Enumeration of Seven-Argument Threshold Functions

Abstract

A tabulation of the 2470 representative threshold functions of seven arguments has been prepared by the author. This paper discusses the methods used in, and the threshold logic implications of, the enumeration. The self-dual classification method of Goto-Takahasi was employed. A lattice was defined on the 8-cube in terms of which all 2-monotonic, canonical, self-dual functions of eight arguments were directly generated. Each such representative function was then treated by a modified form of the Muroga-Toda-Takasu linear programming test-synthesis procedure to obtain minimal 1-realizations. The Chow parameters for each function were calculated, and the final enumeration was ordered lexicographically by these parameters to afford a trivial test-synthesis procedure for n≤7. The enumeration demonstrated that minimal 1-realizations are still integral for n≤7; it corroborated Cobham's result that complete monotonicity is equivalent to 1-realizability, and established hyper-2-monotonicity as a useful characterization, for n≤7. It significantly extended our knowledge of the number of threshold functions and the various symmetry types, the size of weights and threshold required, the number of iterations required by the linear program, and similar statistics.