Nonstable Cycle and Level Sets for Linear Sequential Machines

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Abstract
It is shown that the cycles sets of a linear sequential machine with a constant input having no stable state, are derivable from the cycle sets of that same linear sequential machine with a constant 0 input. It is shown that the level sets are independent of the input. A synthesis procedure and an example are presented. The objective of this short paper is to show the relation between the cycle set and level set of a linear sequential autonomous circuit and the cycle set and level set of a linear sequential circuit with a constant input. It was shown Srinivasan [1] that in response to a constant input, the cycle set (C) obtained, is exactly that obtained when another input was applied repetitively, provided that for both these inputs, there exists a stable state (under an input, the next state equals the present state). We ask now if the levels sets (L) are also identical and, more important, what is the state graph of a linear sequential circuit in response to a constant input under which no stable state exists. The parameters cycle sets (number and length of cycles) [2] and level sets (number of states per level in a tree) [3] are examined to see if a constant input will alter them; only the cycle sets are altered, if at all, by a constant input.

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