential space, i.e., 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.
EXPSPACE is a strict superset of PSPACE, NP, and P. It contains EXPTIME and is believed to strictly contain it, but this is unproven.
Formal definition
In terms of DSPACE and NSPACE,
Examples of problems
Formal languages
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).[1]
Logic
Alur and Henzinger extended linear temporal logic with times (integer) and prove that the validity problem of their logic is EXPSPACE-complete.[2]
Reasoning in the first-order theory of the real numbers with +, ×, = is in EXPSPACE and was conjectured to be EXPSPACE-complete in 1986.[3]
Petri nets
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The coverability problem for Petri Nets is EXPSPACE-complete.[4]
The reachability problem for Petri nets was known to be EXPSPACE-hard for a long time,[5] but shown to be nonelementary,[6] so probably not in EXPSPACE. In 2022 it was shown to be Ackermann-complete.[7][8]
It is EXPSPACE-complete to decide whether the reachability of a given vector addition system is finite.[9]
See also
References
- ^ 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.
- ^ Page Module:Citation/CS1/styles.css has no content.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.
- ^ Page Module:Citation/CS1/styles.css has no content.Ben-Or, Michael; Kozen, Dexter; Reif, John (1986-04-01). "The complexity of elementary algebra and geometry". Journal of Computer and System Sciences. 32 (2): 251–264. doi:10.1016/0022-0000(86)90029-2. ISSN 0022-0000.
- ^ Page Module:Citation/CS1/styles.css has no content.Charles Rackoff (1978). "The covering and boundedness problems for vector addition systems". Theoretical Computer Science: 223–231.
- ^ Page Module:Citation/CS1/styles.css has no content.Lipton, R. (1976). "The Reachability Problem Requires Exponential Space". Technical Report 62. Yale University.
- ^ Page Module:Citation/CS1/styles.css has no content.Wojciech Czerwiński Sławomir Lasota Ranko S Lazić Jérôme Leroux Filip Mazowiecki (2019). "The reachability problem for Petri nets is not elementary". STOC 19.
- ^ Page Module:Citation/CS1/styles.css has no content.Leroux, Jerome (February 2022). "The Reachability Problem for Petri Nets is Not Primitive Recursive". 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE. pp. 1241–1252. arXiv:2104.12695. doi:10.1109/FOCS52979.2021.00121. ISBN 978-1-6654-2055-6.
- ^ Page Module:Citation/CS1/styles.css has no content.Brubaker, Ben (4 December 2023). "An Easy-Sounding Problem Yields Numbers Too Big for Our Universe". Quanta Magazine.
- ^ Page Module:Citation/CS1/styles.css has no content.Schmitz, Sylvain (2016-02-03). "Complexity Hierarchies beyond Elementary". ACM Transactions on Computation Theory. 8 (1): 1–36. arXiv:1312.5686. doi:10.1145/2858784. ISSN 1942-3454.
- Page Module:Citation/CS1/styles.css has no content.Berman, Leonard (1 May 1980). "The complexity of logical theories". Theoretical Computer Science. 11 (1): 71–77. doi:10.1016/0304-3975(80)90037-7.
- Page Module:Citation/CS1/styles.css has no content.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.
