# Quine-McCluskey algorithm

Quine-McCluskey algorithm is a method used for minimisation of Boolean functions. 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.

The method involves two steps:

1. Finding all prime implicants of the function.
2. Using those prime implicants to find the essential prime implicants of the function, as well as using other prime implicants that are necessary to cover the function.

## Example

Minimizing an arbitrary function:

f(A, B, C, D) = [itex]\Sigma[itex](4, 8, 9, 10, 11, 12, 14, 15)
```     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   1
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   1
m15  1 1 1 1   1
```

One can easily form the canonical sum of products expression from this table, simply by summing the minterms where the function evaluates to one:

[itex]f_{A,B,C,D} = A'BC'D' + AB'C'D' + AB'C'D + AB'CD' + AB'CD + ABC'D' + ABCD' + ABCD[itex]

Of course, that's certainly not minimal. So to optimize, all minterms that evaluate to one are first placed in a minterm table:

```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 that digit doesn't matter. Terms that can be combined no more are marked

with a "*".

```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-     |
```

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.

 4 8 9 10 11 12 14 15 m(4,12)* X X m(8,9,10,11)* X X X X m(8,10,12,14) X X X X m(10,11,14,15)* X X X X

Here, each of the essential prime implicants has been starred. If a prime implicant is essential then, as would be expected, it is necessary to include it in the minimized boolean equation. In this case, the EPIs handle all of the minterms, so then, the combined minterms are just summed to give this equation:

[itex]f_{A,B,C,D} = BC'D' + AB' + AC[itex]

The final equation is functionally equivalent to this original (very area-expensive) equation:

[itex]f_{A,B,C,D} = A'BC'D' + AB'C'D' + AB'C'D + AB'CD' + AB'CD + ABC'D' + ABCD' + ABCD[itex]

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