Complexity and Computational Applications
Mason's Rule can grow factorially, because the enumeration of paths in a directed graph grows thusly. To see this consider the complete directed graph on vertices, having an edge between every pair of vertices. There is a path form to for each of the permutations of the intermediate vertices. Thus Gaussian elimination is more efficient in the general case.
Yet Mason's rule characterizes the transfer functions of interconnected systems in a way which is simultaneously algebraic and combinatorial, allowing for general statements and other computations in algebraic systems theory. While numerous inverses occur during Gaussian eliminiation, Mason's rule naturally collects these into a single quasi-inverse. General form is
Where as described above, is a sum of cycle products, each of which typically falls into an ideal (for example, the strictly causal operators). Fractions of this form form a subring of the rational function field. This observation carries over to the noncommutative case, even though Mason's rule itself must then be replaced by Riegle's rule.
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