## Abstract

Fagin et al. characterized those symmetric Boolean functions which can be computed by small AND/OR circuits of constant depth and unbounded fan-in. Here we provide a similar characterization for d-perceptrons-AND/OR circuits of constant depth and unbounded fan-in with a single MAJORITY gate at the output. We show that a symmetric function has small (quasipolynomial, or 2^{log O(1) n} size) d-perceptrons iff it has only poly-log many sign changes (i.e., it changes value log^{O(1)} n times as the number of positive inputs varies from zero to n). A consequence of the lower bound is that a recent construction of Beigel is optimal. He showed how to convert a constant-depth unbounded fan-in AND/OR circuit with poly-log many MAJORITY gates into an equivalent d-perceptron-we show that more than poly-log MAJORITY gates cannot in general be converted to one.

Original language | English (US) |
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Title of host publication | STACS 1993 - 10th Annual Symposium on Theoretical Aspects of Computer Science, Proceedings |

Editors | Patrice Enjalbert, Alain Finkel, Klaus W. Wagner |

Publisher | Springer Verlag |

Pages | 535-544 |

Number of pages | 10 |

ISBN (Print) | 9783540565031 |

DOIs | |

State | Published - 1993 |

Event | 10th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1993 - Wurzburg, Germany Duration: Feb 25 1993 → Feb 27 1993 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 665 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 10th Annual Symposium on Theoretical Aspects of Computer Science, STACS 1993 |
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Country | Germany |

City | Wurzburg |

Period | 2/25/93 → 2/27/93 |

### Bibliographical note

Funding Information:The first author was supported by grants CCR-8812567 and CCR-9008416. The second author was supported by NSF Computer and Computation Theory grants CCR-8922098 and CCR-9207829. The third author was supported in part by the ESPRIT IIBRA Programme of the EC under contract 7141 (ALCOM II).