### MANY-VALUED GALOIS SHANNON-DAVIO TREES AND THEIR COMPLEXITY

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#### Abstract

The idea of Shannon-Davio (S/D) trees for binary logic is a general concept that found applications in the Sum-Of-Product (SOP) minimization and the generation of new diagrams and canonical forms. Extended S/D trees are used to generate forms that include a minimum Galois Field Sum-of-Products (GFSOP) forms. Since there exist many applications of Galois field of quaternary radix especially that GF(4) is considered as an important extension of GF(2), the extension of the S/D trees to GF(4) is presented here. A general formula to calculate the number of Inclusive Forms (IFs) per variable order for an arbitrary Galois field radix and arbitrary number of variables is derived and introduced. A new fast method to count the number of IFs for an arbitrary Galois radix and functions of two variables is also introduced; the IF_{n,2} Triangles. The results introduced in this work can be useful for the creation of an efficient GFSOP minimizer for Galois logic and in other applications such as in reversible logic synthesis.

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