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Counting Complexity Classes over the Reals I: The Additive Case

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Algorithms and Computation (ISAAC 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2906))

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Abstract

We define a counting class #Padd in the Blum-Shub-Smale-setting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class #P introduced by Valiant. We also establish transfer theorems for both directions between the real additive and the discrete setting. Then we characterize in terms of completeness results the complexity of computing basic topological invariants of semi-linear sets given by additive circuits. It turns out that the computation of the Euler characteristic is FP\(_{\rm add}^{\#P_{\rm add}}\)-complete, while for fixed k, the computation of the kth Betti number is FPARadd-complete. Thus the latter is more difficult under standard complexity theoretic assumptions. We use all the above to prove some analogous completeness results in the classical setting.

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Author information

Authors and Affiliations

  1. Faculty of Computer Science, Electrical Engineering, and Mathematics, Paderborn University, D-33095, Paderborn, Germany

    Peter Bürgisser

  2. Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China

    Felipe Cucker

Authors
  1. Peter Bürgisser
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  2. Felipe Cucker
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Editor information

Editors and Affiliations

  1. School of Science and Technology, Kwansei Gakuin University, Sanda, Japan

    Toshihide Ibaraki

  2. Department of Architecture and Architectural Engineering, Kyoto University, 615-8540, Nishikyo-ku, Kyoto, Japan

    Naoki Katoh

  3. Department of Computer Science and Communication Engineering, Kyushu University, 812-8581, Fukuoka, Japan

    Hirotaka Ono

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© 2003 Springer-Verlag Berlin Heidelberg

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Bürgisser, P., Cucker, F. (2003). Counting Complexity Classes over the Reals I: The Additive Case. In: Ibaraki, T., Katoh, N., Ono, H. (eds) Algorithms and Computation. ISAAC 2003. Lecture Notes in Computer Science, vol 2906. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24587-2_64

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  • DOI: https://doi.org/10.1007/978-3-540-24587-2_64

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20695-8

  • Online ISBN: 978-3-540-24587-2

  • eBook Packages: Springer Book Archive

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