## Authors: Arnaud Durand, Anselm Haak and Heribert Vollmer

## Paper Information

Title: | Model-Theoretic Characterizations of Boolean and Arithmetic Circuit Classes of Small Depth |

Authors: | Arnaud Durand, Anselm Haak and Heribert Vollmer |

Proceedings: | LICS PDF files |

Editors: | Anuj Dawar and Erich GrĂ¤del |

Keywords: | finite model theory, descriptive complexity, arithmetic circuits, counting classes |

Abstract: | ABSTRACT. In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes $\textrm{NC}^1$, $\textrm{SAC}^1$ and $\textrm{AC}^1$ as well as their arithmetic counterparts $\#\textrm{NC}^1$, $\#\textrm{SAC}^1$ and $\#\textrm{AC}^1$. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulae of first-order logic, i.e., $\textrm{AC}^0 = \textrm{FO}$, and augment the logical language with an operator for defining relations in an inductive way. Considering slight variations of the new operator, we obtain uniform characterizations of the three just mentioned Boolean classes. The arithmetic classes can then be characterized by functions counting winning strategies in semantic games for formulae characterizing languages in the corresponding Boolean class. |

Pages: | 10 |

Talk: | Jul 10 11:20 (Session 54D) |

Paper: |