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Journal of Engineering and Computer Sciences
Qassim University, Vol. 10, No. 1, pp. 53-88 (January 2017/Rabi' II 1438H)
53
Karnaugh-Map Utilization in Boolean Analysis:
The Case of War Termination
Ali Muhammad Ali Rushdi and Raid Mohammad Salih Badawi
Department of Electrical and Computer Engineering, King Abdulaziz University, P. O. Box 80204,
Jeddah 21589, Saudi Arabia.
{arushdi@kau.edu.sa}
Abstract. This paper advocates and demonstrates the utility of the Karnaugh-map, as a pictorial manual
tool, in the Boolean analysis (BA) of social and political problems, in general, and in problems of peace
research, in particular, as exemplified herein by the problem of war termination. Analysis is performed
for both the appearance and absence of a specified phenomenon for the cases where (a) the logical
remainders (don’t cares) are ignored (actually nullified), and (b) the don’t cares are assigned certain
deliberate but independent values, and (c) faithful representation is used via a partially-defined function
whose asserted part constitutes the definite (certain) causes of the phenomenon while its don’t-care part is
a disjunction of its potential (uncertain) causes. The paper also presents several novel extensions of BA in
which the don’t-care entries in the Karnaugh-map are manipulated in an attempt to make the output
function positive/negative in, independent of, or symmetric in some of its arguments, to assign an
importance metric for each of these arguments, to obtain a threshold representation for it, or to fit a pre-
specified hypothesis. An explanation is given for the relation between BA per se, and two variants
thereof, namely Crisp-set Qualitative Comparative Analysis (csQCA) and Coincidence Analysis (CNA).
Key Words: Boolean analysis, Karnaugh map, Logical remainders (don’t cares), War termination,
Minimization, Novel extensions, Crisp-set Qualitative Comparative Analysis (csQCA), Coincidence
analysis (CNA).
Ali Muhammad Rushdi and Raid Mohammad Badawi
54
1. Introduction
Boolean analysis (BA) was introduced by Flament [1, 2] for application to
questionnaire data [3, 4]. This analysis is a partial-order generalization of scalogram
analysis [5], and is particularly useful for hierarchical data [6], and generally any
type of data that can be dichotomized, i.e., reduced to optimal binary data [7-9].
Boolean analysis is the core essence of Qualitative Comparative Analysis (QCA)
first introduced in the seminal paper of Ragin et al. [10], well established via the
celebrated text of Ragin [11], and further elaborated by Ragin, his associates and
other scholars [12-24]. Qualitative Comparative Analysis has now branched into
three variants, namely crisp-set Qualitative Comparative Analysis (csQCA), multi-
value Qualitative Comparative Analysis (mvQCA), and fuzzy-set Qualitative
Comparative Analysis (fsQCA) [20]. Qualitative Comparative Analysis bridges the
gap between (and embodies some key strengths of) the qualitative and quantitative
approaches in the social and political sciences [18]. Though csQCA has many subtle
differences with rudimentary BA, we find it more convenient herein for the purposes
of this paper to treat the terms BA and csQCA as synonymous. We will also include
under the general umbrella of BA, all pertaining mathematical features and
engineering applications [25-29]. A notable offshoot of BA is Coincidence Analysis
(CNA) [30-33] which is formally more similar to the original work of Flament [1, 2].
This CNA offshoot differs slightly form the mainstream BA (or csQCA) studied
herein. In CNA, a relation is studied between several variables without presuming
which are causes and which are effects, while in mainstream BA a certain outcome
(effect) is studied as a function of certain inputs (causes).
Perhaps the most important advantage of BA is that it addresses explicitly the
concept that there can be multiple causal mechanisms producing the same output
[34], a concept known as ‘multiple conjunctural causation’. Boolean analysis starts
by identifying the outcome (dependent variable) as a Boolean function of n
conditions (independent variables). This function is typically represented in the form
of a truth table whose input domain consists of
2n
lines representing the
2n
possible combinations of the independent variables. These lines or combinations are
usually called configurations. If the outcome has a specific value of 1 or 0 for each
of these configurations, the Boolean function is said to be totally defined or
completely specified. Other possible values for a configuration are
The “indeterminate” value (-) among observed cases, a value that should be
avoided since one is supposed to handle specific outcomes across well-selected
cases. Rihoux & de Meur [19] call this value a don’t care, but we believe that this is
a misnomer that might lead to confusion, and hence we will not adopt it herein. In
fact, we reserve the term “don’t care” for use as an identical term to the term
“logical remainder” to be discussed below.
The “contradiction” value (C), for a configuration that has a “0” outcome for
some observed cases and a “1” outcome for other observed cases. Such a value poses a
logical contradiction that must be resolved before further processing via techniques
Karnaugh-Map Utilization in Boolean Analysis …
55
discussed by Rihoux & de Meur [19] and Jordan et al. [14]. Fortunately, we do not
encounter such a value in our forthcoming analysis based on that of Chan [34].
The “Logical Remainder” value (L) or (R), which is what is called a don’t-
care (d) in digital design and electrical engineering circles [29, 35-39]. This value
(which stands for either ‘1’ or ‘0’ but for nothing else) designates configurations that
have not been observed among the empirical cases. We reiterate that we will
consider Logical Remainders and don’t-cares as synonymous herein.
The aim of this paper is to advocate and demonstrate the use of a pictorial
manual tool (specifically, the Karnaugh-map) in general BA, with a stress on the
Boolean Analysis of political problems pertaining to peace research such as the
celebrated problem of war termination [34, 40-55]. At this point, we present several
observations in order.
1. There is a long history of utilization of pictorial tools in logic, engineering,
and mathematics [56]. These include the Venn diagram, the Carroll map, the
Marquand-Veitch map, and the Karnaugh map [26, 57]. There are subtle differences
among these tools; though sometimes these differences go unnoticed to the extent
that one tool might be given the name of another. For example the ‘Venn diagram’
produced by the ‘visualizer’ tool of the Tosmana software [19] is in fact a Carroll
map [57-58] though there is no significant fault (apart from a historical one
pertaining to giving unfair credit) in naming it a Venn diagram. In Section 2, we will
explain why the Karnaugh-map is our tool of choice and why it is more convenient
to use than the other tools, a fact attested to by its widespread use (unopposed) in
digital design circles [29,37,38,59], and by the existence of a variety of complaints
concerning the use of other tools [58, 60-65].
2. Boolean Analysis or dichotomous QCA (Crisp-set QCA) is typically
designed to address small-N or medium-N situations of less than 30-40 cases [66].
Since a configuration accommodates zero, one or more cases, the number of
configurations is somewhat expected to be comparable to the number of cases, and
hence the number of independent variables (which is the ceiling of (i.e., the smallest
integer greater than or equal to) the natural logarithm of the number of
configurations) is expected to be around six. In fact, in an overview of QCA
applications in political science during the period 2003-2011, Marx, et al. [16]
survey almost five hundred papers to find that the number
n
of conditions
(independent variables) used in each of them ranged from 3 to 10. Table I
summaries their findings concerning the possible values for the number
m
of
papers employing the number of conditions
n
,
3 n
10
. We stress that the
maximum value of
n
is 10, while its average value is
/ 5.287yx
whose ceiling
is 6. These maximum and average values seem as if deliberately set to suit the use of
a Karnaugh map, whose conventional form is conveniently used up to 6 variables
[29, 37], while its variable-entered form is conveniently used up to 12 variables [36,
37, 67-80].
Ali Muhammad Rushdi and Raid Mohammad Badawi
56
3. Rihoux [66] characterizes QCA as a technique based on Boolean algebra
and implemented by a set of computer programs. In fact, there are many such
programs or packages including the Tosmana software [81] and QCA [22-24, 82].
We argue that implementation of QCA via computer programs is not really
warranted for the dichotomous or crisp-set QCA (but could not probably be
dispensed with for the multi-value QCA or fuzzy-set QCA). It is clear from the
previous paragraph that a crisp-set QCA can be easily implemented by the
conventional Karnaugh map (CKM), or if necessary, by its extension, the variable-
entered Karnaugh map (VEKM) [36, 37, 67-80]. In fact, some limited use of the
CKM has been already made by the QCA community [22, 82, 83].
Table (1). Number of conditions (independent variables) in papers overviewed by Marx et al. (2013)
n=Number of conditions
3
4
5
6
7
8
9
10
Total
m=Number of papers m using n
7
29
25
13
9
6
4
1
94
x
n*m
21
116
125
78
63
48
36
10
497
y
4. Rihoux [66] also characterizes QCA as a labor-intensive, interactive, and
creative process. This means that QCA demands a lot of human work and
intervention. Implementing Boolean minimization via a computer program,
definitely saves some time, but this time is negligible compared to the total time of
the whole process. This not-so-important time saving is achieved at the expense of
the loss of the ready insight and full control gained via map use. In fact, with the
iterative nature of QCA, a researcher must make sense out of the solution, interpret
it by re-interrogating the cases, and go back to them to examine each one as a whole
[66]. Working manually all throughout on a map might be preferable to a mixed
type of work jumping between manual and automated work.
5. The seminal idea of Boolean Analysis has now outgrown to several
distinguished variants or threads as shown in Fig. 1. One might go to one extreme of
viewing each of these threads as essentially a replica of any of the others in disguise,
or go to the other extreme of claiming each of these threads to be a stand-alone
discipline marginally or superficially related to the others. Viewing these threads
from a Karnaugh-map perspective might yield fruitful insights about certain striking
similarities among these threads as well as some subtle differences among them.
Such a perspective might remedy the problem that these threads currently exist in
considerable isolation and at best have some minimal interactions among them.
Inspired by the aforementioned observations, we present this paper as a
tutorial exposition of utilization of the Karnaugh map in political science in general,
and in peace research in particular. In some sense, our paper can be thought of as a
computer-science sequel to the seminal paper by Chan [34] that appeared earlier in a
Karnaugh-Map Utilization in Boolean Analysis …
57
political-science journal. Our paper amplifies and clarifies the seminal work in
Chan’s paper and opens wide avenues allowing a variety of useful extensions for it.
The organization of the remainder of this paper is as follows. Section 2
discusses the characteristics of the Karnaugh map and offers a mini tutorial on its
construction and its use in minimization. Section 3 presents a Boolean analysis, from
a Karnaugh-map perspective, of the causes of war termination as reported by Chan
[34]. The analysis is performed for both a Boolean outcome variable and its
complement, and for three cases, namely: (a) when the logical remainders (don’t
cares) are ignored (actually deliberately nullified for an arbitrarily-selected form or
literal of the outcome variable rather than for its complementary form or literal), (b)
when the don’t cares are assigned independent values so as to achieve a certain
objective, typically minimization, and (c) when there is a need for faithful
representation for the outcome via a partially-defined function whose asserted part
constitutes the definite (certain) causes of the phenomenon, while its don’t-care part
is a disjunction of its potential (uncertain) causes. Section 4 demonstrates several
novel extensions of the aforementioned analysis that make the most of the Karnaugh
map by utilizing its don’t-care entries in studying the possibility of making the
outcome function positive or negative in, or independent of, some of its arguments,
assigning an important metric for each of these arguments, rendering the outcome
function partially or totally symmetric in these arguments, finding a threshold
formulation of the outcome function, or making it fit a pre-specified hypothesis.
Section 5 concludes the paper.
Fig. (1). Six distinguished threads or variants of General Boolean Analysis
Ali Muhammad Rushdi and Raid Mohammad Badawi
58
2. On the Karnaugh-map
A classical or conventional Karnaugh map of n variables is in essence, a truth-table
representation of a switching or Boolean function
22
:n
f B B
(1)
where
2
B
is the bivalent Boolean carrier {0,1}. However, the Karnaugh map differs
from an ordinary truth table in two major aspects:
1- The Karnaugh map is a two-dimensional display rather than a one-
dimensional arrangement. Hence, the Karnaugh map makes the most of the human
capability to view two independent dimensions simultaneously.
2- The combinations of the input variables (configurations) are ordered
according to the Gray code (reflected binary code) [26, 37]. This code differs from
(and has a spatial advantage over) the ordinary binary code shown in Fig. 2(a). For
comparison, we display in Fig. 2(b) a 3-bit Gray-code representation for the integers
from 0 to 7. Note that the leftmost bit
1
X
is reflected w.r.t. the continuous lines, the
middle bit
2
X
is reflected w.r.t. the dashed lines, and the rightmost bit
3
X
is
reflected w.r.t. the dotted lines. The Gray ordering gives the map the visual
advantage that neighboring cells are represented by adjacent input variable
combinations, i.e., by binary numbers that differ in only one bit position [26, 29,
37]. The top and bottom rows of the map are viewed as contiguous. Similarly, the
leftmost and rightmost columns are considered adjacent. In that sense, a Karnaugh
map can be imagined to exist on the three-dimensional surface of a torus, albeit
conveniently drawn on a two-dimensional plane [29].
Thanks to these two differences between a Karnaugh map and an ordinary
truth table, the Karnaugh map is a more concise and time- and space-saving tool. It
also provides pictorial insight to many switching-theoretic concepts such as duality,
prime implicants, prime implicates, complementation, differentiation, and utilization
of don’t-care values. Rushdi [37] discusses advantages of the Karnaugh-map over
other graphical representations of sets, events and propositions, including the
ubiquitous Venn diagram, Euler diagram, Carroll diagram, and the Marquand-Veitch
chart. Wheeler [65] laments that the Karnaugh map had not been much explored by
teachers (of mathematics), a regrettable fact that seems to be still valid nowadays. In
fact, a recent article by Mahoney and Vanderpoel [15] explored, admirably how
visual tools (which they called set diagrams) can facilitate the application of
qualitative methods and improve the presentation of qualitative findings. However,
they mainly relied on Venn and Euler diagrams rather than Karnaugh maps.
The straightforward and mechanical conversion of a truth table to a Karnaugh
map is discussed in numerous texts on logic design [29, 38, 84]. As an aid to the
reader, we write in the right upper corner of each cell of the Karnaugh map in Fig.
3(a) the numerical value of the 4-bit binary digit.
Karnaugh-Map Utilization in Boolean Analysis …
59
2
( ) 8 4 2ABCD A B C D
,
(2)
which represents the corresponding line number in the original truth table
(numbered from 0 to 15).
Fig. 2 (a). The 3-bit integers
1 2 3
()X X X
, which represent 0 to 7 in the ordinary binary code. Here
1 2 3 1 2 3
( ) 4 2X X X X X X
Fig. 2 (b). The 3-bit integers
1 2 3
()X X X
representing 0 to 7 in Gray code. The continuous lines is a
mirror for the variable
1
X
, the dashed lines is a mirror for
2
X
, while the dotted lines is
a mirror for
3
X
Ali Muhammad Rushdi and Raid Mohammad Badawi
60
Fig. 2 (c) A one-dimensional Karnaugh map utilizing the Gray ordering in Fig. 1b. Each map cell
differs form an adjacent cell in a single bit. The leftmost and the rightmost cells are
adjacent
The most famous application of the Karnaugh map is its use in minimization,
i.e., obtaining a formula for the map function in sum-of-products (disjunction-of-
conjunctions) form that has a minimum number of prime implicants as the primary
objective and that has a minimum number of literals as the secondary objective.
Dually, the map can also be used for expressing the map function in a product-of-
sums (conjunction of disjunctions) form that has a minimum number of prime
implicates as the primary objective, and again a minimum number of literals as the
secondary objective [29].
The primal or sum-of-products map application heavily depends on the map
capability to visually identify prime implicants. A prime implicant
P
of a given
function
f
is:
(a) An implicant of
f
(i.e.
,P f P f
or
f g P
),
(b) No other term subsumed by
P
(whose set of literals is a subset of the set
of literals of
P
) is an implicant of
f
.
A prime implicant is represented by a (continuous or split) rectangular loop
consisting of
2k
adjacent asserted cells (called 1-enterd cells or 1-cells) or don’t-
care cells (called d-cells), where
k
is a natural number chosen as large as possible.
Generally, the Karnaugh map is used to represent an incompletely-specified
(partially-defined) function. To express the map function one (a) must use prime-
implicant loops to cover each 1-cell (cell entered with 1) at least once, (b) can cover
each of d-cells (cells entered with don’t cares) once or more (independently of the
other d-cells), and (c) never cover any of the 0-cells (cells entered with 0). This
means that the coverage of 1-cells, d-cells, and 0-cells is obligatory, optional
(permissible) and prohibited, respectively. Once a 1-cell is covered, it is changed to
Karnaugh-Map Utilization in Boolean Analysis …
61
a d-cell, i.e., its further covering is still allowed, i.e., such coverage becomes only
permissible and ceases to be mandatory.
The Karnaugh map procedure for minimization consists of two major steps:
Step 1: Algorithmic covering of essential prime implicants:
Here, one checks the most isolated 1-cells first, so as to locate a 1-cell that
can be covered uniquely, i.e., that is covered by a single prime implicant
i
P
and no
more, called an essential prime implicant. The prime implicant is added to the
disjunctive expression for the function
f
and entries of all cells within its loop are
turned into don’t cares. This process is repeated until all uniquely-covered 1-cells, if
any, are exhausted.
Step 2: Heuristic (trial-and-error) covering of the remaining 1-cells:
Such coverage is achieved by as few and large prime implicants as possible.
These non-essential prime implicants are added to the disjunctive expression of
f
.
When
f
is totally covered, the final expression of
f
becomes an irredundant
disjunctive from which might be minimal or not.
Detailed expositions of the Karnaugh map and its minimization procedure are
available in classical textbooks on logic design such as those of Lee [84], Muroga
[29], Hill and Peterson [85], and Roth & Kinney [37]. Some advanced aspects of
map minimization are available in Rushdi [36, 69, 86], Rushdi & Ba-Rukab [87-
89], Rushdi & Ghaleb [90] and Rushdi & Alturki [91]. Moreover, the examples on
using the Karnaugh map in this paper will be explained in ample detail to allow
readers with minimal map knowledge (or even with no previous such knowledge) to
follow and understand them.
Rihoux & de Meur ([19, pp. 59-65]) devote a special section to explain why
Logical Remainders are useful and how they can be utilized in obtaining more
parsimonious minimal formulas. They raise the concern “Isn’t it altogether
audacious to make assumptions about non-observed cases?” and they point out that
this concern is among the critiques targeted at QCA. It might sound strange enough
that the above concern seems to have never been raised in electrical engineering
circles. Electrical engineers used to obtain parsimonious minimal formulas that fill
up the Boolean space well beyond the specified configurations. Their reasoning is
that unspecified configurations are guaranteed to never happen, so though their
formulas specify or assign outcomes for such configurations, practically such
assigned outcomes will never be used or required. Of course, the situation in social
and political sciences might be different, since an unobserved configuration might
not be guaranteed to never happen, as it might be observed later. However, a strong
rebuttal to the above concern can be stated as follows. If one refrains from utilizing
Logical Remainders or don’t-cares for minimization or for any other purpose, one is
still setting certain unknown values to zero, and hence is still making unwarranted
Ali Muhammad Rushdi and Raid Mohammad Badawi
62
or unjustified assumptions about non-observed data. There is no reason to prefer
these assumptions to the ones leading to minimization. In fact, the choice of
nullifying
k
logical remainders can be thought to be equally likely to all other
(2 1)
k
possible choices for the logical remainders, and hence it can be equally
accepted or rejected. We consider all these choices as equally interesting, and
demonstrate how to utilize some of them to achieve certain useful purposes not
excluded to minimization. We will also discuss herein how to obtain a faithful
algebraic reproduction of the observed data that does not arbitrarily assign values to
unobserved data. This faithful representation is in terms of partially-defined
(incompletely-specified) functions. Again, the Karnaugh map will come to our aid in
making such a representation. The spirit of such an incomplete representation is that
“When only partial observations are available, no method can provide definite
answers,” [27]. One should treat the observed data as temporary or tentative, and
should seek more information to update one’s findings. This process should continue
as long as the observed data does not seem to be final. A stopping criterion need to
be found so as to terminate the updating process whenever the data is deemed final
in the sense that further observations are impossible or too costly.
3. Karnaugh-map Analysis of War Termination
We base our analysis in this section on Table II of Chan [34], which is a Boolean
truth table for short wars. This table summarizes observations of important violent
conflicts in modern history all over the world. For simplicity, we designate the
outcome (dependent variable) of short wars by the single literal
S
, and use the
variables
,,A B C
and
D
(as in Chan [34]) to denote Chan’s hypothesized
dichotomous determinants of war duration, namely:
A
‘The war has a ‘big and fast start’ rather than a ‘small and slow one’ (Big
start),
A
Small start,
B
‘The war involves major-minor dyads rather than more evenly-matched
contestants’ (Major-minor),
B
‘Matched contestants’
C
‘The war is multilateral rather than bilateral’ (Multilateral),
C
‘Bilateral’
D
‘The war involves moderately repressive and exclusionary regimes rather
than goes without such involvement’ (Semi-rep.),
D
‘Not Semi-rep.’.
We reproduce the aforementioned Table II of Chan [34] in the form of the
Karnaugh map of Fig. 3(a). The interested reader might wish to consult typical texts
on digital design [29, 38, 84] about the mechanical procedure of conversion from a
truth table to a Karnaugh map. We remind the reader that the integer in the right
Karnaugh-Map Utilization in Boolean Analysis …
63
upper corner of each map cell corresponds to the line number in Table II of Chan
[34], where the line numbers range from 0 to 15.
As stated earlier in the introduction, we analyze this problem for three cases,
namely: (a) when the logical remainders (don’t cares) are ignored (actually
deliberately nullified for an arbitrarily-selected form or literal of the outcome
variable rather than for its complementary form or literal), (b) when the don’t cares
are assigned independent values so as to achieve a certain objective, typically
minimization, and (c) when there is a need for faithful representation for the
outcome via a partially-defined function whose asserted part constitutes the definite
(certain). We devote separate subsections to the aforementioned three cases.
3.1 The Case of Ignoring the Logical Remainders
A first possible expression for the proposition of short wars (
S
) can be obtained by
identifying the six combinations in which the outcome code in Table I of Chan [34]
is asserted namely:
1 ABCD ABCD ABCDS ABC
D ABCD ABCD
.
(3)
Our Equation (3) is identical to Equation (1) in Chan [34] with the single
exception that we use a bar to designate a complemented literal, while such a literal
is depicted therein by a lower–case letter. Figure 3(a) demonstrates Equation (3)
simply as coverage of single cells in the corresponding Karnaugh map. This is called
a minterm expression of (the asserted part of)
S
in digital design texts [29], or a
listing of (asserted) discriminates of
S
in Boolean reasoning texts [26]. Equation
(3) is simplified algebraically in Chan [34] via
Table (2). Interpretation of Results via Verbal Statements.
Class of
Results
Equation
N
o.
Verbal Statement
Nullifyi
ng don't-
cares for
S
1 ABCD ABCD ABCD ABCDS ABCD ABCD
1 ABCD ABCD ABCD ABCDS ABCD ABCD
1 ABCD ABCD ABCD ABCDS ABCD ABCD
Same as (1) of Chan (2003)
(3)
Short War = Small Start AND Matched
contestants AND Multilateral AND Semi-rep.
OR Big Start AND Matched
contestants AND Bilateral AND Semi-rep.
OR Big Start AND Matched
contestants AND Multilateral AND Semi-rep.
OR Big Start AND Major-minor
AND Bilateral AND No Semi-rep.
OR Big Start AND Major-minor
AND Bilateral AND Semi-rep.
OR Big Start AND Major-minor
Ali Muhammad Rushdi and Raid Mohammad Badawi
64
Class of
Results
Equation
N
o.
Verbal Statement
AND Multilateral AND Semi-rep.
2
S ABC AD BCD
Same as (2) of Chan (2003)
(11)
Short War = Big Start AND Major-minor
AND Bilateral
OR Big Start AND Semi-rep.
OR Matched contestants AND
Multilateral AND Semi-rep.
4
S AB B
DA
C CD
Same as (3) of Chan (2003)
(12)
No Short War = Small Start AND Major-
minor
OR Matched contestants AND
No Semi-rep.
OR Small Start AND Bilateral
OR Multilateral AND No
Semi-rep.
Utilizing
don't
cares for
minimiz
ation
3 S AB AD BC
(13)
Short War = Big Start AND Major-minor
OR Big Start AND Semi-rep.
OR Matched contestants AND
Multilateral
5
S AB B
DA
C
(14)
No Short War = Small Start AND Major-
minor
OR Matched contestants AND
No Semi-rep.
OR Small Start AND Bilateral
Faithful
Represe
ntation
(S AD ABC BCD d CD BCD A
(S AD ABC BCD d CD BCD A
B
)D
(17)
Short War = Big Start AND Semi-rep.
OR Big Start AND Major-minor
AND Bilateral
OR Matched contestants AND
Multilateral AND Semi-rep.
OR d(Multilateral AND No Semi-
rep. OR Major-minor AND Bilateral AND
Semi-rep. OR Small Start
AND Matched contestants AND No
Semi-rep.)
No Short War = Big Start AND Matched
Karnaugh-Map Utilization in Boolean Analysis …
65
Class of
Results
Equation
N
o.
Verbal Statement
S AB
C
D ABC
DA
.
B
CD
(ABCD d CD A
)C
(18)
contestants AND Bilateral AND No Semi-rep.
OR Small Start AND Major-
minor AND Bilateral AND No Semi-rep.
OR Small Start AND Matched
contestants AND Bilateral AND Semi-rep.
OR Small Start AND Major-
minor AND Multilateral AND Semi-rep.
OR d( Multilateral AND No
Semi-rep. OR Small Start AND Bilateral.
reasoning texts [26]. Equation (3) is simplified algebraically in Chan [34] via
painstaking and elaborate efforts to find ways for combining minterms. For
example, the fact that the ‘AND’ operation is distributive over the ‘OR’ operation
namely:
( ) ( ) ( )X Y Z X Y X Z
,
(4)
and that ‘OR’ is idempotent, namely:
X X X
,
(5)
can be used to write
()ABCD ABCD ABC D D ABC
,
(6)
( )( ) ,
ABCD ABCD ABCD ABCD
A B B C C D AD
(7)
( ) .ABCD ABCD A A BCD BCD
(8)
For clarity, we underlined the minterms that are used repeatedly since duplicate
instances of them are available due to the idempotency of OR in (5), namely
ABCD ABCD ABCD
(9)
ABCD ABCD ABCD
.
(10)
The result is Equation (2) in Chan (2003), namely
2
S ABC AD BCD
.
(11)
Ali Muhammad Rushdi and Raid Mohammad Badawi
66
3.2 The Case of Assigning Independent Values to the Logical Remainders
The great effort exerted in Chan [34] to go from (3) to (11) via unguided
verbal arguments (and not even via Equations (6)–(8)) has surprisingly reached the
correct result (11). Such an effort can be substantially reduced via the pictorial
insight offered by the Karnaugh map in Fig. 3(b) in which the distributive law is
manifest in combining
2 ( 1)
mm
adjacent cells, and the idempotency law is
exhibited in permitting the resulting loops to be overlapping, i.e., allowing certain
cells to be covered more than once. Now, we come to the most important issue,
which is well known in electrical-engineering circles and seems to be somewhat
unknown, occasionally ignored, or deliberately avoided in some other fields. This
issue pertains to the lines left blank in Table II of Chan [34] and in Figs. 1(a) and
1(b) because they correspond to cases that were never observed. These cases
actually never happened so far, and presumably expected to never happen if
observations are deemed final. The function
, , ,S A B C D
should be identified as an
incompletely specified switching function [29], or equivalently as a partially defined
two-valued Boolean function [27]. The blank lines or cells are now assigned the
values of don’t-cares (
d
), which can be either 0 or 1, but nothing else. These values
are called logical remainders in the QCA literature [19].
Figure 1(c) shows the blank cells entered with don’t-care values
i
d
,
16i
. Contrary to the common practice of assigning the same symbol
''d
to
every don’t-care cell (albeit with an implicit assumption that the
d
values in
different cells are different), we have assigned explicitly different don’t-care values
to different cells to stress that these values are independent, and to deliberately avoid
a potential misconception that they are necessarily the same. The solution obtained
via conventional map minimization techniques [29, 38] is:
3 S AB AD BC
,
(12)
and represents a true minimal representation of
S
, typically called a
‘minimal sum’ [29], or a most compact or parsimonious formula [66]. The
representation in (12) happens to be unique, since each of the three loops in Fig. 1(c)
is an essential prime implicant, due to the fact that it is the only prime implicant
covering a particular cell. The particular cells uniquely covered by essential prime
implicants are each distinguished in Fig. 3(c) with a star in it. Note that the minimal
solution (12) is obtained for the choice
214
0d d d
and
3 5 6 1,d d d
while
the solutions in Fig. 1(a) and Fig. 1(b) implicitly assume that all these
'ds
are
zeros, i.e. that all blank cells in Figs. 1(a) and 1(b) are entered by 0’s. This implicit
assumption in Chan [34] is made explicit when expressing the non-short war
proposition
()S
via Equation (3) in Chan [34], namely
Karnaugh-Map Utilization in Boolean Analysis …
67
4
S AB B
DA
C CD
.
(13)
Equation (13) can be visually obtained via Fig. 4(a), which represents
4
S
as
the complement of
2
S
in Fig. 3(b) with all blank cells therein being assumed 0-
entered and hence with the corresponding cells in Fig. 4(a) being explicitly 1-
entered. The actual minimal solution in this case is shown in Fig. 4(b) (in which
S
is obtained by complementing the entries in Fig.3(c)) namely,
5
S AB BD AC
.
(14)
Again, the expression in (14) happens to be unique, since it consists of prime
implicants that are all essential, thanks to the uniquely-covered starred cells in Fig.
4(b). The only difference between (13) and (14) is the appearance of the extraneous
or the all-
d
implicant
CD
in (13), which makes (13) less parsimonious than (14).
This prime implicant is called absolutely eliminable [29], since it is possible (but
always unnecessary) to cover it. Hence, this prime implicant is definitely useless
[29, 92], as it is never needed in a minimal representation of its implied function.
In passing, we comment on the values of the logical remainders or don’t-
cares in Fig. 3(c). There are six don’t-cares
16
( )d to d
that could each be assigned
one of the values 0 or 1 independently. This means that there are
6
2
possibilities
for selecting the
i
d
values. These values are selected to be the same as
0, 1 6,
iid
(15)
Fig. (3). The truth table for short wars (Table 1 in Chan (2003)) redrawn as a Karnaugh
map with (a) expressing a minterm expansion, 1(b) expressing further
simplification, and (c) expressing true minimization involving don’t cares.
Ali Muhammad Rushdi and Raid Mohammad Badawi
68
in each of Fig. 3(a) and Fig. 3(b). With such an implicit choice, one is making
certain assumptions about unobserved data. The only reason to think of this choice
as the appropriate one is that it entails apparently no action, though, in fact, it
demands the particular action of nullifying specific d values. The alternative choices
made in Fig. 3(c) or Fig. 4(b) are equally likely (or equally objectionable to)
compared with the choice (15). Somehow, some people might think that the choice
of (supposedly) no action is more appealing or more acceptable than other choices.
An intriguing question is why such a (supposedly) no-action choice is equated to a
no-assumption one for a function
S
and not for its complement
S
. Note that one
can equally well apply the (supposedly) no-action choice to
S
rather than to
S
, and
obtain a possibly different result
3.3 The Case of Faithfully Representing the Boolean Function as a Partially
Defined One
A faithful representation of
S
is to express it as a partially-defined function [
35, 36, 39], namely,
()S g d h
,
(16)
where
g
and
h
are called the asserted and don’t-care parts of
S
. Equation
(16) means that
{ 1} { 1}gS
,
(16a)
{ 0} { 0}g h S
.
(16b)
Note that we keep silent about specifying the value of
S
when
{ 0, 1}gh
, i.e.
we think of this value as a don’t-care. The representation (16) has the pictorial
interpretation of Fig. 5. We have already obtained the minimal asserted part of the
function
S
displayed in Fig. 3(b) and given by Equation (11). The minimal don’t-
care part of
S
is obtained in Fig. 6 in which we are obliged to cover every d-cell at
least once and are allowed to cover any 1-cell, possibly repeatedly. The minimal
asserted part of
S
was not obtained in Fig. 4(a) since the d-cells were not replaced
by 0’s but were replaced by 1’s (in an unjustified bias for
S
against
S
). Figure 7
presents Karnaugh-map minimization for the asserted and don’t-care parts of
.S
Now, we obtain the following expressions for
S
and
:S
()S AD ABC BCD d CD BCD ABD
,
(17)
()S ABCD ABCD ABCD ABCD d CD AC
.
(18)
Karnaugh-Map Utilization in Boolean Analysis …
69
Fig. (4). The complementary function for the function in Fig. 2(a), (a) with the blank cells 1-entered,
and (b) with a true minimal coverage
Fig. (5). A Karnaugh-map-like structure expressing in terms of its asserted part and
don’t-care part . Here d means either 0 or 1 (but nothing else).
Fig. (6). The don’t-care part of the function in Fig. 1(c).
Ali Muhammad Rushdi and Raid Mohammad Badawi
70
Note that we have not subscripted the symbols
S
and
S
in (17) and (18), since
each is a faithful representation of its respective function, and since
S
is now indeed the
complement of
S
. This faithful representation defers the question of assigning values of
non-observed configurations, maybe until they are observed. The terms in the asserted
part represent definite or certain causes, while those in the don’t-care part represent
potential or uncertain causes. For example the term
AD
is a definite (certain) cause of
S
while the term
CD
is a potential (uncertain) cause for it.
We seem to have been too much involved in mathematical manipulations to
the extent that we might have forgotten our original problem. To remedy this
potential shortcoming, we use Table II to restate our results verbally. We express
each of the outcomes
S
(short war) and
S
(no short war) in terms of the four
determinants big start/small start, major-minor/matched contestants, multilateral/
bilateral, and finally semi-rep/no semi-rep. Table II interprets various results
obtained when (a) nullifying the don’t-cares of
S
while asserting those of
S
, (b)
utilizing the don’t cares for independent minimizations of
S
and
S
, and (c) using a
faithful representation for each of
S
and
S
.
Fig. (7). The asserted part (a) and the don’t-care part (b) for the complement of the
function in Fig. 1(c).
4. Novel Extensions for Boolean Analysis
The Availability of the Karnaugh-map representations of Fig. 3(c) and fig. 4(b)
allow several useful extensions of the Boolean Analysis discussed in Sec. 3. These
include (a) deciding whether the pertinent function
S
is independent of some of its
(supposed) arguments, (b) assigning an importance metric for each variable, (c)
investigating whether
S
is positive or negative in its arguments, (d) checking
Karnaugh-Map Utilization in Boolean Analysis …
71
whether
S
is partially symmetric in some of its arguments, and hence whether it is
totally symmetric in all of them, and (e) deciding if
S
is a threshold function, and if
yes, obtaining its threshold-function representation. The aforementioned extensions
are discussed in the following subsections.
4.1 Possible Outcome Independence of Certain Arguments
To investigate the possibility of independence of the pertinent outcome function
, , ,S A B C D
of each of its arguments, we construct the Boolean difference
(derivative) [29, 84, 86, 93].
/ ( , , , |1 ) ( , , , )|0 )
XX
S X S A B C D S A B C D
,
(19)
where
X
stands for
, , ,A B C
or
,D
and
represents the XOR operator defined
by the function table of Fig. 8 [94]. Note that, strictly speaking
11d d d
but
d
can be simply renamed as
d
, and
(...| )
X
Sj
stands for the subfunction or
cofactor of
S
obtained by restricting the argument
X
to the value j, with j = 0 or
1. For example
/SA
(1, , , )S B C D
(0, , , )S B C D
.
(20)
Note that
(...|1 )
x
S
can be written as
/SX
and
(...| 0 )
x
S
can be written as
/.SX
The function
S
is independent of
X
if
/SX
is identically equal to 0. Figure
9(a) illustrates a pictorial illustration for computing (19) via map rotation or folding
[78, 84, 86, 93]. The maps of the various Boolean derivatives
Fig. (8). Function table for the XOR operation. Note that
dd
is
d
and not 0 since the input
'ds
are independatnt and not necessarily identical.
Ali Muhammad Rushdi and Raid Mohammad Badawi
72
/ , / , /S A S B S C
and
/SD
are shown in Figs. 9(b)-9(e).
None of these derivatives can be made identically zero by any choice of the don’t
cares (in fact, each of the maps in Figs. 9(b)-9(e) has at least one cell that is entered
with 1). Hence, the function
S
is not vacuous in (independent of) any of its
arguments. The arguments (independent variables) for
S
chosen in Chan [34] are
definitely justified from a mathematical point of view, since the outcome function
cannot be made independent of any of them. However, the possibility of existence of
other arguments affecting
S
is not mathematically ruled out, but could be excluded
via the purely historical reasoning given by Chan [34].
4.2 Importance Metrics for the Arguments
There are many ways to devise importance metrics for the arguments
(independent variables) that measure the relative importance of each argument or its
relative power in influencing the outcome. We will adopt herein a simple metric
given by the Banzhaf index [95]
x
I
= The weight of the partial derivative of
S
w.r.t.
X
= The number of asserted minterms of
/SX
. (21)
Since the function
S
and hence the functions
( / )SX
, where
{ , , , }X A B C D
are incompletely specified, we do not obtain definite values for
the importance metric
x
I
. However, we can obtain an upper or a lower bound (i.e.,
a maximum or a minimum value) for
x
I
when all the don’t cares in the map for
( / )SX
are set to 1 and 0, respectively: These bounds are
37
A
I
,
(22a)
26
B
I
,
(22b)
16
C
I
,
(22c)
17
D
I
.
(22d)
We cannot use (22) to rank the variables
,,A B C
and
D
according to their
importance (influence on the outcome). There are
6
2
equally likely ways for
assigning the
'ds
in Fig. 9(a) and hence in Figs. 9(b)-9(e). For each of these
assignments, one can get specific values of
,,
A B C
III
, and
D
I
, and hence a
definite ranking of
, , ,A B C
and
D
. For example, with the d’s assignment in Fig.
Karnaugh-Map Utilization in Boolean Analysis …
73
3(c), one obtains
5, 3, 1
A B C D
I I I I
, which amounts to saying that
A
is
the most important (influential) input variable,
D
is the least important one, while
B
and
C
are of intermediate and same importance.
4.3. Whether the Function is Positive/Negative in Its Arguments
The function
S
is said to be positive in its argument
X
when
(1 )
X
S
(0 )
X
S
,
(23)
and if so, the dependence of
S
on the variable
X
can be expressed solely
in terms of the uncomplemented literal
X
. The function
S
is said to be negative
in
X
when
(0 )
X
S
(1 )
X
S
,
(24)
Fig. (9). (a) Map rotation or folding to obtain a map of the derivative w.r.t. A, (b-e) Maps for
Boolean differences (derivatives) of
S
w.r.t. each of its four arguments.
and if so, the dependence of
S
on the variable
S
can be made in terms of
the complemented literal
X
alone. For either case, the function
S
is a unate
function and the variable
X
is a mono-form or a pure variable. Otherwise, the
variable
X
is necessarily biform (appears both complemented and
uncomplemented) in any formula for
S
and function
S
is binate. It is clear from
Fig. 3(c) that
S
could be made positive in
,,AC
and
D
because we could obtain
3
SS
in (12) to involve only the uncomplemented literals
,,AC
and
D
,
respectively. Equation (12) involves a mixed-polarity in the variable
B
, where both
the complemented literal
B
and the uncomplemented one
B
appear. Note that we
can neither
Ali Muhammad Rushdi and Raid Mohammad Badawi
74
a. make
S
positive in
B
(because there is a 0 in cell 7 within the
B
domain, and a 1 in cell 3, the adjacent cell in the
B
domain).
b. make
S
negative in
B
(because there is a 0 in cell 8 within the
B
domain, and a 1 in cell 12, the adjacent cell in the
B
domain).
The failure to make
S
of a single-polarity in
B
means that
S
cannot
become a unate function (one of fixed-polarity in each of its arguments). Unateness
of the function is a sufficient but not necessary condition for it to be a threshold
function [84, 96]. This means that there is still some possibility but no guarantee to
express
S
as a threshold function, which we are going to explore in subsection 4.5.
4.4. Symmetry Considerations
The function
S
is partially symmetric in its arguments
A
and
B
if and
only if
//S AB S AB
,
(25)
where each of
/S AB
or
/S AB
is called a Boolean quotient or ratio [26],
a context [82], a subfunction [35, 36], or a restriction[25], namely
1, 0
/]
AB
S AB S
,
(26)
0, 1
/]
AB
S AB S
.
(27)
Figure 10(a) shows that each of
/S AB
and
/S AB
is represented by one
quarter of the original Karnaugh map in Fig. 3(c). The cell-wise comparison
between these two quarter maps indicates the impossibility of making them equal
since the ‘1’ in cell 11 is not equal to the ‘0’ in cell 7. Hence,
S
cannot be made
partially symmetric in
A
and
B
. However,
S
can be made partially symmetric in
B
and
C
by selecting
26
1dd
and
30d
. Likewise,
S
can be made
partially symmetric in
B
and
D
by choosing
45
1dd
. Similarly,
S
can be
made symmetric in
C
and
D
by making
24
dd
,
30d
, and
56
1dd
.
The net result is that
S
can be made symmetric in the three arguments
,,BC
and
D
by choosing
2 4 5 6 1d d d d
and
30d
. Figure 11 demonstrates a
unique minimal coverage of
S
under these conditions as
Karnaugh-Map Utilization in Boolean Analysis …
75
( ) ( )S A B C D BCD BCD BCD
,
(28)
The above expression (28) for
S
is symmetric in
,,BC
and
D
in the sense that
any two of these three variables can be interchanged without affecting the function.
Fig. (10). (a) Comparisson of
/S AB
and
/S AB
showing that they cannot be made equal.
(b) Comparison of
/S BC
and
/S BC
showing that they can be made equal
4.5. Possibility of Expressing the Function as a Threshold One
The function
S
can be made a threshold function if we find weights
A
W
,
B
W
,
,
C
W
and
D
W
together with a threshold
T
such that [29, 91, 95, 96]
{ 1} { }
A B C D
S W A W B W C W D T
,
(29)
or equivalently
{ 0} { }
A B C D
S W A W B W C W D T
,
(30)
where the plus sign
()
holds its converntional meaning of arithmatic
addition. Figure 11 gives detailed inequalities needed to achive (29) or (30). The
inequalities in cells (7) and (3) can be combined to give
B C D C D
W W W T W W
,
(31)
Ali Muhammad Rushdi and Raid Mohammad Badawi
76
or
0
B
W
,
(32)
while those in cells 8 and 12 yield
A A B
W T W W
,
(33)
0
B
W
,
(34)
Note that (32) and (34) are in obvious contradiction. This means that
S
cannot be made threshold. However, many of its subfucnctions or contexts are
threshold. For example, the subfunction
/SB
is preresented by the middle two
columns in Fig. 12 and can be given by the non-unique threshold representation
{ / 1} {4 2}
A C D
S B W W W
,
(35)
and its sister subfunction
/SB
is represented by the first and last columns
in Fig. 12, and can be given by the non-unique threshold representation.
{ / 1} {4 2}
A B C
S B W W W
.
(36)
5. Discussion
The Karnaugh map can be used to successfully verify or recover many results
obtained earlier by available software packages. For example, the four results
displayed in Figs. 3-2 to 3-5 of Rihoux & de Meur [19] as Carroll diagrams (or
Venn diagrams, as they are called therein) have simple and insightful Karnaugh map
representations that are easy to obtain manually in a very short time. We used a
running example dealing with the problem of war termination to demonstrate the
utility of the Karnaugh map in Boolean analysis. Our methods are easily applicable
in many other QCA problems such as those in Berg-Schlosser, et al. [97], Breuer
[98], Delreux [99], Ishida, et al. [100], Marx & Duşa [101], Ragin [102], Valtonen,
et al. [103] and Zeng [104].
The BA implemented herein via the Karnaugh map corresponds to the
csQCA variant, in which the Karnaugh map represents a single effect (map output)
in terms of several causes (map variables, presumably independent). The initial
work of Flament [1, 2] and the subsequent methodology of Coincidence Analysis
[30-33] might similarly be achieved by a Karnaugh map representing a function
relating all pertinent (and maybe non-pertinent) variables [105]. The map variables
initially stand on equal footing, as the researcher cannot distinguish some of them as
causes and others as effects. It is the job of the analysis to make a distinction
Karnaugh-Map Utilization in Boolean Analysis …
77
between effects and causes as well as express the effects in terms of causes. An
ambitious sequel to the current work is to make a detailed comparison of the BA
variants in Fig. 1 from a Karnaugh-map perspective. A possible outcome is to obtain
a unified BA analysis that encompasses all the variants in one common setting. A
relatively simpler task to follow the current work, is to extend this work to the
largest problems in social and political sciences that require Boolean Analysis. As
stated earlier, such problems involve up to 10 independent variables and hence
cannot be handled by the conventional Karnaugh map. A useful plan is to use the
variable-entered Karnaugh map [36, 37, 67-80] to analyze some such problems for a
number of independent variables ranging from 7 to 10. There are several candidate
papers to utilize in this endeavor [106-121]. Insight provided by the map is
remarkably beneficial in achieving better understanding of the underlying problems.
6. Conclusions
This paper advocated more utilization of the Karnaugh map in the Boolean Analysis
used in social, political, economic, managerial and engineering sciences. The paper
made its point by a detailed exposition of a prominent problem in peace research,
namely that of the causes of war termination. As a bonus, the paper demonstrated
the Karnaugh map utility for achieving important purposes other than minimization.
The paper strived to review all existing concepts in ample details, while occasionally
contributing some novel concepts and methods. To help the readers pursue the topic
further in its original sources, we not only cited most recent papers, but we also
referenced older publications that we felt are of paramount importance, seminal
contribution or everlasting impact.
Acknowledgement
This work was funded by the Deanship of Scientific Research (DSR), King
Abdulaziz University, Jeddah. Therefore, the authors acknowledge, with thanks, the
DSR for their financial and technical support. The authors are also indebted to two
anonymous reviewers for their helpful comments.
Ali Muhammad Rushdi and Raid Mohammad Badawi
78
Fig. (11). The function made symmetric in B, C, and D
Fig. (12). Detailed inequalities needed to achieve (30) or (31).
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{arushdi@kau.edu.sa}
