Stochastic combinational networks

A new model is defined for combinational networks of probabilistic logic gates which differs from the earlier models of von Neumann and McCulloch in the way in which timing is handled. This differences makes the new model compatible with ordinary deterministic switching theory in that the outputs of any network containing feedforward connections, but not feedback loops, depend only on the network's inputs after a short, fixed time delay, not on any preceding inputs. A general method of analyzing arbitrary combinational networks within the new model is developed using a formalism based on the algebra of stochastic matrices. Methods of simplifying network analysis under certain special circumstances are also given. Probabilistic generalizations of several concepts from ordinary deterministic switching theory are developed in some detail. These include the complement of a function, the dual of a function, a fixed variable, a dummy variable, a symmetric function, and an associative function. De Morgan's law and the principle of duality are also shown to have probabilistic generalizations. Several methods are next developed for synthesizing one-output combinational networks within the new model. Included are generalizations of several basic methods from deterministic switching theory such as the and-or, or-and, and Shannon expansion methods. Several methods of multiple-output network synthesis are also developed, including a set of cascade networks as well as generalizations of the various one-output network synthesis methods. Finally, the concept of a sequential network is introduced, and the relationships between combinational networks, sequential networks, and stochastic sequential machines are developed in some detail.