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

Publisher | Springer Verlag |

Pages | 251-264 |

Number of pages | 14 |

Volume | 744 LNAI |

ISBN (Print) | 9783540573708 |

Publication status | Published - 1993 |

Externally published | Yes |

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) |
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Volume | 744 LNAI |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

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

City | Tokyo |

Period | 93/11/8 → 93/11/10 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings*(Vol. 744 LNAI, 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.

**On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions.** / Sakurai, Akito.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings.*vol. 744 LNAI, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 744 LNAI, Springer Verlag, pp. 251-264, 4th Workshop on Algorithmic Learning Theory, ALT 1993, Tokyo, Japan, 93/11/8.

}

TY - GEN

T1 - On the VC-dimension of depth four threshold circuits and the complexity of boolean-valued functions

AU - Sakurai, Akito

PY - 1993

Y1 - 1993

N2 - 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 ((h2/3)+nh(log h)(1+o(1)) and lower bounded by (1/2)((h2/4)+nh)(log h)(1 − o(1)). We also consider the problem of determining complexity c3(N) of Boolean-valued functions defined on N-pointsets in Rn, 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)

AB - 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 ((h2/3)+nh(log h)(1+o(1)) and lower bounded by (1/2)((h2/4)+nh)(log h)(1 − o(1)). We also consider the problem of determining complexity c3(N) of Boolean-valued functions defined on N-pointsets in Rn, 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)

UR - http://www.scopus.com/inward/record.url?scp=84899003901&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84899003901&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84899003901

SN - 9783540573708

VL - 744 LNAI

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 251

EP - 264

BT - Algorithmic Learning Theory - 4th International Workshop, ALT 1993, Proceedings

PB - Springer Verlag

ER -