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On the Approximation of Interval Functions

  • Klaus Meer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3732)

Abstract

Many problems in interval arithmetic in a natural way lead to a quantifier elimination problem over the reals. By studying closer the precise form of the latter we show that in some situations it is possible to obtain a refined complexity analysis of the problem. This is done by structural considerations of the special form of the quantifiers and its implications for the analysis in a real number model of computation. Both can then be used to obtain as well new results in the Turing model. We exemplify our approach by dealing with different versions of the approximation problem for interval functions.

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References

  1. 1.
    Basu, S., Pollack, R., Roy, M.F.: On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM 43(6), 1002–1045 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)Google Scholar
  3. 3.
    Cucker, F., Matamala, M.: On digital nondeterminism. Mathematical Systems Theory 29, 635–647 (1996)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Cucker, F., Shub, M., Smale, S.: Complexity separations in Koiran’s weak model. Theoretical Computer Science 133, 3–14 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants, and multidimensional determimants. Birkhäuser, Basel (1994)CrossRefGoogle Scholar
  6. 6.
    Gustafson, S.Å., Kortanek, K.O.: Semi-infinte programming and applications. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming: The State of the Art, pp. 132–157. Springer, Heidelberg (1983)Google Scholar
  7. 7.
    Koiran, P.: A weak version of the Blum-Shub-Smale model. In: 34th Annual IEEE Symposium on Foundations of Computer Science, pp. 486–495 (1993)Google Scholar
  8. 8.
    Kreinovich, V., Lakeyev, A.V., Rohn, J., Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht (1997)Google Scholar
  9. 9.
    Kreinovich, V., Meer, K.: Complexity results for the range problem in interval arithmetic (in preparation)Google Scholar
  10. 10.
    Koshelev, M., Longpré, L., Taillibert, P.: Optimal Enclusure of Quadratic Interval Functions. Reliable Computing 4, 351–360 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lakeyev, A.V., Noskov, S.I.: A description of the set of solutions of a linear equation with interval defined operator and right-hand side. Russian Acad. Sci. Dokl. Math. 47(3), 518–523 (1993)MathSciNetGoogle Scholar
  12. 12.
    Meer, K.: On the complexity of quadratic programming in real number models of computation. Theoretical Computer Science 133, 85–94 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Meer, K.: On a refined analysis of some problems in interval arithmetic using real number complexity theory. Reliable Computing 10(3), 209–225 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Plaisted, D.A.: New NP-hard and NP-complete polynomial and integer divisibility problems. Theoretical Computer Science 31, 125–138 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rohn, J.: Enclosing solutions of linear interval equations is NP-hard. Computing 53, 365–368 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sturmfels, B.: Introduction to resultants. Application of Computational Algebraic Geometry. In: Cox, D.A., Sturmfels, B. (eds.) Proc. of Symposia in Applied Mathematics, vol. 53, pp. 25–39. American Mathematical Society, Providence (1998)Google Scholar
  17. 17.
    Woźniakowski, H.: Why does information-based complexity use the real number model? Theoretical Computer Science 219, 451–465 (1999)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Klaus Meer
    • 1
  1. 1.Department of Mathematics and Computer ScienceSyddansk UniversitetOdense MDenmark

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