Feedback in Asynchronous Sequential Circuits

In this paper we attempt to realize any flow table of a desired sequential circuit so as to minimize the feedback index (number of feedback loops) of the resulting asynchronous sequential switching circuit. It is shown that any flow table M, which describes the circuit action of a normal fundamental mode asynchronous sequential circuit, can be realized as a fundamental mode circuit with feedback index [log2S], where S is the maximum number of stable states in any input column of M. The realization requires no inverters (assuming double rail inputs) and thus can be shown to be a minimum transistor, as well as a minimal feedback index, realization for any transistor-diode circuit realization of M. Nonfundamental mode realizations of normal mode flow tables are also considered, and it is shown that such a realization can be found with only one feedback loop, assuming ideal inertial delays, for any given flow table. However, no ideal inertial delay realization has been discovered which does not contain internal feedback. Finally, minimum feedback realizations of non-normal and nonfundamental mode flow tables are briefly considered.