On the Number of Multivalued Switching Functions Realizable by Cascades

Butler [6] gave recurrence relations for computing the number N
r
(n) of r-valued (r ≥ 2) switching functions of n variables realizable by a cascaded network of n - 1 r-valued cells. The cascades are assumed to have a fixed-input variable assignment. In this correspondence similar recurrence relations are presented for the number M
r
(n) of such cascade realizable functions that depend on all n input variables. In particular, for the ternary valued case, explicit formulas for N
3
(n) and M
3
(n) are given. Some asymptotic properties of N
r
(n) and M
r
(n) are also derived, which show that their growth is asymptotically exponential as n increases.