Encoding Symbolic Inputs for Multi-Level Logic Implementation

This paper presents an approach for symbolic minimization of combinational logic to be implemented in multi-level form. We consider the problem of finding optimal binary encodings for the different values of a symbolic input variable. An optimal encoding is one that leads to a minimal multi-level implementation of the resulting Boolean logic. Our approach is based on finding encodings that resultin large common divisors among a set of expressions. We develop a theoretical foundation that permit* us to view all possiblecommon algebraic divisors resulting from all possible encodings of the symbolic variable. We determine necessary and sufficient conditions that the encoding must satisfy for a given common divisor to exist in an encoded implementation. Determining the encoding that results in large common divisors is shown to be equivalent to solving a face embedding problem, for which efficient heuristics exist. We increase the powerof our approach by extending it to detect non-algebraic (Boolean) divisors. An interesting aspect is that an initial multi-level decomposition is produced as a by-product of the encoding process.