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.
ASJC Scopus subject areas
- コンピュータ サイエンス（全般）