BOOM-a heuristic Boolean minimizer

We present a two-level Boolean minimization tool (BOOM) based on a new implicant generation paradigm. In contrast to all previous minimization methods, where the implicants are generated bottom-up, the proposed method uses a top-down approach. Thus instead of increasing the dimensionality of implicants by omitting literals from their terms, the dimension of a term is gradually decreased by adding new literals. Unlike most other minimization tools like ESPRESSO, BOOM does not use the definition of the function to be minimized as a basis for the solution, and thus the original coverage influences the solution only indirectly through the number of literals used. Most minimization methods use two basic phases introduced by Quine-McCluskey, known as prime implicant (PI) generation and the covering problem solution. Some more modern methods, like ESPRESSO, combine these two phases, reducing the number of PIs to be processed. This approach is also used in BOOM, where the search for new literals to be included into a term aims at maximum coverage of the output function. The function to be minimized is defined by its on-set and off-set, listed in a truth table. Thus the don't care set, often representing the dominant part of the truth table, need not be specified explicitly. The proposed minimization method is efficient above all for functions with a large number of input variables while only few care terms are defined. The minimization procedure is very fast, hence if the first solution does not meet the requirements, it can be improved in an iterative manner. The method has been tested on several different kinds of problems, like the MCNC standard benchmarks or larger problems generated randomly.

1. Introduction

The problem of two-level minimization of Boolean functions is usually solved for functions with less than 100 variables. This is due to two mutually dependent facts: larger circuits are not so common, and no minimization method for larger problems has been available so far. However, Boolean minimization is by no means limited to the area of switching circuit design, where it was first identified [8]. The new implementation technologies of digital circuits, e.g., multi-level custom design, FPGAs, and above all PLAs, require this minimization in some phase. Minimization problems with a large number of variables are encountered in many modern application areas like design of online real-time control systems, design of built-in self-test equipment for VLSI circuits, in the area of artificial intelligence, in software engineering, etc. These problems are mostly characterized by a limited number of input states for which the output value is determined (care states). On the other hand, the number of don't care states then reaches astronomical values, and the quality of a minimization method is thus determined by its ability to take advantage of their existence without enumerating them. An efficient minimization method must in addition be able to cope with the existence of a large number of prime implicants (PIs) of the given function, most of which are not needed for the minimum solution.