A new approach to generate fixed-polarity Reed–Muller expansions for completely and incompletely specified functions
Maki K. Habib · International Journal of Electronics · 2002
Generally, the application of Exclusive-OR (ExOR) logic design suffers from a lack of straightforward methods. In this paper, based on the Boolean matrix representation, the author introduces a new approach associated with a fast, simple, straightforward and computationally effective algorithm to obtain the minimum Reed–Muller ExOR expansion with fixed polarity. The new algorithm can be applied to completely as well as incompletely specified functions and it applies no constraints on the number of variables within any given switching function. The proposed minimization approach is based on the Boolean matrix representation and minterm separation operation to generate a fixed polarity Reed–Muller expansion that has a minimum number of non-zero valued terms in the final expansion while having a minimum numberof total literals. This is done efficiently and directly without involving exhaustive search procedures. For the case of an incompletely specified function, the algorithm tries to deduce the best selection of the ‘Don’t care’ term values while searching for a minimal fixed polarity Reed–Muller expansion. Demonstration examples with analysis results are illustrated and a comparison with other algorithms is introduced. In all the examples from the literature, the solutions were either the same or better than those of other methods. In addition, the proposed algorithm enables rapid processing, efficiently treating the incompletely specified functions, with no logical constraint on the number of function variables. The significant advantages of the approach are its convenience for computer implementation and its ability to deal with problems of very high dimension.