Abstract
Tarski-Seidenberg principle plays a key role in real algebraic geometry and its applications. It is also constructive and some efficient quantifier elimination algorithms appeared recently. However, the principle is wrong for first-order theories involving certain real analytic functions (e.g., an exponential function). In this case a weaker statement is sometimes true, a possibility to eliminate one sort ofq uantifiers (either ∀ or ∃). We construct an algorithm for a cylindrical cell decomposition ofa closed cube In ⊂ Rn compatible with a semianalytic subset S ⊂ In, defined by analytic functions from a certain broad finitely defined class (Pfaffian functions), modulo an oracle for deciding emptiness of such sets. In particular the algorithm is able to eliminate one sort ofq uantifiers from a first-order formula. The complexity of the algorithm and the bounds on the output are doubly exponential in O(n2).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Basu, S., Pollack, R., Roy, M.-F., On the combinatorial and algebraic complexity ofq uantifier elimination, Journal of the ACM, 43, 1996, 1002–1045.
Blum, L., Cucker, F., Shub, M., Smale, S., Complexity and Real Computation, Springer-Verlag, 1997
Collins, G.E., Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Lecture Notes in Computer Science, 33, 1975, 134–183.
Davenport, J.H., Heintz, J., Real quantifier elimination is doubly exponential,J. Symbolic Comput., 5, 1988, 29–35.
Dries van den, L., Tame Topology and O-minimal Structures, LMS Lecture Note Series, 248, Cambridge University Press, Cambridge, 1998.
Gabrielov, A., Projections ofsemi- analytic sets, Functional Anal. Appl., 2,1968, 282–291.
Gabrielov, A., On complements ofsu banalytic sets and existential formulas for analytic functions, Invent. Math., 125, 1996, 1–12.
Gabrielov, A., Frontier and closure ofa semi-Pfaffian set, Discrete Comput.Geom., 19, 1998, 605–617.
Gabrielov, A., Vorobjov, N., Complexity ofcy lindrical decompositions of sub-Pfaffian sets, J. Pure and Appl. Algebra, 164 (2001), 179–197.
Gabrielov, A., Vorobjov, N., Complexity ofst ratifications of semi-Pfaffian sets,Discrete Comput. Geom., 14, 1995, 71–91.
Grigoriev, D., Complexity ofd eciding Tarski algebra, J. Symbolic Comput., 5, 1988, 65–108.
Grigoriev, D., Vorobjov, N., Solving systems ofp olynomial inequalities in subexponential time, J. Symbolic Comput., 5, 1988, 37–64.
Khovanskii, A., On a class ofsy stems oft ranscendental equations, Soviet Math. Dokl., 22, 1980, 762–765.
Khovanskii, A., Fewnomials, AMS Transl. Math. Monographs, 88, AMS, 1991.
Milnor, J. W.Topology from the differentiable viewpoint, Univ. Pres, Virginia, 1965
Osgood, W. F., On functions of several complex variables,Trans. Amer. Math. Soc., 17, 1916
Renegar, J., On the computational complexity and the first order theory of reals. Parts I–III, J. Symbolic Comput., 13, 1992, 255–352.
Vorobjov, N., The complexity ofde ciding consistency ofs ystems of polynomial in exponent inequalities, J. Symbolic Comput., 13, 1992, 139–173.
Wilkie, A., A general theorem oft he complement and some new o-minimal structures,Selecta Mathematica, New Series, 5, 1999, 397–421
W¨uthrich, H.R., Ein Entscheidungsvefahren f¨ur die Theorie der reellabgeschlossenen K¨orper, Lecture Notes in Computer Science, 43, 1976, 138–162.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pericleous, S., Vorobjov, N. (2003). New Complexity Bounds for Cylindrical Decompositions of Sub-Pfaffian Sets. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds) Discrete and Computational Geometry. Algorithms and Combinatorics, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55566-4_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-55566-4_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62442-1
Online ISBN: 978-3-642-55566-4
eBook Packages: Springer Book Archive