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A Karnaugh map (KM or K-map) is a diagram that can be used to simplify a Boolean algebra expression. Maurice Karnaugh introduced it in 1953 [ 1 ] [ 2 ] as a refinement of Edward W. Veitch 's 1952 Veitch chart , [ 3 ] [ 4 ] which itself was a rediscovery of Allan Marquand 's 1881 logical diagram [ 5 ] [ 6 ] (aka.
Converting a Karnaugh map to a Zhegalkin polynomial. The figure shows a function of three variables, P(A, B, C) represented as a Karnaugh map, which the reader may consider as an example of how to convert such maps into Zhegalkin polynomials; the general procedure is given in the following steps:
Example: The map method usually is done by inspection. The following example expands the algebraic method to show the "trick" behind the combining of terms on a Karnaugh map: Minterms #3 and #7 abut, #7 and #6 abut, and #4 and #6 abut (because the table's edges wrap around). So each of these pairs can be reduced.
Although more practical than Karnaugh mapping when dealing with more than four variables, the Quine–McCluskey algorithm also has a limited range of use since the problem it solves is NP-complete. [22] [23] [24] The running time of the Quine–McCluskey algorithm grows exponentially with the number of variables.
While this example was simplified by applying normal algebraic methods [= (′ +)], in less obvious cases a convenient method for finding minimal PoS/SoP forms of a function with up to four variables is using a Karnaugh map. The Quine–McCluskey algorithm can solve slightly larger problems.
"For more than three variables, the basic illustrative form of the Venn diagram is inadequate. Extensions are possible, however, the most convenient of which is the Karnaugh map, to be discussed in Chapter 6." [13] (p 64) In Chapter 6, section 6.4 "Karnaugh map representation of Boolean functions" they begin with:
In the early days, logic design involved manipulating the truth table representations as Karnaugh maps. The Karnaugh map-based minimization of logic is guided by a set of rules on how entries in the maps can be combined. A human designer can typically only work with Karnaugh maps containing up to four to six variables.
For the case when the Boolean function is specified by a circuit (that is, we want to find an equivalent circuit of minimum size possible), the unbounded circuit minimization problem was long-conjectured to be -complete in time complexity, a result finally proved in 2008, [4] but there are effective heuristics such as Karnaugh maps and the ...