Fixed polarity Reed-Muller expressions (FPRMs) exhibit several useful properties that make them suitable for many practical applications. This paper presents an exact minimization algorithm for FPRMs for incompletely specified functions. For an n-variable function with α unspecified minterms there are 2n+α distinct FPRMs, and a minimum FPRM is one with the fewest product terms. To find a minimum FPRM the algorithm requires to determine an assignment of the incompletely specified minterms. This is accomplished by using the concept of integer-valued functions in conjunction with an extended truth vector and a weight vector. The vectors help formulate the problem as an assignment of the variables of integer-valued functions, which are then efficiently manipulated by using multi-terminal binary decision diagrams for finding an assignment of the unspecified minterms. The effectiveness of the algorithm is demonstrated through experimental results for code converters, adders, and randomly generated functions.

A generalized Reed-Muller expression (GRM) is obtained by negating some of the literals in a positive polarity Reed-Muller expression (PPRM). There are at most 2(n2)^(n-1) different GRMs for an n-variable function. A minimum GRM is one with the fewest products. This paper presents certain properties and an exact minimization algorithm for GRMs. The minimization algorithm uses binary decision diagrams. Up to five variables, all the representative functions of NP-equivalence classes were generated and minimized. Tables compare the number of products necessary to represent four-and five-variable functions for four classes of expressions: PPRMs, FPRMs, GRMs and SOPs. GRMs require, on the average, fewer products than sum-of-products expressions (SOPs), and have easily testable realizations.

Recently, various efficient algorithms for solving combinatorial optimization problems using BDD-based set manipulation techniques have been developed. Minato proposed O-suppressed BDDs (ZBDDs) which is suitable for set manipulation, and it is utilized for various search problems. In terms of practical limits of space, however, there are still many search problems which are solved much better by using conventional branch-and-bound techniques than by using BDDs or ZBDDs, while the ability of conventional branch-and-bound approaches is limited by computation time. In this paper, an extension of APPLY operation, named APPRUNE (APply + PRUNE) operation, is proposed, which performs APPLY operation (ZBDD construction) and pruning simultaneously in order to reduce the required space for intermediate ZBDDs. As a prototype, a specific algorithm of APPRUNE operation is shown by assuming that the given condition for pruning is a threshold function, although it is expected that APPRUNE operation will be more effective if more sophisticated condition are considered. To reduce size of ZBDDs in intermediate steps, this paper also pay attention to the number of cared variables. As an application, an exact-minimization algorithm for generalized Reed-Muller expressions (GRMs) is implemented. From experimental results, it is shown that time and memory usage improved 8.8 and 3.4 times, respectively, in the best case using APPRUNE operation. Results on generating GRMs of exact-minimum number of not only product terms but also literals is also shown.

This paper presents an optimization method for pseudo-Kronecker expressions of p-valued input two-valued output functions by using multi-place decision diagrams for p2 and p4. A conventional method using extended truth tables requires memory of O (3n) to simplify an n-variable expression, and is only practical for functions of up to n14 variables when p2. The method presented here utilizes multi-place decision diagrams, and can optimize considerably larger problems. Experimental results for up to n39 variables are shown.