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

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.