Bounds on the Number of Threshold Functions

It has been conjectured [1] that the number R
_n
of threshold functions of n arguments has the limiting form: Lim
_n→∞
log
₂
R
_n
/n
²
= const. Bounds previously obtained [2], [3] show that such a constant would have to lie between ⅓ and one. In the present note this constant is shown to have a lower bound of ½.
¹
The result is extended to the number R
_n
^m
of threshold functions defined on m minterms of n arguments and suggests the more general form in the limit of large n, m/n. {log
_m/n
R
_n
^m
/n} = const. with the same limits for the constant, providing that the minterms are spread out in a certain sense.