# On the VC-dimension of depth four threshold circuits and the complexity of Boolean-valued functions

Akito Sakurai

4 被引用数 (Scopus)

## 抄録

We consider the problem of determining the VC-dimension δ3(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 → ∞, δ3(h) 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 the complexity C3(N)(c3(N)) of Boolean functions defined on N-pointsets of vertices of n-dimensional hypercube (Boolean-valued functions defined on N-pointsets in Rn, 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 → ∞, C3(N) is upper bounded by √32( N log N)(1 + o(1)) and lower bounded by √6( N log N)(1 - o(1)) and c3(N) is upper bounded by √16( N log N)(1 + o(1)) + 4n2 - 2n and lower bounded by √6( N log N)(1 - o(1)) + ( 9 4)n2 - ( 3 2)n.

本文言語 English 109-127 19 Theoretical Computer Science 137 1 https://doi.org/10.1016/0304-3975(94)00163-D Published - 1995 1月 9 はい

## ASJC Scopus subject areas

• 理論的コンピュータサイエンス
• コンピュータ サイエンス（全般）

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