### Abstract

We consider the problem of determining VC-dimension (Formula Found) of depth four n-input 1-output threshold circuits with h elements. Best known asymptotic lower bounds and upper bounds are proved, that is, when (Formula Found) is upper bounded by ((h^{2}/3)+nh(log h)(1+o(1)) and lower bounded by (1/2)((h^{2}/4)+nh)(log h)(1 − o(1)). We also consider the problem of determining complexity c_{3}(N) of Boolean-valued functions defined on N-pointsets in R^{n}, measured by the number of threshold elements, with which we can construct a depth four circuit to realize the functions. We also show the best known upper and lower bounds, that is, when N → ∞, the complexity is upper bounded by (Formula Found) and lower bounded by (Formula Found)

Original language | English |
---|---|

Title of host publication | Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings |

Editors | Klaus P. Jantke, Shigenobu Kobayashi, Etsuji Tomita, Takashi Yokomori |

Publisher | Springer Verlag |

Pages | 251-264 |

Number of pages | 14 |

ISBN (Print) | 9783540573708 |

DOIs | |

Publication status | Published - 1993 Jan 1 |

Event | 4th Workshop on Algorithmic Learning Theory, ALT 1993 - Tokyo, Japan Duration: 1993 Nov 8 → 1993 Nov 10 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 744 LNAI |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 4th Workshop on Algorithmic Learning Theory, ALT 1993 |
---|---|

Country | Japan |

City | Tokyo |

Period | 93/11/8 → 93/11/10 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

## Fingerprint Dive into the research topics of 'On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions'. Together they form a unique fingerprint.

## Cite this

*Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings*(pp. 251-264). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 744 LNAI). Springer Verlag. https://doi.org/10.1007/3-540-57370-4_52