A nonlinear integer programming approach for the minimization of Boolean expressions

A novel approach is suggested in this paper for the minimization of Boolean expressions. This is particularly useful in logic synthesis, since it leads to simpler logic circuit implementations. Although the proposed method is general, emphasis is given on Exclusive-or Sum Of Products (ESOPs) functions. A transformation is derived to convert the problem from the Boolean algebra area to the classical algebraic area. The resulting problem becomes a nonlinear, integer program and an original branch-and-bound procedure with several relaxations is developed for its solution. The suggested methodology is especially suitable for the minimization of incompletely specified functions, which is a difficult problem in the Boolean area. Numerical examples are provided to demonstrate the applicability and performance of the approach and possible future directions are described. The resulting nonlinear problems can sometimes be very demanding but their challenging solutions could solve open ESOP problems.