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Real quantifier elimination is doubly exponential. (English) Zbl 0663.03015
The authors show that quantifier elimination over the first-order theory of real-closed fields can require doubly-exponential space (and hence time) and show that this doubly-exponential behaviour is intrinsic to the problem. This result has already been proved by Weispfenning by a completely different method in 1985, but the method of the paper is of independent interest.
Reviewer: Li Xiang

03C10Quantifier elimination and related topics
68Q25Analysis of algorithms and problem complexity
12L05Decidability related to field theory
68W30Symbolic computation and algebraic computation
12D99Real and complex fields
Full Text: DOI
[1] Ben-Or, M.; Kozen, D.; Reif, J.: The complexity of elementary algebra and geometry. J. comput. Syst. sci 32, 251-264 (1986) · Zbl 0634.03031
[2] Collins, G. E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. Springer lecture notes in computer science 33, 134-183 (1975) · Zbl 0318.02051
[3] Davenport, J. H.: Computer algebra for cylindrical algebraic decomposition. TRITA-NA-8511, NADA, KTH, Stockholm, sept. 1985 (1985)
[4] Heintz, J.: Definability and fast quantifier elimination in algebraically closed fields. Theor. comp. Sci. 26, 239-277 (1983) · Zbl 0546.03017
[5] Mccallum, S.: An improved projection operation for cylindrical algebraic decomposition. Proc. EUROCAL 85 2, 277-278 (1985)
[6] Mccallum, S.: An improved projection operation for cylindrical algebraic decomposition. Computer science tech. Report 548 (1985)
[7] Tarski, A.: A decision method for elementary algebra and geometry. (1951) · Zbl 0044.25102
[8] Weispfenning, V.: The complexity of linear problems in fields. Manuscript, 1985. (1985) · Zbl 0576.06017