Karnaugh maps: representations for configurations

Manipulating a large number of configurations is sometimes tedious. Plotting 8, 16 or 32 states takes a lot of place and is difficult to have an integrated view on how the various configural states are articulated.

A well-adapted data representation is a necessary step towards a better understanding of the underlying dynamics of psychological phenomena. For this purpose we use a technique of computer science that usually helps to derive logical functions from truth tables: the Karnaugh map  [Leussler HamLeussler Ham1979,Lagasse, Courvoisier RichardLagasse 1977].

Karnaugh maps originate from the field of logic and computer science. It is a technique used to determine logical equations from truth table; its purpose consists of minimization of prime implicants. It is vastly employed for designing digital circuit [Lagasse, Courvoisier RichardLagasse 1977].

Instead of writing a straight truth table, one disposes a matrix-like structure of binary values taken by each variable. On the rows and columns are placed the values of the independent variables and inside each cell is put the value of the dependent variable.

Here are the structures of two- three- and four-variable Karnaugh maps (figures 2.2, 2.3 and 2.4 respectively; inside the cells are written the binary combinations of variables).

Table 2.2: Structural Karnaugh map of two variables

	V2=0	V2=1
V1=0	00	01
V1=1	10	11

Table 2.3: Structural Karnaugh map of three variables

	V2=0 V3=0	V2=0 V3=1	V2=1 V3=1	V2=1 V3=0
V1=0	000	001	011	010
V1=1	100	101	111	110

Table 2.4: Structural Karnaugh map of four variables

	V3=0 V4=0	V3=0 V4=1	V3=1 V4=1	V3=1 V4=0
V1=0 V2=0	0000	0001	0011	0010
V1=0 V2=1	0100	0101	0111	0110
V1=1 V2=1	1100	1101	1111	1110
V1=1 V2=0	1000	1001	1011	1010

Using two variables is straightforward: it is just an ordinary contingency table. When using three and more variables, one has to put combinations of variables on rows and columns. For three-variable combinations one variable is placed on rows (one for each of its two binary values) and the two other variables are placed in columns. But here comes a little trick: on the Karnaugh map values do not follow the usual order 00, 01, 10 and 11; instead they follow the Gray code order, for which each combination differ from its neighbor by the change of only one value. Therefore it consists of 00, 01, 11 and 10. The same procedure is taken for 4 variables, except that on rows there are two variables and their 4 possible combinations.

Inside the Karnaugh map the sequence of configurations does not follow the usual progressive order; it is arranged so that any two adjacent configurations differ from each other by the change of only one variable. For example, neighboring configurations 100 (#4) and 000 (#0) differ only by the third variable. Moreover the disposition of the map makes the sides also adjacent (top with bottom, left with right); although "far apart" states 000 (#0) and 010 (#2), differ only on the second variable.

Inside each cell of the Karnaugh map are put the value of the "dependent" variable, the one for which a function is sought. On figure 2.1 there is a truth table for which a simple compact function must be found. The Karnaugh map representation on the right helps to discover it (finding the equation is a matter of groups of cells having a value of 1 by adjacent pairs, 4-tuple in line or in square, or even 8-tuple in a contiguous block, which makes half the Karnaugh map; logical equations are then formed by a collection of "AND" "OR" connectors using values of group cells. Although we do not go into details here, the boolean function of figure 2.1 is $X = V3\cdot \overline{V4} + V2\cdot V3 + V1\cdot V3$).

Figure 2.1: Example of a logical function described by a Karnaugh map

We did not intend to use Karnaugh map to find logical equations, although the kinetic logic approach [ThomasThomas1979,Thomas D'AryThomas D'Ary1990] derives equations from it. Instead we had this simple yet powerful idea: to twist the primary use of Karnaugh map - a graphical technique for finding logical equations - into a graphical display of data. We first used it to plot frequencies of psychological configurations and then displayed its dynamics.

Let's illustrate the technique with configurations composed of familiarity of situations (BFAM), emotionality (BEMOT) and stress (BSTRE) for our subject. The usual linear arrangement of frequencies yields a table such as table 2.5. Integrated results would not be the main characteristics of this representation.

Table 2.5: Frequency of BFAM, BEMOT, BSTRE configuration

Config (bool)	Config (decimal)	Frequency
000	0	6
001	1	5
010	2	0
011	3	9
100	4	190
101	5	23
110	6	7
111	7	20
missing	99	1
Total		261

Let's now put the configuration frequencies in a Karnaugh map, as in table 2.6. The result is more compact and clearer to read. We will see how a micro-macro perspective emerges from such representation.

Table 2.6: Structural Karnaugh map of BFAM, BEMOT, BSTRE

	BEMOT=0 BSTRE=0	BEMOT=0 BSTRE=1	BEMOT=1 BSTRE=1	BEMOT=1 BSTRE=0	Total
BFAM=0	6	5	9	0	20
BFAM=1	190	23	20	7	240
Total	196	28	29	7	260