Optimum Functional Decomposition Using Encoding

In this paper, we revisit the classical problem of functional decomposition [1, 2] that arises so often in logic synthesis. One basic problem that has remained largely unaddressed to the best of our knowledge is that of decomposing a function such that the resulting sub-functions are simple, i.e., have small number of cubes or literals. In this paper, we show how to solve this problem optimally. We show that the problem is intimately related to the encoding problem, which is also of fundamental importance in sequential synthesis, especially state-machine synthesis. We formulate the optimum decomposition problem using encoding. In general, an input-output encoding formulation has to be employed. However, for field-programmable gate array architectures that use look-up tables, the input encoding formulation suffices, provided we use minimum-length codes. The last condition is really not a constraint, since each extra code bit means that an extra table has to be used (and that could be expensive). The unused codes are used as don´t cares for simplifying the sub-functions. We compare the original implementation of functional decomposition, which ignores the encoding problem, with the new version that uses encoding while doing decomposition. We obtain an average improvement of over 20% on a set of standard benchmarks for look-up table architectures.