In order to accurately use the Quine-McCluskey, the function needs to be given as a sum of minterms if the Boolean function is not in minterm form, the minterm expansion can be found to determine a minimum sum-of-products SOP expression for a function. During the first step of the method, all prime implicants of a function are systematically formed by combining minterms. These minterms are represented in a binary notation and combined as follows:. If two variables differ in exactly one variable, the two minterms will combine together. To find all prime implicants, all possible pairs of minterms should be compared and combined whenever possible.

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The Quine—McCluskey algorithm QMC , also known as the method of prime implicants , is a method used for minimization of Boolean functions that was developed by Willard V. Quine in [1] [2] and extended by Edward J. McCluskey in Samson and Burton E. Mills in [10] [6] and by Raymond J.

Nelson in Abrahams and John G. Nordahl [12] as well as Albert A. Mullin and Wayne G. Kellner [13] [14] [15] [16] proposed a decimal variant of the method. The Quine—McCluskey algorithm is functionally identical to Karnaugh mapping , but the tabular form makes it more efficient for use in computer algorithms, and it also gives a deterministic way to check that the minimal form of a Boolean function has been reached.

It is sometimes referred to as the tabulation method. 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. For a function of n variables the number of prime implicants can be as large as 3 n ln n , e. Functions with a large number of variables have to be minimized with potentially non-optimal heuristic methods, of which the Espresso heuristic logic minimizer was the de facto standard in Step two of the algorithm amounts to solving the set cover problem ; [26] NP-hard instances of this problem may occur in this algorithm step.

We encode all of this information by writing. One can easily form the canonical sum of products expression from this table, simply by summing the minterms leaving out don't-care terms where the function evaluates to one:. So to optimize, all minterms that evaluate to one are first placed in a minterm table. Don't-care terms are also added into this table names in parentheses , so they can be combined with minterms:.

At this point, one can start combining minterms with other minterms. If two terms differ by only a single digit, that digit can be replaced with a dash indicating that the digit doesn't matter. For instance and can be combined to give , indicating that both minterms imply the first digit is 1 and the next two are 0.

When going from Size 2 to Size 4, treat - as a third bit value. For instance, and can be combined to give , as can and to give , but and cannot because is implied by while is not. Trick: Match up the - first. Note: In this example, none of the terms in the size 4 implicants table can be combined any further.

In general this process should be continued sizes 8, 16 etc. None of the terms can be combined any further than this, so at this point we construct an essential prime implicant table. Along the side goes the prime implicants that have just been generated, and along the top go the minterms specified earlier. The don't care terms are not placed on top—they are omitted from this section because they are not necessary inputs. To find the essential prime implicants, we run along the top row.

This prime implicant is essential. This means that m 4,12 is essential. So we place a star next to it. The second prime implicant can be 'covered' by the third and fourth, and the third prime implicant can be 'covered' by the second and first, and neither is thus essential. If a prime implicant is essential then, as would be expected, it is necessary to include it in the minimized boolean equation.

In some cases, the essential prime implicants do not cover all minterms, in which case additional procedures for chart reduction can be employed. The simplest "additional procedure" is trial and error, but a more systematic way is Petrick's method. In the current example, the essential prime implicants do not handle all of the minterms, so, in this case, the essential implicants can be combined with one of the two non-essential ones to yield one equation:.

From Wikipedia, the free encyclopedia. The American Mathematical Monthly. Bell System Technical Journal. Retrieved Proceedings of the London Mathematical Society. In Peirce, Charles Sanders ed. Studies in Logic. History and Philosophy of Logic. Archived from the original on Bulletin of the American Mathematical Society. Abstracts of Papers: The Journal of Symbolic Logic. April June Association for Symbolic Logic.

Jellison Funeral Home and Cremation Services. Springfield, Illinois, USA. Archived PDF from the original on In his book , Caldwell dates this to November as a teaching memorandum. Operations Using Decimal Symbols". Switching Circuits and Logical Design. They discussed the method independently and then collaborated in preparing a class memorandum: P. Abraham and J. For the first major treatise of the decimal method in this book, it is sometimes misleadingly known as "Caldwell's decimal tabulation".

Fisher, Harvey I. Transactions of the Illinois State Academy of Science. A residue test for Boolean functions. Transactions of the Illinois State Academy of Science, vol. The Journal of Symbolic Logic Review. Caldwell […]. In this book, the author gives credit to Mullin and Kellner for development of the manipulations with the decimal numbers. November Some sources list the authors as " P. Abraham " and " I.

December Minimierung Boolescher Funktionen". Written at Jena, Germany. Elektronisches Rechnen und Regeln in German. License no. Order no. Abraham und I. Nordahl in [ Caldwell ]. A second edition exists as well.

Digital Logic Design 4 ed. This work does not cite the prior art on decimal methods. Some NP-complete set covering problems. The complexity of minimizing disjunctive normal form formulas Master's thesis. University of Aarhus. Troy; Carroll, Bill D. David Digital Logic Circuit Analysis and Design 2 ed. Prentice Hall. Journal of Computer and System Sciences. Acta Informatica in German. Namespaces Article Talk. Views Read Edit View history.

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## Everything About the Quine-McCluskey Method

In previous chapter, we discussed K-map method, which is a convenient method for minimizing Boolean functions up to 5 variables. But, it is difficult to simplify the Boolean functions having more than 5 variables by using this method. Quine-McClukey tabular method is a tabular method based on the concept of prime implicants. This tabular method is useful to get the prime implicants by repeatedly using the following Boolean identity. If there is a change in only one-bit position, then take the pair of those two min terms. It consists of set of rows and columns. Prime implicants can be placed in row wise and min terms can be placed in column wise.

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## Quine–McCluskey algorithm

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