Abstract
Frequently in threshold logic it is necessary to realize a switching function f of n variables in a two or more stages realization, both because it is not possible to realize f in a single gate and for reliability considerations. This means that the given function must be decomposed into a certain number of threshold functions, each of them having a number of variables less than n. When a geometric representation of switching functions is considered, the problem is equivalent to that of looking for functions whose patterns (on-sets) cover, in an adequate way, the on-set of the function f to be realized. In this paper a heuristic covering approach has been introduced which provides a manageable method for searching for a two stage realization of f. Furthermore a class of switching functions whose patterns possess a particular geometric shape (shell structure) has been considered. It has been proved that such functions (shell functions) are linearly separable and that one of their separating systems strictly depends on the centre and on the diameter of the shell structure. Gates implementing shell functions have been considered and their use has been revealed more convenient than that of majority gates in the solution of the problem of network synthesis. To clarify this advantage an example has been discussed in detail.
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Arcelli, C., Massarotti, A. A geometric synthesis method to realize a switching function. Kybernetik 13, 155–163 (1973). https://doi.org/10.1007/BF00270510
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DOI: https://doi.org/10.1007/BF00270510