An Overview of Recent Developments in Big Boolean Equations

TL;DR

This paper surveys and compares algebraic and map methods for solving large Boolean equations, highlighting their efficiency, usability, and solution types, with a focus on recent developments and atomic decomposition techniques.

Contribution

It provides a comprehensive review and comparison of algebraic and map methods, introducing recent advances like atomic decomposition for solving big Boolean equations.

Findings

Map techniques can produce more compact solutions than algebraic methods.

Map methods are at least as efficient as algebraic techniques, sometimes superior.

Different solution types offer various advantages in solving Boolean equations.

Abstract

Methods of solving big Boolean equations can be broadly classified as algebraic, tabular, numerical and map methods. The most prominent among these classes are the algebraic and map methods. This paper surveys and compares these two types of methods as regards simplicity, efficiency, and usability. The paper identifies the main types of solutions of Boolean equations as subsumptive general solutions, parametric general solutions and particular solutions. In a subsumptive general solution, each of the variables is expressed as an interval based on successive conjunctive or disjunctive eliminants of the original function. In a parametric general solution each of the variables is expressed via arbitrary parameters which are freely chosen elements of the underlying Boolean algebra. A particular solution is an assignment from the underlying Boolean algebra to every pertinent variable that makes the Boolean equation an identity. The paper offers a tutorial exposition, review, and comparison of the three types of solutions by way of an illustrative example. This example is also viewed from the perspective of a recently developed technique that relies on atomic decomposition. Map techniques are demonstrated to be at least competitive with (and occasionally superior to) algebraic techniques, since they have a better control on the minimality of the pertinent function representations, and hence are more capable of producing more compact solutions.