Geodesic
Synthesis of combinational circuits by means of bi-decomposition of Boolean functions
Prikladnaâ diskretnaâ matematika, no. 2 (2023), pp. 95-105

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The problem of combinational circuits synthesis in the basis of two-input gates is considered. Those gates are AND, OR, NAND and NOR. A method for solving this problem by means of Boolean functions bi-decomposition is suggested. The method reduces the problem to the search for a weighted two-block cover of the orthogonality graph of ternary matrice rows representing the given Boolean function by complete bipartite subgraphs (bi-cliques). Each bi-clique in the obtained cover is assigned in a certain way with a set of variables that are the arguments of the function. This set is the weight of the bi-clique. Each of those bi-cliques defines a Boolean function whose arguments are the variables assigned to it. The functions obtained in such a way constitute the required decomposition. The process of combinational circuit synthesis consists in successively applying bi-decomposition to the functions obtained. The method for two-block covering the orthogonality graph of ternary matrice rows is described.
Keywords: synthesis of combinational circuits, Boolean function, decomposition of Boolean functions, ternary matrix, complete bipartite subgraph, two-block cover. 
@article{PDM_2023_2_a7,
     author = {Yu. V. Pottosin},
     title = {Synthesis of combinational circuits by means of bi-decomposition of {Boolean} functions},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {95--105},
     publisher = {mathdoc},
     number = {2},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PDM_2023_2_a7/}
}
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Yu. V. Pottosin. Synthesis of combinational circuits by means of bi-decomposition of Boolean functions. Prikladnaâ diskretnaâ matematika, no. 2 (2023), pp. 95-105. http://geodesic.mathdoc.fr/item/PDM_2023_2_a7/