Reading the Sampling Theorem in Multiple-Valued Logic: A Journey from the (Shannong) Sampling Theorem to the Shannon Decomposition Rule

Signals described by functions of continuous and discrete variables can be uniformly studied in a group theoretic framework. This paper presents a consideration which shows that in the case of multiple-valued (MV) functions, the notion of bandwidth relates to the concept of essential variables. Sampling conditions convert into requirements for periodicity and regularity in the truth-vectors of MV functions. Due to that, by starting from the sampling theorem, we derive generalized Shannon decomposition rules for MV functions that include the classical Shannon decomposition rule in binary-valued logic as a particular case. It follows from these considerations that the sampling theorem provides a regular way for the decomposition of a MV function into subfunctions of smaller numbers of variables. In circuit synthesis, this allows decomposition of a network to realize a function into subnetworks realizing subfunctions depending on subsets of variables, where the cardinality of the subsets is determined by the bandwidth selected. As there are no convergence problems, the sampling theorem for discrete functions can be formulated in terms of a class of Fourier-like transforms with certain properties provided. Due to that, different decompositions of a given function can be determined by selecting various Fourier-like transforms.