In a Gray code, any number differs from its predecessor (the Gray code representation of the number minus 1) by a change in exactly one bit.
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In its simplest form, a Karnaugh map is a graphical shortcut to solving problems expressed in Boolean algebra. Boolean algebra is a form of mathematics that uses two values to perform computations and create expressions. This type of algebra is one of the fundamental concepts behind computer science and digital circuit design, and the Karnaugh map was first developed to help solve certain problems without using long computations. The map in its modern form was developed by physicist Maurice Karnaugh in 1953.
Karnaugh maps are designed to shift the burden of solving certain problems away from computations and toward pattern recognition. These maps are also used to help sift visual information and discern meaningful organizations. Since humans are naturally skilled at pattern recognition, the use of Karnaugh maps quickly sped up certain aspects of circuit design. One of the Karnaugh map’s particular strengths is in finding and showing possible solutions to race hazards, which are flaws in a system caused by timing issues. Race hazards are of particular concern to software developers, network architects and computer security specialists, as they can severely compromise and damage systems.
A Karnaugh map is generally built as a rectangular diagram divided into rows and columns. Karnaugh maps, also known as Veitch diagrams or KV-maps, are essentially truth tables — tables that show all the valid combinations for a particular set of values. A Karnaugh or KV-map can be built with any number of variables, but standard practice is to hold the number of variables in the table to six or less. These variables are usually expressed in Gray code, a number system that expresses binary values, or a system using only 0 and 1.
The real value of the Karnaugh map lies in its simplicity. Since a Karnaugh map is presented as essentially a grid, the structure is straightforward and easy to understand at a glance. The grid structure also allows for simplified arrangement of like variables, which means that terms can be grouped and regrouped as needed to resolve potential issues. In addition, due to the structure of the Karnaugh map, any grouping of variables leads to an easy representation of the changes in variables. Adjacent variables are separated by a change in only one variable, which further simplifies operations. This holds true no matter how many variables are used. As a result, the Karnaugh map remains a simple and useful tool for many designers and engineers working in digital circuit design and information theory.