Exotic Quantifiers, Complexity Classes, and Complete Problems

(Extended Abstract)
  • Peter Bürgisser
  • Felipe Cucker
Conference paper

DOI: 10.1007/978-3-540-73420-8_20

Volume 4596 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Bürgisser P., Cucker F. (2007) Exotic Quantifiers, Complexity Classes, and Complete Problems. In: Arge L., Cachin C., Jurdziński T., Tarlecki A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg

Abstract

We define new complexity classes in the Blum-Shub-Smale theory of computation over the reals, in the spirit of the polynomial hierarchy, with the help of infinitesimal and generic quantifiers. Basic topological properties of semialgebraic sets like boundedness, closedness, compactness, as well as the continuity of semialgebraic functions are shown to be complete in these new classes. All attempts to classify the complexity of these problems in terms of the previously studied complexity classes have failed. We also obtain completeness results in the Turing model for the corresponding discrete problems. In this setting, it turns out that infinitesimal and generic quantifiers can be eliminated, so that the relevant complexity classes can be described in terms of usual quantifiers only.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Peter Bürgisser
    • 1
  • Felipe Cucker
    • 2
  1. 1.Dept. of Mathematics, University of Paderborn, D-33095 PaderbornGermany
  2. 2.Dept. of Mathematics, City University of Hong KongHong Kong