Citation key |
BC-Counting-Complexity-Classes-Over-The-Reals-I-The-Additive-Case |

Author |
Peter Bürgisser and Felipe Cucker |

Title of Book |
In Proceedings of ISAAC 2003 |

Pages |
625-634 |

Year |
2003 |

Number |
2906 |

Publisher |
Springer |

Series |
Lecture Notes in Computer Science |

Abstract |
We define a counting class $\#P_add$ 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 $FP^#P_add_add$-complete, while for fixed $k$, the computation of the $k$-th Betti number is $FPAR_add$-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. |