Use the Quine-McCluskey method to minimize the following function f(A, B, C, D)

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in 1956. It 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. The method involves two steps: Finding all prime implicants of the function. Use those prime implicants in a prime implicant chart to find the essential prime implicants of the function, as well as other prime implicants that are necessary to cover the function. Minimizing an arbitrary function: A B C D f m0 0 0 0 0 0 m1 0 0 0 1 0 m2 0 0 1 0 0 m3 0 0 1 1 0 m4 0 1 0 0 1 m5 0 1 0 1 0 m6 0 1 1 0 0 m7 0 1 1 1 0 m8 1 0 0 0 1 m9 1 0 0 1 x m10 1 0 1 0 1 m11 1 0 1 1 1 m12 1 1 0 0 1 m13 1 1 0 1 0 m14 1 1 1 0 x m15 1 1 1 1 1 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: Of course, that's certainly not minimal. 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, so they can be combined with minterms: Number of 1s Minterm Binary Representation 1 m4 0100 m8 1000 2 m9 1001 m10 1010 m12 1100 3 m11 1011 m14 1110 4 m15 1111 At this point, one can start combining minterms with other minterms. If two terms vary by only a single digit changing, that digit can be replaced with a dash indicating that the digit doesn't matter. Terms that can't be combined any more are marked with a "*". When going from Size 2 to Size 4, treat '-' as a third bit value. Ex: -110 and -100 or -11- can be combined, but not -110 and 011-. (Trick: Match up the '-' first.) Number of 1s Minterm 0-Cube Size 2 Implicants Size 4 Implicants 1 m4 0100 m(4,12) -100* m(8,9,10,11) 10--* m8 1000 m(8,9) 100- m(8,10,12,14) 1--0* -- -- -- m(8,10) 10-0 -- 2 m9 1001 m(8,12) 1-00 m(10,11,14,15) 1-1-* m10 1010 -- -- m12 1100 m(9,11) 10-1 -- -- -- -- m(10,11) 101- -- 3 m11 1011 m(10,14) 1-10 -- m14 1110 m(12,14) 11-0 -- 4 m15 1111 m(11,15) 1-11 -- m(14,15) 111- Note: In this example, none of the terms in the size 4 implicants table can be combined any further. Be aware that this processing should be continued otherwise (size 8 etc.). [edit]Step 2: prime implicant chart 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. 4 8 10 11 12 15 => A B C D m(4,12)* X X -100 => - 1 0 0 m(8,9,10,11) X X X 10-- => 1 0 - - m(8,10,12,14) X X X 1--0 => 1 - - 0 m(10,11,14,15)* X X X 1-1- => 1 - 1 - Here, each of the essential prime implicants has been starred - 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, one can combine the essential implicants with one of the two non-essential ones to yield one of these two equations:

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