computational complexity

, Volume 4, Issue 1, pp 12–36 | Cite as

A note on Rabin's width of a complete proof

  • José L. Montaña
  • Luis M. Pardo
  • Tomás Recio


We introduce and analyze the concept of generic width of a semialgebraic set, showing that it gives lower bounds for decisional complexities. By means of the computation of the generic width we are able to solve rigorously the complexity problems posed by M.O. Rabin in [10], such as optimization of linear mappings on finite sets. We show that the results on the generic width can also be applied to obtain lower bounds for problems which in general do not admit a linear mapping description, such as optimization of polynomial mappings on finite sets, existence of a real root, finite selection and subset decision, or the direct oriented-convex hull problem introduced by J. Jaromczyk in [8].

Key words

Algebraic complexity theory decisional complexity semialgebraic sets width of a complete proof generic width of a semialgebraic set 

Subject classifications

68Q25 68Q40 14P20 14P10 


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  1. [1]
    M. Ben-Or, Lower bounds for algebraic computation trees. InProc. Fifteenth Ann. ACM Symp. Theor. Comput., 1983, 80–86.Google Scholar
  2. [2]
    R. Benedetti andJ. J. Risler,Real algebraic and semialgebraic geometry. Hermann, Paris, 1990.Google Scholar
  3. [3]
    J. Bochnak, M. Coste andM.-F. Roy,Géométrie algébrique réelle. Ergebnisse der Math., 3.Folge, Band12, Springer-Verlag, Berlin, Heidelberg, New York, 1987.Google Scholar
  4. [4]
    J. Bochnak, Sur la factorialité des anneaux de fonctions de Nash.Comment. Math. Helv. 52 (1977), 211–218.Google Scholar
  5. [5]
    L. Bröcker, Minimale Erzeugung von Positivbereich.Geom. Dedicata 16 (1984), 335–350.Google Scholar
  6. [6]
    L. Bröcker, Spaces of orderings and semialgebraic sets. InQuadratic and Hermitian Forms, CMS Conf. Proc. 4, Providence, Amer. Math. Soc. (1984), 231–248.Google Scholar
  7. [7]
    M. Coste, Ensembles Semi-algébriques. InGéométrie Algébrique Réelle et Formes Quadratiques, ed.J. L. Colliot-Thélene, M. Coste, L. Mahé, andM.-F. Roy. Lecture Notes in Mathematics959, Springer-Verlag, Berlin, Heidelberg, New York, 1982, 109–139.Google Scholar
  8. [8]
    J. Jaromczyk, An extension of Rabin's complete proof concept. InMath. Found. of Comp. Sci. 1981, ed.J. Gruska andM. Chytill. Lecture Notes in Computer Science118, Springer-Verlag, Berlin, Heidelberg, New York, 1981, 321–326.Google Scholar
  9. [9]
    J. L. Montaña, L. M. Pardo andT. Recio, The non-scalar model of complexity in computational geometry. InProc. MEGA'90, ed.C. Traverso andT. Mora. Progress in Mathematics94, Birkhäuser Boston, 1991, 347–362.Google Scholar
  10. [10]
    M. O. Rabin, Proving simultaneous positivity of linear forms.J. Comput. System Sci. 6 (1972) 639–650.Google Scholar
  11. [11]
    T. Recio, Una Descomposición de un Conjunto Semialgebraico. InActas del V Congreso de la Agrupación de Matemáticos de Expresión Latina, CSIC, Publicaciones del Instituto Jorge Juan, Madrid, 1978, 217–221.Google Scholar
  12. [12]
    J. J. Risler, Sur l'anneau des fonctions de Nash globales.Ann. Scien. Ecole Norm. Sup., 4éme série,8 (1975), 365–378.Google Scholar
  13. [13]
    J.T. Schwartz,Differential Geometry and Topology Notes on Mathematics and its Applications, Gordon and Breach, 1968.Google Scholar
  14. [14]
    V. Strassen, Algebraic Complexity Theory. InHandbook of Theoretical Computer Science, ed.J. van Leeuwen. Elsevier, Amsterdam, 1990, 633–673.Google Scholar
  15. [15]
    F.F. Yao, Computational Geometry. InHandbook of Theoretical Computer Science, ed.J. van Leeuwen. Elsevier, Amsterdam, 1990, 343–391.Google Scholar

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • José L. Montaña
    • 1
  • Luis M. Pardo
    • 1
  • Tomás Recio
    • 1
  1. 1.Dept. de Matemáticas, Est. y Comp. Facultad de CienciasUniversidad de CantabriaSantanderSpain

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