Iterative Methods for Multi-Valued Logical Equation System Solving while Digital System Simulating

Abstract

The article is devoted to the analysis of methods for solving systems of multivalued logical equations by iteration methods. Iterative methods for solving such systems of equations are a mathematical description of the main process of functional-logical simulation, which is used at the design stage of digital systems to check the correctness of the design. Consideration of multivalued (finite-valued) values of logical signals at the outputs of blocks and elements of digital systems is explained by the fact that in a number of cases, to analyze the correctness of time relationships when simulating of digital system hardware, several-valued representations of binary logical signals are used, as well as those that recently, the development of logical elements that implement four or more significant logic is underway. Based on the analysis of the structure of the system of logical equations used in digital equipment simulation, using graph and logical models, an analysis of the existence of solutions and their number is carried out. Iterative methods of simple and generalized iteration are analyzed, the relationship between the number of solutions to a system of equations and its graph representation, reflecting a given connection scheme of elements of a digital system hardware, is shown. For the generalized iteration method, variants with different structures of the iteration trace are considered, in particular, it is shown that with a certain structure of the iteration trace, the generalized iteration turns into a simple iteration or Seidel iteration. It is shown that the generalized iteration most adequately describes the process of simulating the switching of logical signals in digital system hardware. The correspondence between various variants of functional-logical simulation of digital systems and the used method of iterative solution of systems of logical equations is shown.