Quine–McCluskey algorithm

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The Quine–McCluskey algorithm (or the method of prime implicants) is a method used for minimization of boolean functions which was developed by W.V. Quine and Edward J. McCluskey. 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:

  1. Finding all prime implicants of the function.
  2. 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.

Contents

[edit] Complexity

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-hard: the runtime of the Quine-McCluskey algorithm grows exponentially with the input size. It can be shown that for a function of n variables the upper bound on the number of prime implicants is 3n/n. If n = 32 there may be over 6.5 * 1015, prime implicants. Functions with a large number of variables have to be minimized with potentially non-optimal heuristic methods, of which the Espresso heuristic logic minimizer is the de-facto world standard.

[edit] Example

[edit] Step 1: finding prime implicants

Minimizing an arbitrary function:

f(A,B,C,D) =\sum m(4,8,10,11,12,15) + d(9,14) \,
     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:

fA,B,C,D = A'BC'D' + AB'C'D' + AB'CD' + AB'CD + ABC'D' + ABCD

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-     |

[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
m(4,12)* X X -100
m(8,9,10,11) X X X 10--
m(8,10,12,14) X X X 1--0
m(10,11,14,15)* X X X 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 is thus neither an 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:

f_{A,B,C,D} = BC'D' + AB' + AC \
f_{A,B,C,D} = BC'D' + AD' + AC \

Both of those final equations are functionally equivalent to this original (very area-expensive) equation:

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

[edit] See also

[edit] External links

 The R implementation is exhaustive and it offers complete and exact solutions. It processes up to 20 input variables.
  • [1] , a Java program to display the boolean expression ..... by Manoranjan Sahu
  • [2], an applet for a step by step analyze of the QMC- algorithm by Christian Roth
  • [3] SourceForge.net C++ program implementing the algorithm.
  • Perl Module