Implication Theory and Algorithm for Reachability Matrix Model

A reachability matrix M is a binary matrix with the reflexive and transitive property, i.e., M + I = M, and M2 = M, where I is the identity matrix. The entries of the matrix M are shown to form a multilevel implication structure derived using the transitivity property. The fundamental implication matrix P that defines this structure is derived. The matrix Q of the transitive closure of P, the complete implication matrix, is defined. It is proved that Q = p2. The problem of efficiently filling the partially filled reachability matrix is considered. An algorithm for determining all of the implied values of the unknown elements of the partially filled reachability matrix M derived from a supplied value is proposed. The algorithm requires 0(n2) computer time and 0(n2) storage, where n is the size of the matrix M. Use of the algorithm to the interpretive structural modeling (ISM) process makes it possible to do a flexible and an efficient transitive embedding.