What is a Karnaugh Map?

A Karnaugh map is a graphical representation of a logical function. The Karnaugh map of a logical function is to fill the minimum terms in the minimum term expression of this function into a grid chart, which is called a Karnaugh chart.

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The graphical representation of the logical relationship of a combination circuit can be traced back to the Venn diagram invented by British logician John Venn in 1881 to deal with the logical relationship between sets in set theory, Helm Ha Helmut Hasse effectively used the Hass diagram used by Vogt in 1895 to represent the finite partial order set in the theory of order. Edward W. Veitch in 1952 converted the Venn diagram The circle in Chinese was redrawn into a rectangle and the Veitch diagram was invented. However, these maps are not as good as the digital logic and faults of the Karnaugh map or K-map improved by the Wichi diagram in 1953, as telecommunication engineer Maurice Karnaugh of Bell Labs in the United States in 1953. It is widely used in many fields such as diagnosis. ^[2]

The characteristics of Karnaugh maps make Karnaugh maps have an important property: they can intuitively find the combination of adjacent minimum terms from the graph. The theoretical basis of the merger is the union theorem AB + AB inverse = A. E.g,

According to the definition of theorem AB + AB inverse = A and adjacent minimum terms, two adjacent minimum terms can be merged into an AND term and eliminate a reciprocal variable. For example, the 4-variable minimum terms ABCD and ABC reverse D can be merged into ABD; A reverse BCD and A reverse BC reverse D can be merged into A reverse BD; and the terms A reverse BD and ABD are adjacent AND terms, so the same reasoning can be used to further merge two adjacent AND terms into a BD.

Karnaugh map

Logical functions can be simplified by not using minimum terms. If F adopts ANDor expression, the function can be expanded into the smallest term in the process of filling the Karnaugh map.

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2 ^ n (n-number of variables) small cells representing the smallest items, arranged in a rectangle.

Fill all the smallest terms contained in the function with "1" in the corresponding cells of the variable Karnaugh map.

1. Several definitions

1. Given logical function as standard AND-OR expression

When the logical function is a standard AND-OR expression, just find the small square corresponding to the smallest item in the expression on the Kano chart and fill in 1 with the remaining small squares filled with 0 to get Karnaugh map of functions.

2. Logical functions are general AND-OR expressions

When the logical function is a general "AND-OR" expression, a corresponding Carnot diagram can be made according to the commonality of "AND" and the superposition of "OR".

When filling in the Kano chart of this function, just find the small squares corresponding to the terms AB, CD, A · BC on the 4-variable Kano chart and fill in 1 to get the Kano chart of the function .

When the expression of the logic function is in other forms, it can be transformed into the above form before making a Kano graph.

For the convenience of description, the small squares filled with 1 on the Kano chart are usually called 1 squares, and the small squares filled with 0 are called 0 squares. Zero squares are sometimes represented by spaces.

An important feature of the Karnaugh map is that it graphically and clearly reflects the neighbor relationship of the smallest terms. When four small squares form a large square, or form a row (column), or are at the ends of two adjacent rows (columns), or at the four corners, the smallest items in the table can be merged. Variables. When eight small squares form a large square, or two adjacent rows (columns), or two side rows (columns), the smallest items represented can be merged, and three variables can be eliminated after the merge. So far, taking the 3 and 4 variable Karnaugh maps as examples, the merging method of 2, 4, and 8 minimum terms has been discussed. By analogy, it is not difficult to obtain the merging rule of the smallest term in the Karnaugh map of n variables.

To sum up, the merging rule of the smallest term in the Karnaugh map of n variables is as follows:

(1) The number of small squares in the Carnot circle must be 2 ^ m, where m is an integer less than or equal to n.

(2) The 2 ^ m small squares in the Carnot circle have a certain arrangement rule. Specifically, they contain m different variables and (nm) the same variables.

(3) The smallest term corresponding to the 2 ^ m small squares in the Carnot circle can be represented by the "and" term of (nm) variables, and the "and" term is composed of the same variables among these smallest terms.

(4) When m = n, the Kano circle surrounds the entire Kano graph, which can be represented by 1, that is, the sum of all minimum terms of n variables is 1.