Quine–McCluskey algorithm

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. 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 $3^n/\backslash sqrt\{n\}$ , 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.{{update inline|date=May 2017|reason=The reference correctly describes the situation in 1995. We need to expand this to include the changes of the past twenty years, however.}} For one natural class of functions $f$, the precise complexity of finding all prime implicants is better-understood: Milan Mossé, Harry Sha, and Li-Yang Tan discovered a near-optimal algorithm for finding all prime implicants of a formula in conjunctive normal form.{{Cite journal |last1=Mossé |first1=Milan |last2=Sha |first2=Harry |last3=Tan |first3=Li-Yang |date=2022 |title=A Generalization of the Satisfiability Coding Lemma and Its Applications |journal=DROPS-IDN/V2/Document/10.4230/LIPIcs.SAT.2022.9 |series=Leibniz International Proceedings in Informatics (LIPIcs) |volume=236 |pages=9:1–9:18 |language=en |publisher=Schloss Dagstuhl – Leibniz-Zentrum für Informatik |doi=10.4230/LIPIcs.SAT.2022.9|doi-access=free |isbn=978-3-95977-242-6 }}

Step two of the algorithm amounts to solving the set cover problem; NP-hard instances of this problem may occur in this algorithm step.

In this example, the input is a Boolean function in four variables, $f\; :\backslash \{0,1\backslash \}^4\; \backslash to\; \backslash \{0,1\backslash \}$ 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

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

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). The summation symbol $\backslash sum$ denotes the logical sum (logical OR, or disjunction) of all the terms being summed over.

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

:

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{{sub|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 (names in parentheses), so they can be combined with minterms:

:

At this point, one can start combining minterms with other minterms in adjacent groups; as in, we compare minterms in nth group with (n+1)th group. So for the m4 minterm in with only one Number of 1s, we compare it to m9, m10, and m12 which have two Number of 1s.

If two terms differ by only a single digit, that digit is replaced with a dash indicating that the digit doesn't matter. For instance 1000 and 1001 can be combined to give 100-, indicating that both minterms imply the first digit is 1 and the next two are 0. Terms that can't be combined any more are marked with an asterisk ({{color|red|*}}).

:

When going from Size 2 to Size 4, treat - as a third bit value. Match up the -'s first. The terms represent products and to combine two product terms they must have the same variables. One of the variables should be complemented in one term and uncomplemented in the other. The remaining variables present should agree. So to match two terms the -'s must align and all but one of the other digits must be the same. For instance, -110 and -100 can be combined to give -1-0, as can -110 and -010 to give --10, but -110 and 011- cannot since the -'s do not align. -110 corresponds to BCD' while

011- corresponds to A'BC, and BCD' + A'BC is not equivalent to a product term.

:

Note: In this example, none of the terms in the size 4 implicants table can be combined any further. In general, this process is continued in sizes that are powers of 2 (sizes 8, 16 etc.) until no more terms can be combined.

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 (these are the ones that have been marked with a "{{color|red|*}}" in the previous step), 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 look for columns with only one "✓". If a column has only one "✓", this means that the minterm can only be covered by one prime implicant. This prime implicant is essential.

For example: in the first column, with minterm 4, there is only one "✓". This means that m(4,12) is essential (hence marked by {{color|blue|^#}}). Minterm 15 also has only one "✓", so m(10,11,14,15) is also essential. Now all columns with one "✓" are covered. The rows with minterm m(4,12) and m(10,11,14,15) can now be removed, together with all the columns they cover.

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:

:f{{sub|A,B,C,D}} = BC'D' + AB' + AC

or

:f{{sub|A,B,C,D}} = BC'D' + AD' + AC

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

:f{{sub|A,B,C,D}} = A'BC'D' + AB'C'D' + AB'C'D + AB'CD' + AB'CD + ABC'D' + ABCD' + ABCD.

The pseudocode below recursively computes the prime implicants given the list of minterms of a boolean function. It does this by trying to merge all possible minterms and filtering out minterms that have been merged until no more merges of the minterms can be performed and hence, the prime implicants of the function have been found.

// Computes the prime implicants from a list of minterms.

// each minterm is of the form "1001", "1010", etc and can be represented with a string.

function getPrimeImplicants(list minterms) is

primeImplicants ← empty list

merges ← new boolean array of length equal to the number of minterms, each set to false

numberOfmerges ← 0

mergedMinterm, minterm1, minterm2 ← empty strings

for i = 0 to length(minterms) do

for c = i + 1 to length(minterms) do

minterm1 ← minterms[i]

minterm2 ← minterms[c]

// Checking that two minterms can be merged

if CheckDashesAlign(minterm1, minterm2) && CheckMintermDifference(minterm1, minterm2) then

mergedMinterm ← MergeMinterms(minterm1, minterm2)

if primeImplicants Does Not Contain mergedMinterm then

primeImplicants.Add(mergedMinterm)

numberOfMerges ← numberOfMerges + 1

merges[i] ← true

merges[c] ← true

// Filtering all minterms that have not been merged as they are prime implicants. Also removing duplicates

for j = 0 to length(minterms) do

if merges[j] == false && primeImplicants Does Not Contain minterms[j] then

primeImplicants.Add(minterms[j])

// if no merges have taken place then all of the prime implicants have been found so return, otherwise

// keep merging the minterms.

if numberOfmerges == 0 then

return primeImplicants

else

return getPrimeImplicants(primeImplicants)

In this example the CheckDashesAlign and CheckMintermDifference functions perform the necessary checks for determining whether two minterms can be merged. The function MergeMinterms merges the minterms and adds the dashes where necessary. The utility functions below assume that each minterm will be represented using strings.

function MergeMinterms(minterm1, minterm2) is

mergedMinterm ← empty string

for i = 0 to length(minterm1) do

//If the bits vary then replace it with a dash, otherwise the bit remains in the merged minterm.

if minterm[i] != minterm2[i] then

mergedMinterm ← mergedMinterm + '-'

else

mergedMinterm ← mergedMinterm + minterm1[i]

return mergedMinterm

function CheckDashesAlign(minterm1, minterm2) is

for i = 0 to length(minterm1) do

// If one minterm has dashes and the other does not then the minterms cannot be merged.

if minterm1[i] != '-' && minterm2[i] == '-' then

return false

return true

function CheckMintermDifference(minterm1, minterm2) is

// minterm1 and minterm2 are strings representing all of the currently found prime implicants and merged

// minterms. Examples include '01--' and '10-0'

m1, m2 ← integer representation of minterm1 and minterm2 with the dashes removed, these are replaced with 0

// ^ here is a bitwise XOR

res ← m1 ^ m2

return res != 0 && (res & res - 1) == 0

The pseudocode below can be split into two sections:

1. Creating the prime implicant chart using the prime implicants
2. Reading the prime implicant chart to find the essential prime implicants.

The prime implicant chart can be represented by a dictionary where each key is a prime implicant and the corrresponding value is an empty string that will store a binary string once this step is complete. Each bit in the binary string is used to represent the ticks within the prime implicant chart. The prime implicant chart can be created using the following steps:

1. Iterate through each key (prime implicant of the dictionary).
2. Replace each dash in the prime implicant with the \d character code. This creates a regular expression that can be checked against each of the minterms, looking for matches.
3. Iterate through each minterm, comparing the regular expression with the binary representation of the minterm, if there is a match append a "1" to the corresponding string in the dictionary. Otherwise append a "0".
4. Repeat for all prime implicants to create the completed prime implicant chart.

When written in pseudocode, the algorithm described above is:

function CreatePrimeImplicantChart(list primeImplicants, list minterms)

primeImplicantChart ← new dictionary with key of type string and value of type string

// Creating the empty chart with the prime implicants as the key and empty strings as the value.

for i = 0 to length(primeImplicants) do

// Adding a new prime implicant to the chart.

primeImplicantChart.Add(primeImplicants[i], "")

for i = 0 to length(primeImplicantChart.Keys) do

primeImplicant ← primeImplicantChart.Keys[i]

// Convert the "-" to "\d" which can be used to find the row of ticks above.

regularExpression ← ConvertToRegularExpression(primeImplicant)

for j = 0 to length(minterms) do

// If there is a match between the regular expression and the minterm than append a 1 otherwise 0.

if regularExpression.matches(minterms[j]) then

primeImplicantChart[primeImplicant] += "1"

else

primeImplicantChart[primeImplicant] += "0"

// The prime implicant chart is complete so return the completed chart.

return primeImplicantChart

The utility function, ConvertToRegularExpression, is used to convert the prime implicant into the regular expression to check for matches between the implicant and the minterms.

function ConvertToRegularExpression(string primeImplicant)

regularExpression ← new string

for i = 0 to length(primeImplicant) do

if primeImplicant[i] == "-" then

// Add the literal character "\d".

regularExpression += @"\d"

else

regularExpression += primeImplicant[i]

return regularExpression

Using the function, CreatePrimeImplicantChart, defined above, we can find the essential prime implicants by simply iterating column by column of the values in the dictionary, and where a single "1" is found then an essential prime implicant has been found. This process is described by the pseudocode below.

function getEssentialPrimeImplicants(Dictionary primeImplicantChart, list minterms)

essentialPrimeImplicants ← new list

mintermCoverages ← list with all of the values in the dictionary

for i = 0 to length(ticks) do

mintermCoverage ← ticks[i]

for j = 0 to length(mintermCoverage) do

if mintermCoverage[j] == "1" then

essentialPrimeImplicants.Add(primeImplicantChart.Keys[i])

return essentialPrimeImplicants

Using the algorithm above it is now possible to find the minimised boolean expression, by converting the essential prime implicants into the canonical form ie. -100 -> BC'D' and separating the implicants by logical OR. The pseudocode assumes that the essential prime implicants will cover the entire boolean expression.

• Blake canonical form
• Buchberger's algorithm – analogous algorithm for algebraic geometry
• Petrick's method
• Qualitative comparative analysis (QCA)

{{reflist|refs=

{{cite journal |last=Quine |first=Willard Van Orman |author-link=Willard Van Orman Quine |date=October 1952 |title=The Problem of Simplifying Truth Functions |jstor=2308219 |journal=The American Mathematical Monthly |volume=59 |issue=8 |pages=521–531 |doi=10.2307/2308219}}

{{cite journal |last=Quine |first=Willard Van Orman |author-link=Willard Van Orman Quine |date=November 1955 |title=A Way to Simplify Truth Functions |jstor=2307285 |journal=The American Mathematical Monthly |volume=62 |issue=9 |pages=627–631 |doi=10.2307/2307285 |hdl=10338.dmlcz/142789}}

{{cite journal |last=McCluskey |first=Edward Joseph Jr.|author-link=Edward Joseph McCluskey, Jr. |date=November 1956 |title=Minimization of Boolean Functions |journal=Bell System Technical Journal |volume=35 |issue=6 |pages=1417–1444 |doi=10.1002/j.1538-7305.1956.tb03835.x |url=https://archive.org/details/bstj35-6-1417 |access-date=2014-08-24 |hdl=10338.dmlcz/102933}}

{{cite book |last1=Nelson |first1=Victor P. |first2=H. Troy |last2=Nagle |first3=Bill D. |last3=Carroll |first4=J. David |last4=Irwin |author-link4=J. David Irwin |date=1995 |title=Digital Logic Circuit Analysis and Design |url=https://books.google.com/books?id=8V5TAAAAMAAJ |publisher=Prentice Hall |page=234 |edition=2 |access-date=2014-08-26 |isbn=978-0-13463894-2}}

Logic Friday program

{{cite book |first=William J. |last=Masek |title=Some NP-complete set covering problems |publisher=unpublished |date=1979}}

{{cite thesis |type=Master's thesis |first=Sebastian Lukas Arne |last=Czort |title=The complexity of minimizing disjunctive normal form formulas |institution=University of Aarhus |date=1999}}

{{cite journal |first1=Christopher |last1=Umans |author-link1=Christopher Umans |first2=Tiziano |last2=Villa |first3=Alberto Luigi |last3=Sangiovanni-Vincentelli |author-link3=Alberto Luigi Sangiovanni-Vincentelli |title=Complexity of two-level logic minimization |journal=IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems |volume=25 |number=7 |pages=1230–1246 |date=2006-06-05 |doi=10.1109/TCAD.2005.855944 |s2cid=10187481}}

{{cite journal |first=Vitaly |last=Feldman |title=Hardness of Approximate Two-Level Logic Minimization and PAC Learning with Membership Queries |journal=Journal of Computer and System Sciences |volume=75 |pages=13–25 [13–14] |date=2009 |doi=10.1016/j.jcss.2008.07.007|doi-access=free }}

{{cite journal |first=James F. |last=Gimpel |title=A Method for Producing a Boolean Function Having an Arbitrary Prescribed Prime Implicant Table |journal=IEEE Transactions on Computers |volume=14 |pages=485–488 |date=1965 |doi=10.1109/PGEC.1965.264175}}

{{cite journal |first=Wolfgang Jakob |last=Paul |author-link=:de:Wolfgang Paul (Informatiker) |title=Boolesche Minimalpolynome und Überdeckungsprobleme |language=de |journal=Acta Informatica |volume=4 |issue=4 |pages=321–336 |date=1974 |doi=10.1007/BF00289615 |s2cid=35973949}}

{{cite book |first1=Paul William |last1=Abrahams |first2=John "Jack" G. |last2=Nordahl |title=The Modified Quine–McCluskey Reduction Procedure |date=November 1955 |location=Electrical Engineering Department, Massachusetts Institute of Technology, Massachusetts, USA |type=Class memorandum}} (4 pages) (NB. Some sources list the authors as "{{citeref|Caldwell|1958|P. W. Abraham|style=plain}}" and "{{citeref|Kämmerer|1969|I. G. Nordahl|style=plain}}", the title is also cited as "{{citeref|Caldwell|1958|Modified McCluskey–Quine Reduction Procedure|style=plain}}".)

{{cite web |title=Welcome to the memorial page for John "Jack" G Nordahl June 14, 1933 ~ November 20, 2017 (age 84) |publisher=Jellison Funeral Home and Cremation Services |url=https://www.jellisonfuneralhome.com/obituary/JohnJack-Nordahl |access-date=2020-05-05 |url-status=live |archive-url=https://web.archive.org/web/20200505151004/https://www.jellisonfuneralhome.com/obituary/JohnJack-Nordahl |archive-date=2020-05-05}}

{{cite book |title=Switching Circuits and Logical Design |chapter=5.8. Operations Using Decimal Symbols |first=Samuel Hawks |last=Caldwell |author-link=Samuel Hawks Caldwell |version=5th printing September 1963 |date=1958-12-01 |orig-date=February 1958 |edition=1st |publisher=John Wiley & Sons Inc. |publication-place=New York, USA |location=Watertown, Massachusetts, USA |isbn=0-47112969-0 |lccn=58-7896 |pages=162–169 |quote-page=166 |quote=[...] It is a pleasure to record that this treatment is based on the work of two students during the period they were studying Switching Circuits at the Massachusetts Institute of Technology. They discussed the method independently and then collaborated in preparing a class memorandum: {{citeref|Abrahams|Nordahl|1955|P. W. Abraham and J. G. Nordahl|style=plain}} [...]}} (xviii+686 pages) (NB. For the first major treatise of the decimal method in this book, it is sometimes misleadingly known as "Caldwell's decimal tabulation".)

{{cite journal |title=A Residue Test for Boolean Functions |first1=Albert Alkins |last1=Mullin |author-link1=Albert Alkins Mullin |first2=Wayne G. |last2=Kellner |journal=Transactions of the Illinois State Academy of Science |volume=51 |number=3–4 |date=1958 |publication-place=Springfield, Illinois, USA |location=University of Illinois, Urbana, USA and Electrical Engineering Department, Massachusetts Institute of Technology, Massachusetts, USA |type=Teaching memorandum |s2cid=125171479 |pages=14–19 |url=https://ilacadofsci.com/wp-content/uploads/2016/03/051-16-print.pdf |access-date=2020-05-05 |url-status=live |archive-url=https://web.archive.org/web/20200505163915/https://ilacadofsci.com/wp-content/uploads/2016/03/051-16-print.pdf |archive-date=2020-05-05}} [https://ilacadofsci.com/archives/9279] (6 pages) (NB. In {{citeref|Caldwell|1958|his book|style=plain}}, Caldwell dates this to November 1955 as a teaching memorandum. Since Mullin dates their work to 1958 in {{citeref|Mullin|1960|another work|style=plain}} and Abrahams/Nordahl's {{citeref|Abrahams|Nordahl|1955|class memorandum|style=plain}} is also dated November 1955, this could be a copy error.)

{{cite journal |title=Two Applications of Elementary Number Theory |first=Albert Alkins |last=Mullin |author-link=Albert Alkins Mullin |journal=Transactions of the Illinois State Academy of Science |volume=52 |number=3–4 |date=1960-03-15 |orig-date=1959-09-19 |location=University of Illinois, Urbana, USA |publication-place=Springfield, Illinois, USA |editor-first1=Harvey I. |editor-last1=Fisher |editor-first2=George E. |editor-last2=Ekblaw |editor-first3=F. O. |editor-last3=Green |editor-first4=Reece |editor-last4=Jones |editor-first5=Francis |editor-last5=Kruidenier |editor-first6=John |editor-last6=McGregor |editor-first7=Paul |editor-last7=Silva |editor-first8=Milton |editor-last8=Thompson |pages=102–103 |url=https://ilacadofsci.com/wp-content/uploads/2016/03/052-15-print.pdf |access-date=2020-05-05 |url-status=live |archive-url=https://web.archive.org/web/20200505143916/https://ilacadofsci.com/wp-content/uploads/2016/03/052-15-print.pdf |archive-date=2020-05-05}} [https://ilacadofsci.com/archives/9203][https://archive.org/details/transactionsofil5219unse][https://archive.org/stream/transactionsofil5219unse/transactionsofil5219unse_djvu.txt] (2 pages)

{{cite journal |title=Albert A. Mullin and Wayne G. Kellner. A residue test for Boolean functions. Transactions of the Illinois State Academy of Science, vol. 51 nos. 3 and 4, (1958), pp. 14–19. |type=Review |last=McCluskey |first=Edward Joseph Jr.|author-link=Edward Joseph McCluskey, Jr. |journal=The Journal of Symbolic Logic |volume=25 |issue=2 |date=June 1960 |doi=10.2307/2964263 |page=185 |jstor=2964263 |s2cid=123530443 |quote-page=185 |quote=[...] The results of this paper are presented in the more readily available {{citeref|Caldwell|1958|book|style=plain}} by S. H. Caldwell [...]. In this book, the author gives credit to {{citeref|Mullin|Kellner|1958|Mullin and Kellner|style=plain}} for development of the manipulations with the decimal numbers.}} (1 page)

{{cite book |title=Digitale Automaten – Theorie, Struktur, Technik, Programmieren |language=de |chapter=I.12. Theorie: Minimierung Boolescher Funktionen |first=Wilhelm |last=Kämmerer |author-link=:de:Wilhelm Kämmerer |editor-first1=Hans |editor-last1=Frühauf |editor-link1=:de:Hans Frühauf |editor-first2=Wilhelm |editor-last2=Kämmerer |editor-first3=Kurz |editor-last3=Schröder |editor-first4=Helmut |editor-last4=Winkler |edition=1 |date=May 1969 |publisher=Akademie-Verlag GmbH |publication-place=Berlin, Germany |location=Jena, Germany |volume=5 |id=License no. 202-100/416/69. Order no. 4666 ES 20 K 3. |series=Elektronisches Rechnen und Regeln |pages=98, 103–104 |url=https://books.google.com/books?id=jkcgAQAAIAAJ&q=P.+W.+Abraham+I.+G.+Nordahl |quote-page=98 |quote=[...] 1955 wurde das Verfahren auf die bequemere dezimale Schreibweise umgestellt ({{citeref|Abrahams|Nordahl|1955|P. W. Abraham und I. G. Nordahl|style=plain}} in [{{citeref|Caldwell|1958|Caldwell|style=plain}}]). [...]}} (NB. A second edition 1973 exists as well.)

{{cite journal |first=Hugh |last=McColl |author-link=Hugh McColl (mathematician) |date=1878-11-14 |title=The Calculus of Equivalent Statements (Third Paper) |journal=Proceedings of the London Mathematical Society |volume=s1-10 |issue=1 |doi=10.1112/plms/s1-10.1.16 |pages=16–28 |url=https://academic.oup.com/plms/article-abstract/s1-10/1/16/1503062|url-access=subscription }}

{{cite journal |first=Archie |last=Blake |title=Canonical expressions in Boolean algebra |series=Abstracts of Papers |journal=Bulletin of the American Mathematical Society |date=November 1932 |page=805}}

{{cite book |last=Blake |first=Archie |date=1938 |orig-date=1937 |title=Canonical Expressions in Boolean Algebra |type=Dissertation |publisher=University of Chicago Libraries |location=Chicago, Illinois, USA |edition=Lithographed |page=36 |url=https://books.google.com/books?id=gqRYAAAAMAAJ |quote-page=36 |quote=[...] this method was known to Peirce and his students [...] It is mentioned at several places in Studies in Logic, by members of the Johns Hopkins University, 1883 [...]}} (ii+60 pages)

{{cite journal |first=Archie |last=Blake |title=Corrections to Canonical Expressions in Boolean Algebra |journal=The Journal of Symbolic Logic |issn=0022-4812 |publisher=Association for Symbolic Logic |volume=3 |issue=2 |date=June 1938 |doi=10.2307/2267595 |jstor=2267595 |url=https://projecteuclid.org/euclid.jsl/1183385465 |pages=112–113|s2cid=5810863 |url-access=subscription }}

{{cite book |last1=Samson |first1=Edward Walter |last2=Mills |first2=Burton E. |date=April 1954 |title=Circuit Minimization: Algebra and Algorithms for New Boolean Canonical Expressions |publisher=Air Force Cambridge Research Center |id=Technical Report AFCRC TR 54-21 |location=Bedford, Massachusetts, USA}}

{{cite journal |last=Nelson |first=Raymond J. |date=June 1955 |title=Simplest Normal Truth Functions |journal=The Journal of Symbolic Logic |publisher=Association for Symbolic Logic |volume=20 |issue=2 |doi=10.2307/2266893 |jstor=2266893 |pages=105–108|s2cid=32920372 }} (4 pages)

{{cite book |last=Ladd |first=Christine |author-link=Christine Ladd |date=1883 |chapter=On the algebra of logic |editor-first=Charles Sanders |editor-last=Peirce |editor-link=Charles Sanders Peirce |title=Studies in Logic |location=Boston, USA |publisher=Little, Brown & Company |pages=17–71 |quote-page=23 |quote=[...] If the reduction [of an expression to simplest form] is not evident, it may be facilitated by taking the negative of the expression, reducing it, and then restoring it to the positive form. [...]}}

{{cite journal |title=Classroom Reduction of Boolean Functions |first=Daniel C. |last=Fielder |journal=IEEE Transactions on Education |publisher=IEEE |volume=9 |issue=4 |date=December 1966 |issn=0018-9359 |eissn=1557-9638 |doi=10.1109/TE.1966.4321989 |bibcode=1966ITEdu...9..202F |pages=202–205}}

{{cite journal |title=McColl and Minimization |journal=History and Philosophy of Logic |volume=31 |number=4 |pages=337–348 |publisher=Taylor & Francis |issn=1464-5149 |date=November 2010 |orig-date=2010-10-27 |doi=10.1080/01445340.2010.517387 |first=Frank Markham |last=Brown |s2cid=216590641 |url=https://www.researchgate.net/publication/233139704 |access-date=2020-04-15 |url-status=live |archive-url=https://web.archive.org/web/20200415201838/https://www.researchgate.net/publication/233139704_McColl_and_Minimization |archive-date=2020-04-15}}

{{cite conference |title=Investigation on Quine McCluskey Method: A Decimal Manipulation Based Novel Approach for the Minimization of Boolean Function |first1=Alak |last1=Majumder |first2=Barnali |last2=Chowdhury |first3=Abir J. |last3=Mondal |first4=Kunj |last4=Jain |location=Department of Electronics & Communication, Engineering National Institute of Technology, Arunachal Pradesh Yupia, India |date=2015-01-30 |orig-date=2015-01-09 |type=Conference paper |conference=2015 International Conference on Electronic Design, Computer Networks & Automated Verification (EDCAV), Shillong, India |doi=10.1109/EDCAV.2015.7060531 |pages=18–22 |url=https://www.researchgate.net/publication/301408055 |access-date=2020-05-08 |url-status=live |archive-url=https://web.archive.org/web/20200508200648/https://www.researchgate.net/publication/301408055_Investigation_on_Quine_McCluskey_method_A_decimal_manipulation_based_novel_approach_for_the_minimization_of_Boolean_function

|archive-date=2020-05-08}} [https://www.researchgate.net/profile/Alak_Majumder2/publication/301408055_Investigation_on_Quine_McCluskey_method_A_decimal_manipulation_based_novel_approach_for_the_minimization_of_Boolean_function/links/5948a1bba6fdcc70635a3706/Investigation-on-Quine-McCluskey-method-A-decimal-manipulation-based-novel-approach-for-the-minimization-of-Boolean-function.pdf] (NB. This work does not cite the prior art on decimal methods.) (5 pages)

{{cite book |title=Digital Logic Design |chapter=3.17 Decimal approach to Quine–McCluskey simplification of Boolean algebra |first1=Brian |last1=Holdsworth |first2=Clive |last2=Woods |edition=4 |date=2002 |publisher=Newnes Books / Elsevier Science |isbn=((0-7506-4588-2)) |pages=65–67 |chapter-url=https://books.google.com/books?id=o7enSwSVvgYC&pg=PA65 |url=https://books.google.com/books?id=o7enSwSVvgYC |access-date=2020-04-19}} (519 pages) [https://web.archive.org/web/20200419213939/http://s2.bitdownload.ir/Ebook/Electronics/Holdsworth%20-%20Digital%20Logic%20Design%204e%20HQ%20(Newnes,%202002).pdf]

{{cite journal |title=On the number of prime implicants |first1=Ashok K. |last1=Chandra |first2=George |last2=Markowsky |journal=Discrete Mathematics |volume=24 |number=1 |date=1978 |pages=7–11 |doi=10.1016/0012-365X(78)90168-1 |doi-access=free }}

}}

• {{cite book |title=A new approach to the design of switching circuits |author-first=Herbert Allen |author-last=Curtis |chapter=Chapter 2.3. McCluskey's Method |publisher=D. van Nostrand Company, Inc. |date=1962 |location=Princeton, New Jersey, USA |edition=1 |series=The Bell Laboratories Series |s2cid=57068910 |oclc=1036797958 |isbn=0-44201794-4 |id={{ISBN|978-0-44201794-1}}. ark:/13960/t56d6st0q |pages=90–160}} (viii+635 pages) (NB. This book was reprinted by Chin Jih in 1969.)
• {{cite journal |first=Olivier |last=Coudert |title=Two-level logic minimization: an overview |journal=Integration, the VLSI Journal |issn=0167-9260 |volume=17-2 |date=October 1994 |issue=2 |pages=97–140 |doi=10.1016/0167-9260(94)00007-7 |url=https://oliviercoudert.files.wordpress.com/2020/11/int94.pdf |access-date=2020-05-10 |url-status=live |archive-url=https://web.archive.org/web/20200510214647/http://www.ocoudert.com/papers/pdf/int94.pdf |archive-date=2020-05-10}} (47 pages)
• {{cite arXiv |title=Modified Quine-McCluskey Method |first1=Vitthal |last1=Jadhav |first2=Amar |last2=Buchade |date=2012-03-08 |class=cs.OH |eprint=1203.2289}} (4 pages)
• {{cite web |title=All about Quine-McClusky |first=Jack |last=Crenshaw |date=2004-08-19 |website=embedded.com |url=http://www.embedded.com/electronics-blogs/programmer-s-toolbox/4025004/All-about-Quine-McClusky |access-date=2020-05-10 |url-status=live |archive-url=https://web.archive.org/web/20200510215306/https://www.embedded.com/all-about-quine-mcclusky/ |archive-date=2020-05-10}}
• {{cite journal |last1=Tomaszewski |first1=Sebastian P. |last2=Celik |first2=Ilgaz U. |last3=Antoniou |first3=George E. |title=WWW-based Boolean function minimization |journal=International Journal of Applied Mathematics and Computer Science |volume=13 |issue=4 |date=December 2003 |orig-date=2003-03-05, 2002-04-09 |pages=577–584 |url=http://matwbn.icm.edu.pl/ksiazki/amc/amc13/amc13414.pdf |access-date=2020-05-10 |url-status=live |archive-url=https://web.archive.org/web/20200510214937/http://matwbn.icm.edu.pl/ksiazki/amc/amc13/amc13414.pdf |archive-date=2020-05-10}} [https://www.researchgate.net/publication/228862329_WWW-based_Boolean_function_minimization][https://archive.today/20180115131301/http://matwbn.icm.edu.pl/ksiazki/amc/amc13/amc13414.pdf] (7 pages)
• {{cite journal |first=Adrian |last=Duşa |date=2008-10-01 |orig-date=September 2007 |title=A mathematical approach to the boolean minimization problem |journal=Quality & Quantity |volume=44 |pages=99–113 |doi=10.1007/s11135-008-9183-x |s2cid=123042755 |id=Article number: 99 (2010)}} [https://web.archive.org/web/20120313031147/http://www.compasss.org/files/WPfiles/Dusa2007.pdf] (22 pages)
• {{cite web |title=Enhancing Quine-McCluskey |first=Adrian |last=Duşa |location=University of Bucharest |date=2007 |url=http://www.compasss.org/wpseries/Dusa2007b.pdf |access-date=2020-05-12 |url-status=live |archive-url=https://web.archive.org/web/20200512134710/http://www.compasss.org/wpseries/Dusa2007b.pdf |archive-date=2020-05-12}} (16 pages) (NB.

• [http://sites.google.com/site/simpogical/download Tutorial] Tutorial on Quine-McCluskey and Petrick's method.
• For a fully worked out example visit: http://www.cs.ualberta.ca/~amaral/courses/329/webslides/Topic5-QuineMcCluskey/sld024.htm

{{DEFAULTSORT:Quine-Mccluskey Algorithm}}

Category:Boolean algebra

Category:Willard Van Orman Quine