Circuit Decomposition Research Articles

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This article treats the tree circuit synthesis problem for families F= {f1, f2, ···, fm} of Boolean functions fj of the same three variables. In addition to the development of criteria for determining the most economical of the three possible tree circuit decompositions: (j= 1, 2,···, m) of a given family, certain upper bounds BT(3, m) are obtaie···,d on the tree circuit cost T( F) of such a fmily; these have the property tha regardless of the mebers of F, the in-equality T(F)?BT(3, m) holds and furthermore, a family Fists in which actualy attains this upper bound. These upper bounds or estimates are known to have a wide application in switching theory generally, and in particular in the theory of tree circuits and in the decomposition theory of Boolean functions.

The decomposition of stochastic automata

Finite-state stochastic sequential systems which are interconnections of two or more component systems exhibit a special structural property in their transition matrices as a consequence of the statistical independence of the next states of the component systems given the present states and the input to the interconnection. Along with the concept of a “lumpable” stochastic matrix, this structural property is of central importance to the decomposition theory presented. These ideas allow a generalization to the stochastic case of the definitions used by Hartmanis in his study of the decomposition of conventional switching circuits. With these definitions we are able to prove decomposition theorems for stochastic automata which reduce to those of Hartmanis in the deterministic case.

Matrix Analysis of Oriented Graphs with Irreducible Feedback Loops

A generating function is obtained for the nonrepetitive closed loops of a network with unidirectional elements; from this is derived an analogous function for the irreducible loops. Criteria for the existence of loops are then established and the size of the smallest loop present determined; an asymptotic evaluation is made of the number of irreducible loops in a completely connected network. Further application of the generating function permits estimation of bounds for both irreducible and composite closed loops of a given order; less rigorous bounds are found by two perturbation techniques. The generating function is reformulated in terms of determinant-like quantities and application made to small networks. Transformations preserving the irreducible loops of a system are discussed at length, following a delineation of the meaning of loop equivalence. Methods employed include elimination of branches, condensation of nodes, and decomposition of circuits, with criteria for their utilization being set forth. Finally, connection is made between the analysis of the present paper and those prevalent in more avowedly topological treatments.