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  • Mathematical apparatus for synthesis of k-valued digital circuits based on linear algebra

    The mathematical basics of the non-classical approach to the logical synthesis of k-valued digital structures, based on the replacement of the classic mathematical apparatus of logic synthesis (Boolean algebra) to the proposed mathematical apparatus - linear algebra are considered. The limiting properties of Boolean algebra, preventing further improve the technical ICs characteristics are discussed. One of the possible directions of overcoming these properties is the transition to a multi-valued logic. A new in a logical synthesis concept of linear synthesis of digital structures is proposed. A strong definition of linear algebra and its basic properties are shown. The logic synthesis process of two valued and multi-valued digital structures in linear algebra, including the formation of bases of a linear space, the original representation of the implemented logical function, obtaining decomposition of logic function according to the system of basis vectors are considered.

    Keywords: multi-valued logic, Boolean algebra, linear algebra, linear logic synthesis, linear digital structures, bases of the linear space, decomposition of logic functions, logical variable, logical synthesis, digital logical elements