Characteristic Logic Equations

DERIVATION OF THB CHARACfl!RISTIC Loo,c EQUATION The characteristic logic equation expresses the necessary and sufficient conditions to be satisfied by the entries in an interconnection matrix M8,. that interconnects two hierarchical digraphs A* and B* for which M,18 = 0. Digraphs A* and B* are described by the reachability matrices M,1,1a nd M88, respectively. Each of these matrices can be assumed to have the following properties. a) The indices of the matrix are ordered in blocks corresponding to levels in the hierarchical relation, beginning with the highest level and ending with the lowest. b) As a consequence of a), every entry to the right of the main diagonal is 0. c) By definition, the main diagonal is filled with I's. d) By definition, the Boolean square of the matrix is identical to the matrix. e) Because of d), the matrix is its own transitive closure [I]. Reachability for the digraph c• that results from the interconnection is described by the reachability matrix M cc, which can be assumed to have the following properties. I) Mee can be written in the form