2-EXPTIME

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

In terms of DTIME,

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

Comparison with other complexity classes

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 ${\displaystyle 2^{2^{2^{n^{k}}}}}$. This can be generalized to higher and higher time bounds.

Examples

Examples of algorithms that require at least double-exponential time 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]
• Quantifier elimination on real closed fields takes doubly exponential time (see Cylindrical algebraic decomposition). Thus, deciding whether a first-order formula over the real numbers is in 2-EXPTIME. But it was shown to be EXPSPACE and was conjectured to be EXPSPACE-complete in 1986. ^[5]
• Calculating the complement of a regular expression ^[6]

2-EXPTIME-complete problems

Logic

The satisfiability problem for CTL ⁺ (Computation tree logic) is 2-EXPTIME-complete ^[7] . The satisfiability problem of ATL* (alternating-time temporal logic) is 2-EXPTIME-complete ^[8] .

Implicational Relevance Logic is 2-EXPTIME-complete ^[9] .

The satisfiability problem for propositional dynamic logic with intersection (IPDL) is 2-EXPTIME-complete ^[10] .

Planning

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. ^[11]

Synthesis

LTL (linear temporal logic) synthesis (deciding whether a reactive module satisfying an LTL specification) is 2EXPTIME-complete ^[12] .

See also

• Double exponential function

References

1. Christos Papadimitriou, Computational Complexity (1994), ISBN   978-0-201-53082-7. Section 20.1, corollary 3, page 495.
2. 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
3. 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.
4. 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 .
5. 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.
6. 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.
7. 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.
8. Schewe, Sven (2008). Aceto, Luca; Damgård, Ivan; Goldberg, Leslie Ann; Halldórsson, Magnús M.; Ingólfsdóttir, Anna; Walukiewicz, Igor (eds.). "ATL* Satisfiability Is 2EXPTIME-Complete". Automata, Languages and Programming. Berlin, Heidelberg: Springer: 373–385. doi:10.1007/978-3-540-70583-3_31. ISBN   978-3-540-70583-3.
9. Schmitz, Sylvain (2016). "Implicational Relevance Logic Is 2-Exptime-Complete". The Journal of Symbolic Logic. 81 (2): 641–661. ISSN   0022-4812.
10. Lange, Martin; Lutz, Carsten (2005). "2-Exptime Lower Bounds for Propositional Dynamic Logics with Intersection". The Journal of Symbolic Logic. 70 (4): 1072–1086. ISSN   0022-4812.
11. Jussi Rintanen (2004). "Complexity of Planning with Partial Observability" (PDF). Proceedings of International Conference on Automated Planning and Scheduling. AAAI Press: 345–354.
12. Pnueli, A.; Rosner, R. (1989-01-03). "On the synthesis of a reactive module". Proceedings of the 16th ACM SIGPLAN-SIGACT symposium on Principles of programming languages. POPL '89. New York, NY, USA: Association for Computing Machinery: 179–190. doi:10.1145/75277.75293. ISBN   978-0-89791-294-5.