### Abstract

We consider the problem of determining the VC-dimension δ_{3}(h) 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 h → ∞, δ_{3}(h) 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 the complexity C_{3}(N)(c_{3}(N)) of Boolean functions defined on N-pointsets of vertices of n-dimensional hypercube (Boolean-valued functions defined on N-pointsets in R^{n}, respectively), 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 → ∞, C_{3}(N) is upper bounded by √32( N log N)(1 + o(1)) and lower bounded by √6( N log N)(1 - o(1)) and c_{3}(N) is upper bounded by √16( N log N)(1 + o(1)) + 4n^{2} - 2n and lower bounded by √6( N log N)(1 - o(1)) + ( 9 4)n^{2} - ( 3 2)n.

Original language | English |
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Pages (from-to) | 109-127 |

Number of pages | 19 |

Journal | Theoretical Computer Science |

Volume | 137 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1995 Jan 9 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)