On the Number of Multivalued Switching Functions Realizable by Cascades

Abstract

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.