Bi-decomposition of multi-valued logical functions and its applications

The bi-decomposition of multi-valued logical (MVL) functions, including disjoint and non-disjoint cases, is considered. Using a semi-tensor product, an MVL function can be expressed in its algebraic form. Based on this form, straightforward verifiable necessary and sufficient conditions are provided for each case, respectively. The constructive proofs also lead to constructing corresponding decompositions. Using these results, the implicit function theorem (IFT) of k -valued functions, as a special bi-decomposition, is obtained. Finally, as an application, the normalization of dynamic–algebraic (D–A) Boolean networks is investigated using IFT of k -valued functions.