2-EXPTIME

In computational complexity theory, the complexity class 2-EXPTIME (sometimes called 2-EXP) is the set of all decision problems solvable by a deterministic Turing machine in O(2^{2^p(n)}) time, where p(n) is a polynomial function of n.

In terms of DTIME,

{\mathsf {2{\mbox{-}}EXPTIME}}=\bigcup _{k\in \mathbb {N} }{\mathsf {DTIME}}\left(2^{2^{n^{k}}}\right).

We know

P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE ⊆ 2-EXPTIME ⊆ ELEMENTARY.

2-EXPTIME can also be reformulated as the space class AEXPSPACE, the problems that can be solved by an alternating Turing machine in exponential space. This is one way to see that EXPSPACE ⊆ 2-EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine.^[1]

2-EXPTIME is one class in a hierarchy of complexity classes with increasingly higher time bounds. The class 3-EXPTIME is defined similarly to 2-EXPTIME but with a triply exponential time bound $2^{2^{2^{n^{k}}}}$ . This can be generalized to higher and higher time bounds.

Examples

Examples of algorithms that require at least 2-EXPTIME include:

Each decision procedure for Presburger arithmetic provably requires at least doubly exponential time^[2]
Computing a Gröbner basis over a field. In the worst case, a Gröbner basis may have a number of elements which is doubly exponential in the number of variables. On the other hand, the worst-case complexity of Gröbner basis algorithms is doubly exponential in the number of variables as well as in the entry size.^[3]
Finding a complete set of associative-commutative unifiers^[4]
Satisfying CTL⁺ (which is, in fact, 2-EXPTIME-complete)^[5]
Quantifier elimination on real closed fields takes doubly exponential time (see Cylindrical algebraic decomposition).
Calculating the complement of a regular expression^[6]

2-EXPTIME-complete problems

Generalizations of many fully observable games are EXPTIME-complete. These games can be viewed as particular instances of a class of transition systems defined in terms of a set of state variables and actions/events that change the values of the state variables, together with the question of whether a winning strategy exists. A generalization of this class of fully observable problems to partially observable problems lifts the complexity from EXPTIME-complete to 2-EXPTIME-complete.^[7]

References

^ Christos Papadimitriou, Computational Complexity (1994), ISBN 978-0-201-53082-7. Section 20.1, corollary 3, page 495.
^ Fischer, M. J., and Michael O. Rabin, 1974, ""Super-Exponential Complexity of Presburger Arithmetic. Archived 2006-09-15 at the Wayback Machine" Proceedings of the SIAM-AMS Symposium in Applied Mathematics Vol. 7: 27–41
^ Dubé, Thomas W. (August 1990). "The Structure of Polynomial Ideals and Gröbner Bases". SIAM Journal on Computing. 19 (4): 750–773. doi:10.1137/0219053.
^ Kapur, Deepak; Narendran, Paliath (1992), "Double-exponential complexity of computing a complete set of AC-unifiers", [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science, pp. 11–21, doi:10.1109/LICS.1992.185515, ISBN 0-8186-2735-2, S2CID 206437926.
^ Johannsen, Jan; Lange, Martin (2003), "CTL⁺ is complete for double exponential time", in Baeten, Jos C. M.; Lenstra, Jan Karel; Parrow, Joachim; Woeginger, Gerhard J. (eds.), Proceedings of the 30th International Colloquium on Automata, Languages and Programming (ICALP 2003) (PDF), Lecture Notes in Computer Science, vol. 2719, Springer-Verlag, pp. 767–775, doi:10.1007/3-540-45061-0_60, ISBN 978-3-540-40493-4, archived from the original (PDF) on 2007-09-30, retrieved 2006-12-22.
^ Gruber, Hermann; Holzer, Markus (2008). "Finite Automata, Digraph Connectivity, and Regular Expression Size" (PDF). Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2008). Vol. 5126. pp. 39–50. doi:10.1007/978-3-540-70583-3_4.
^ Jussi Rintanen (2004). "Complexity of Planning with Partial Observability" (PDF). Proceedings of International Conference on Automated Planning and Scheduling. AAAI Press: 345–354.