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Counting Complexity Classes over the Reals I: The Additive Case

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

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.

Keywords

  • Turing Machine
  • Complexity Class
  • Euler Characteristic
  • Betti Number
  • Algebraic Topology

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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

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