Using variable-entered Karnaugh maps to produce compact parametric general solutions of Boolean equations

Abstract

The variable-entered Karnaugh map (VEKM) is shown to be the natural map for representing and manipulating general ‘big’ Boolean functions that are not restricted to the switching or two-valued case. The VEKM is utilized herein in producing a compact general solution of a system of Boolean equations. It serves as a powerful manual tool for function inversion, implementation of the solution procedure, handling don't-care conditions and minimization of the final expressions. The rules of using the VEKM are semi-algebraic and collective in nature, and hence are much easier to state, remember and implement than are the tabular and per-cell rules of classical maps. As a result, the maps used are significantly smaller than those required by classical methods. As an offshoot, the paper contributes some pictorial insight into the representation of ‘big’ Boolean algebras and functions. It also predicts the correct number of particular solutions of a Boolean equation, and produces an exhaustive list of particular solutions. Details of the method are carefully explained and further demonstrated via an illustrative example.

Keywords:

• Boolean equations
• parametric general solutions
• particular solutions
• variable-entered Karnaugh maps
• finite Boolean algebras

2011 AMS Subject Classifications :

• 03G05
• 94C10
• 06E30
• 03E35
• 08B05

C.R. Category :

• B.6.3