In computational complexity theory, a nonelementary problem[1] is a problem that is not a member of the class ELEMENTARY. As a class it is sometimes denotes as NONELEMENTARY.
Examples of nonelementary problems that are nevertheless decidable include:
the problem of regular expression equivalence with complementation[2]
decision problem for monadic second-order logic over trees[3]
decision problem for term algebras[4]
satisfiability of the W. V. O. Quine's fluted fragment of first-order logic[5]
References
Vorobyov, Sergei; Voronkov, Andrie (1998), "Complexity of Nonrecursive Logic Programs with Complex Values", Proceedings of the Seventeenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS '98), New York, NY, USA: ACM, pp. 244–253, CiteSeerX 10.1.1.39.8822, doi:10.1145/275487.275515, ISBN 978-0-89791-996-8.
Stockmeyer, Larry J. (1974), The Complexity of Decision Problems in Automata Theory and Logic (PDF), Ph.D. dissertation, Massachusetts Institute of Technology
Libkin, Leonid (2006), "Logics for unranked trees: an overview", Logical Methods in Computer Science, 2 (3): 3:2, 31, arXiv:cs.LO/0606062, doi:10.2168/LMCS-2(3:2)2006, MR 2295773.
Vorobyov, Sergei (1996), "An improved lower bound for the elementary theories of trees", Automated Deduction — Cade-13: 13th International Conference on Automated Deduction New Brunswick, NJ, USA, July 30 – August 3, 1996, Proceedings, Lecture Notes in Computer Science, 1104, Springer, pp. 275–287, CiteSeerX 10.1.1.39.1499, doi:10.1007/3-540-61511-3_91, ISBN 978-3-540-61511-8.
"Quine's Fluted Fragment is Non-Elementary" (PDF).
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