Quine–McCluskey algorithm

The Quine–McCluskey algorithm (or the method of prime implicants) is a method used for minimization of Boolean functions that was developed by Willard V. Quine^[1]^[2] and extended by Edward J. McCluskey.^[3] 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.

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 running time of the Quine–McCluskey algorithm grows exponentially with the number of variables. For a function of n variables the number of prime implicants can be as large as 3ⁿln(n), e.g. for 32 variables there may be over 534 * 10¹² 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 was the de-facto standard in 1995.^[4]

Example

Input

In this example, the input is a Boolean function in four variables, $f:\{0,1\}^{4}\to \{0,1\}$ which evaluates to $1$ on the values $4,8,10,11,12$ and $15$ , evaluates to an unknown value on $9$ and $14$ , and to $0$ everywhere else (where these integers are interpreted in their binary form for input to $f$ for succinctness of notation). The inputs that evaluate to $1$ are called 'minterms'. We encode all of this information by writing

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

This expression says that the output function f will be 1 for the minterms $4,8,10,11,12$ and $15$ (denoted by the 'm' term) and that we don't care about the output for $9$ and $14$ combinations (denoted by the 'd' term).

Step 1: finding prime implicants

First, we write the function as a table (where 'x' stands for don't care):

	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:

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

which is 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
1	m8	1000
2	m9	1001
	m10	1010
	m12	1100
3	m11	1011
3	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 an asterisk (*). When going from Size 2 to Size 4, treat '-' as a third bit value. For instance, -110 and -100 can be combined, as well as -110 and -11-, but -110 and 011- cannot. (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	—
	—	—	m(8,12)	1-00	—
2	m9	1001	m(9,11)	10-1	m(10,11,14,15)	1-1-*
	m10	1010	m(10,11)	101-	—
	—	—	m(10,14)	1-10	—
	m12	1100	m(12,14)	11-0	—
3	m11	1011	m(11,15)	1-11	—
3	m14	1110	m(14,15)	111-	—
4	m15	1111		—	—

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.).

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.

	⇒	A	B	C	D
m(4,12)*	⇒	—	1	0	0
m(8,9,10,11)	⇒	1	0	—	—
m(8,10,12,14)	⇒	1	—	—	0
m(10,11,14,15)*	⇒	1	—	1	—

To find the essential prime implicants, we run along the top row. We have to look for columns with only 1 "X". If a column has only 1 "X", this means that the minterm can only be covered by 1 prime implicant. This prime implicant is essential.

For example: in the first column, with minterm 4, there is only 1 "X". This means that m(4,12) is essential. So we place a star next to it. Minterm 15 also has only 1 "X", so m(10,11,14,15) is also essential. Now all columns with 1 "X" are covered.

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 equation:

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

^[5]

or

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

Both of those final equations are functionally equivalent to the original, verbose equation:

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

References

↑ Quine, Willard Van Orman (October 1952). "The Problem of Simplifying Truth Functions". The American Mathematical Monthly. 59 (8): 521–531. doi:10.2307/2308219. JSTOR 2308219.
↑ Quine, Willard Van Orman (November 1955). "A Way to Simplify Truth Functions". The American Mathematical Monthly. 62 (9): 627–631. doi:10.2307/2307285. JSTOR 2307285.
↑ McCluskey, Jr., Edward J. (November 1956). "Minimization of Boolean Functions". Bell System Technical Journal. 35 (6): 1417–1444. doi:10.1002/j.1538-7305.1956.tb03835.x. Retrieved 2014-08-24.
↑ Nelson, Victor P.; et al. (1995). Digital Logic Circuit Analysis and Design. Prentice Hall. p. 234. Retrieved 2014-08-26.
↑ Logic Friday program

External links

Quine-McCluskey Solver, by Hatem Hassan.
Quine-McCluskey algorithm implementation with a search of all solutions, by Frédéric Carpon.
Modified Quine-McCluskey Method, by Vitthal Jadhav, Amar Buchade.
All about Quine-McClusky, article by Jack Crenshaw comparing Quine-McClusky to Karnaugh maps
Karċma 3, A set of logic synthesis tools including Karnaugh maps, Quine-McCluskey minimization, BDDs, probabilities, teaching module and more. Logic Circuits Synthesis Labs (LogiCS) - UFRGS, Brazil.
A. Costa BFunc, QMC based boolean logic simplifiers supporting up to 64 inputs / 64 outputs (independently) or 32 outputs (simultaneously)
Python Implementation by Robert Dick, with an optimized version.
Python Implementation for symbolically reducing Boolean expressions.
Quinessence, an open source implementation written in Free Pascal by Marco Caminati.
QCA an open source, R based implementation used in the social sciences, by Adrian Duşa
A series of two articles describing the algorithm(s) implemented in R: first article and second article. The R implementation is exhaustive and it offers complete and exact solutions. It processes up to 20 input variables.
minBool an implementation by Andrey Popov.
QMC applet, an applet for a step by step analyze of the QMC- algorithm by Christian Roth
C++ implementation SourceForge.net C++ program implementing the algorithm.
Perl Module by Darren M. Kulp.
Tutorial Tutorial on Quine-McCluskey and Petrick's method (pdf).
Petrick C++ implementation (including Petrick) based on the tutorial above
C program Public Domain console based C program on SourceForge.net.
Tomaszewski, S. P., Celik, I. U., Antoniou, G. E., "WWW-based Boolean function minimization" INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS AND COMPUTER SCIENCE, VOL 13; PART 4, pages 577-584, 2003.
For a fully worked out example visit: http://www.cs.ualberta.ca/~amaral/courses/329/webslides/Topic5-QuineMcCluskey/sld024.htm
An excellent resource detailing each step: Olivier Coudert "Two-level logic minimization: an overview" INTEGRATION, the VLSI journal, 17-2, pp. 97–140, October 1994
The Boolean Bot: A JavaScript implementation for the web: http://booleanbot.com/
open source gui QMC minimizer
Computer Simulation Codes for the Quine-McCluskey Method, by Sourangsu Banerji.

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[Quine_1952-1] Quine, Willard Van Orman (October 1952). "The Problem of Simplifying Truth Functions". The American Mathematical Monthly. 59 (8): 521–531. doi:10.2307/2308219. JSTOR 2308219.

[Quine_1955-2] Quine, Willard Van Orman (November 1955). "A Way to Simplify Truth Functions". The American Mathematical Monthly. 62 (9): 627–631. doi:10.2307/2307285. JSTOR 2307285.

[McCluskey_1956-3] McCluskey, Jr., Edward J. (November 1956). "Minimization of Boolean Functions". Bell System Technical Journal. 35 (6): 1417–1444. doi:10.1002/j.1538-7305.1956.tb03835.x. Retrieved 2014-08-24.

[Nelson_1995-4] Nelson, Victor P.; et al. (1995). Digital Logic Circuit Analysis and Design. Prentice Hall. p. 234. Retrieved 2014-08-26.

[Logic_Friday-5] Logic Friday program