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Exotic Quantifiers, Complexity Classes, and Complete Problems

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Automata, Languages and Programming (ICALP 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4596))

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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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Author information

Authors and Affiliations

  1. Dept. of Mathematics, University of Paderborn, D-33095 Paderborn, Germany

    Peter Bürgisser

  2. Dept. of Mathematics, City University of Hong Kong, Hong Kong

    Felipe Cucker

Authors
  1. Peter Bürgisser
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  2. Felipe Cucker
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Editor information

Lars Arge Christian Cachin Tomasz Jurdziński Andrzej Tarlecki

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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. https://doi.org/10.1007/978-3-540-73420-8_20

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  • DOI: https://doi.org/10.1007/978-3-540-73420-8_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73419-2

  • Online ISBN: 978-3-540-73420-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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