Boolean minimization

Abstract

The set \(B_{{\sigma _1}, \ldots ,{\sigma _k}}^{n,{i_1}, \ldots ,{i_k}} = \left\{ {\left( {{\alpha _1}, \ldots ,{\alpha _n}} \right) \in {B^n}:{\alpha _{{i_1}}} = {\sigma _1}, \ldots ,{\alpha _{{i_k}}} = {\sigma _k}} \right\}\) is called a face of the cube B ⁿ . The set {i _l, ..., i _n } is called the direction of the face, the number k is called the rank of the face, and n−k is called the dimension of the face \(B_{{\sigma _1}, \ldots ,{\sigma _k}}^{n,{i_1}, \ldots ,{i_k}}\). The code of the face \(G = B_{{\sigma _1}, \ldots ,{\sigma _k}}^{n,{i_1}, \ldots ,{i_k}}\) is the vector \(\tilde \gamma \left( G \right) = \left( {{\gamma _1}, \ldots ,{\gamma _n}} \right)\) such as \({\gamma _{{i_1}}} = {\sigma _1}, \ldots ,{\gamma _{{i_k}}} = {\sigma _k}\) and the other coordinates are “−”. For example, \(\tilde \gamma \left( {B_{01}^{n,1,3}} \right) = \left( {0 - 1 - } \right)\). A one-dimensional face is called an edge of the n-cube.