A collection of logic gates forms a combinational circuit if the outputs can be described as boolean functions of the current input values. Optimizing combinational circuitry, for instance, by reducing the number of gates (the area) or by reducing the length of the signal paths (the delay), is an overriding concern in the design of integrated circuits.

The accepted wisdom is that combinational circuits must have acyclic (i.e., loop-free or feed-forward) topologies. In fact, the idea that "combinational" and "acyclic" are synonymous terms is so thoroughly ingrained that many textbooks provide the latter as a definition of the former. And yet simple examples suggest that this is incorrect. In this dissertation, we advocate the design of cyclic combinational circuits (i.e., circuits with loops or feedback paths). We demonstrate that circuits can be optimized effectively for area and for delay by introducing cycles.

On the theoretical front, we discuss lower bounds and we prove that certain families of functions can be implemented by cyclic circuits with 50% fewer gates than is possible with acyclic circuits. On the practical front, we describe an efficient approach for analyzing cyclic circuits, and we provide a general framework for synthesizing such circuits. On trials with industry-accepted benchmark circuits, cyclic optimizations led to significant improvements in area and delay in nearly all cases. Based on these results, we suggest that it is time to re-write the definition: in both theory and practice, combinational might well mean cyclic.

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