Anatomy of Boolean-function simplification

Abstract

The existing methods for the simplification of Boolean functions are usually classified according to their mathematical background (e.g. consenus method) or their procedural tools (e.g. diagram methods) or are named in honour of scientists (e.g. Quine-McCluskey method). To make the methods more transparent, with respect to the judgment of their effectiveness in solving practical problems, it is suggested that they be classified according to the strategy of prime implicant generation. It is shown that the known methods can be assigned to one of the following basic strategies for the generation of prime implicants: (a) building-up from smaller implicants; (b) expansion of the function along literals or variables; (c) separation of the vector space into 1-subspaces and O-subspaces; (d) systematic inspection of all possible implicants; (e) heuristic methods. The strategy is shown to be a governing factor for the effectiveness of minimisation procedures, particularly if, owing to the complexity of the problems, only approximately minimal solutions can be obtained.