UNIV. BEOGRAD. PuDL. ELEKTROTEHN. FAK.
Ser. Mat. Fiz. Nil 634-NiI 677 (1979), 156-163.
PARTIAL ABSORPTION
659.
FUNCTION
Jozo J, Dujmovic
ABSTRACT. The partial absorption function plays an important role in extended continuous logic and is very often applied during the complex criteria synthesis process.
In this paper the basic variants of the partial absorption function are defined and analyzed. A simple procedure for selecting parameters of the partial absorption function
is proposed. This procedure is based on the use of special tables, which are also presented
in the paper.
1. Let the unit interval I: = [0, 1] and the logic variables xE/, yEI be given.
In extended continuous logic [1] two basic elementary operations are defined:
quasi-conjunction Llc (whose parameter is the conjunction degree c), and quasidisjunction '\ld (whose parameter is the disjunction degree d). Using these elementary
operations in the way shown in Fig. 1, the partial absorption function ot: 12-+1
is defined having the following two variants:
DC-variant:
(x, y) f-+ (I - W2) [WI x '\ld (1-
CD-variant:
(x, y) f-+ (I - W\)[Wz x Llc(I - Wz)y] '\ld WI x,
WI' WzE(O,
I);
1/2<c:;;;'
)(
I;
WI)y] Llc W2 x,
1/2:;;;,d:;;;,1.
)(
y
z
y
DC - variant
z
CD- variant
Figure
1
In both variants, the main variable, x, controls the influence of the auxiliary
variable y on the output z. This influence can be partially absorbed (disabled) in
a well-controlled manner. In the extreme case when c=d= I the absorption is total,
i.e. for the DC-variant it follows that z=(x V y) 1\ x=x, and for the CD-variant
it follows that z=(xl\y)Vx=x.
Since four parameters (Wb W2, C, d) are used in the partial absorption function, it is of practical interest to investigate in what way these parameters influence
the properties of the partial absorption function. Furthermore, we need a proce156
Partial
absorption
157
function
dure for selecting the appropriate values of the parameters Wi> Wz, c and d, based
on the desired properties of the function. These problems are considered below.
2. Using weighted power means [2], both variants of the partial absorption
function may be uniformly expressed in the following way:
z = {(I - a)[bx't + (1 - b) y'l ]'2/'1 + ax'2 }1/'2= : IX(x, y),
where either '1<1,
'z~I,
or 'I~I,
'z<l,
a=WI'
b=Wz
for '1<1,
a=Wz'
b=W1 for 'I~I,
and
'z~I,
(CD-variant),
'z<l,
(DC-variant),
and, according to [1], the relations between parameters c, d and 'i> '2 are given as
.
follows:
1
c=2-3
1
dx
x'+y'
J 0J( ~
o
I/r
)
dy
in the DC-variant),
(,=rl in the CD-variant, and '='2
1
1
X' + Y' I/r dy
(
J 0J ~ )
o
d = - 1+ 3
dx
\
(r='2 in the CD-variant, and '='1 in the DC-variant).
Since x, y, and z represent the truth values of corresponding statements, the
cases y = 0 and y = 1 deserve special attention:
zo: = IX(x, 0)=x[(I-a)br2/'1
+o]l/r2, '1>0
Zl: = IX(x, 1) = {(I -~) [bx" + (1 - b)]'2/'t+ oxr2}1/r2.
In the case y=O it follows zo;;;;x, so that the resulting truth value z is obtained by
decreasing the initial truth value x by the "attenuation factor" ~- (expressed as
a percentage):
zo=x(1 +0.01 a-),
a-<o.
Therefore, the factor a- is defined as follows:
~-: = 100{[(I-a)b'2/'t
= 100 (01/r2- 1),
+0]I/r2 -I},
'1>0
rl ~O.
In the case y= I the increase in the resulting truth value z is not a linear function
of x. In this case it is convenient to introduce the "equivalent amplification factor"
a+, using the relation
1
J[O.OIa++(I-O.OI~+)x]dx=J
o
1
IX(x, I)dx.
0
158
J. J. Dujrnovic
Thus yields the definition
)
8+:
= 100[2!
o
A typical geometric
interpretation
IX(x, I)dx-I].
of factors
8- and 8+ is shown in Fig. 2.
z
x
o
Figure 2
3. In applications related to the synthesis of system evaluation criteria, the
evaluator is always able to specify the desired values of 8- and 8+. Therefore, it is
convenient to develop a procedure for determining the values of parameters W1>W2,
c, d (or a, b, '1> '2) directly from the values given for 8- and 8+. Obviously, the
simplest way to determine W1> W2, c, and d from the desired pair (8-,8+) is based
on the use of tables. The following four characteristic cases were considered:
AR-case: ,) = 1,
RA -case: ,) =,<1,
'2=,<1}
'2 = I
d= 1/2,
Ij2<c
~I
RR-case: ,) =,<1, '2=r>l}
c=dE(~,
1]
RR-case: ,) =r> I, '2 =,< 1
(, and f denote parameters of the weighted power means which correspond to the
conjunction degrees c and I-c respectively). The cases AR and RA have the restrictive
property 18- 1>8+, while the cases RR andRR are more general: 18- 1~8+. In the RA
case for ~~O it follows 8- = -100 (1- WI), If the conjunction is used as an operator
(, = - 00), then in the AR case it follows 8- = -100 (1- W 1),and for RR and RR case
8- = O. In all four cases, = - 00 yields 8+ = O. Although these four cases generally
have different properties, it was shown that the differences among partial absorption
functions having similar (8-, 8+)-pairs are not significant. Therefore, only the tables
for the AR-case are presented in this paper. The tables contain (8-, 8+)-pairs, where
8-<0 and 8+>0. From these tables it is possible to determine the weights W1>W2E
E{IO%, 20%,...,
90%} and the quasi-conjunction
~cE{C--,
C-, C-+, CA,
C+-, C+, C++, C} for the desired (8-, 8+)-pair.
159
Partial absorption function
c..-
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
-84.4 -74.2
79.2
70.4
--64.5
61.7
-55.0
53.0
-45.6
44.2
-36.4
35.4
-27.2
26.6
-18.1
17.8
-9.0
8.9
0.2
-77.9
68.7
-67.8
61.2
-58.6
53.7
-49.7
46.1
-41.1
38.6
-32.6
31.0
-24.3
23.3
-16.1
15.6
-8.0
7.9
0.3
-70.6
58.6
-61.0
52.3
-52.3
45.9
-44.2
39.5
-36.4
33.1
-28.8
26.6
-21.4
20.1
-14.2
13.5
-7.0
6.8
0.4
-62.6
48.9
-53.6
43.7
-45.8
38.4
-38.5
33.1
-31.6
27.8
-24.9
22.4
-18.5
17.0
-12.2
11.4
-6.0
5.8
0.5
-53.8
39.6
-45.8
35.4
-38.9
31.2
-32.6
27.0
-26.6
22.7
-21.0
18.3
-15.5
13.9
-10.2
9.4
-5.1
4.8
0.6
-44.3
30.7
-37.5
27.6
-31.7
24.3
-26.4
21.1
-21.6
-16.9
-12.5
17.7
14.4
11.0
-8.2
7.4
-4.0
3.8
0.7
-34.1
22.3
-28.7
20.0
-24.2
-20.1
17.7
15.4
-16.4
13.0
-12.8
10.6
-9.4
8.1
-6.2
5.5
-3.0
2.8
0.8
-23.4
14.4
-19.6
-16.4
-13.6
12.9
11.5
10.0
-11.0
8.5
-8.6
-6.3
-4.1
-2.0
0.9
-12.0
6.9
-10.0
6.2
-8.3
5.5
-6.9
4.8
-5.6
4.1
-4.3
0.1
0.2
0.3
0.4
0.5
0.1
-86.5
76.6
-75.7
68.3
-65.5
60.0
-55.7
51.6
0.2
-82.1
64.4
-70.7
57.6
~.6 -51.1
50.7
43.8
0.3
-76.7
53.4
-65.1
47.9
-55.1
42.3
0.4
-70.2
43.3
-58.6
39.0
0.5
-62.5
34.2
0.6
WI
W2
0.1
6.9
1.9
5.3
3.6
3.4
-3.2
2.6
-2.1
1.8
-1.0
0.9
0.6
0.7
0.8
0.9
-46.1
43.2
.,.-36.7 -27.3
34.7
26.2
-18.1
17.6
-9.0
8.9
-42.0
36.8
-33.2
29.7
-24.6
22.5
-16.3
15.2
-8.1
7.7
-46.1
36.7
-37.6
30.9
-29.6
25.1
-21.8
19.1
-14.3
13.0
-7.1
6.7
-49.1
34.6
-40.7
30.1
-33.0
25.5
-25.8
20.7
-18.9
15.9
-12.4
10.9
-6.1
5.6
-51.4
30.9
-42.6
27.5
-35.0
24.0
-28.2
20.4
-21.9
-16.0
-10.4
12.9
16.7
8.9
-5.1
4.6
-53.4
26.0
-43.2
23.5
-35.4
21.0
-28.9
18.4
-23.1
15.7
-17.8
12.9
-13.0
10.0
-8.4
6.9
-4.1
3.6
0.7
-42.8
18.5
-34.0
16.8
-27.6
15.0
-22.3
13.2
-17.7
11.3
-13.6
9.4
-9.8
7.3
-6.3
5.1
-3.1
0.8
-30.4
11.7
-23.8
10.7
-19.1
9.6
-15.3
8.5
-12.1
7.3
-9.2
6.0
-6.6
4.7
-4.3
3.3
-2.1
1.8
0.9
-16.2
5.6
-12.5
5.1
-9.9
4.6
-7.9
4.1
-6.2
3.5
-4.7
2.9
-3.4
2.3
-2.2
1.6
-1.0
0.9
c-
WI
W2
2.7
J. J. Dujmovic
160
C-+
wI
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
W2
0.1
-87.8
72.2
-76.9
64.5
{)6.5
57.0
-56.4
49.2
-46.6
41.4
-37.0
33.5
-27.5
25.4
-18.2
17.2
-9.1
8.7
0.2
-85.1
58.2
-73.2
52.4
-62.5
46.5
-52.4
40.5
-42.9
34.3
-33.7
27.9
-24.9
21.4
-16.4
14.6
-8.1
7.6
0.3
-81.6
46.7
{)8.8
42.3
-57.9
37.7
-48.0
33.0
-38.9
28.2
-30.3
23.1
-22.2
17.9
-14.5
12.3
-7.1
6.5
0.4
-77.1
37.1
{)3.6
33.7
-52.7
30.2
-43.1
26.6
-34.6
22.8
-26.7
18.8
-19.4
14.7
-12.6
10.2
{).1
5.4
0.5
-71.3
28.8
-57.4
26.3
-46.7
23.7
-37.7
20.9
-29.9
18.0
-22.9
15.0
-16.5
11.7
-10.6
8.3
-5.2
4.4
0.6
{)3.8
21.6
-49.9
19.8
-39.8
17.9
-31.7
15.9
-24.9
13.7
-18.9
11.5
-13.5
9.1
-8.6
6.4
-4.1
3.5
0.7
-54.0
15.3
-40.8
14.0
-31.9
12.7
-25.0
11.3
-19.4
9.9
-14.6
8.3
-10.3
6.6
{).5
4.7
-3.1
2.6
0.8
-41.0
9.6
-29.8
8.9
-22.8
8.1
-17.6
7.2
-13.4
6.3
-10.0
5.3
-7.0
4.2
-4.4
3.0
-2.1
1.7
0.9
-23.6
4.6
-16.4
4.2
-12.2
3.8
-9.3
3.5
-7.0
3.0
-5.1
2.6
-3.6
2.1
-2.2
1.5
-1.1
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
-88.8
64.0
-77.9
57.8
----{)7.4 -57.2
45.0
51.5
-47.1
38.2
-37.3
31.3
-27.7
24.0
-18.3
16.5
-9.1
8.5
0.2
-87.2
49.4
-75.5
45.0
--64.4
40.4
-53.9
35.7
-44.0
30.6
-34.5
25.4
-25.3
19.8
-16.6
13.8
-8.1
7.3
0.3
-85.3
38.7
-72.5
35.5
----{)0.9 -50.3
32.1
28.5
-40.5
24.7
-31.4
20.6
-22.8
16.3
-14.8
11.5
-7.2
6.2
0.4
-82.8
30.3
----{)8.8
27.9
-56.7
25.4
-46.1
22.6
-36.6
19.7
-28.0
16.6
-20.1
13.2
-12.9
9.4
----{).2
5.1
0.5
-79.4
23.3
--64.2
21.6
-51.7
19.7
-41.3
17.7
-32.3
15.5
-24.3
13.1
-17.3
10.5
-11.0
7.6
-5.2
4.2
0.6
-74.8
17.4
-58.2
16.1
-45.7
14.8
-35.7
13.3
-27.4
11.7
-20.3
10.0
-14.3
8.0
-8.9
5.8
-4.2
3.3
0.7
----{)8.1 -50.4
11.4
12.2
-38.2
10.5
-29.0
9.5
-21.8
8.4
-16.0
7.1
-11.0
5.8
----{).8
4.2
-3.2
2.4
-39.6
7.2
-28.7
6.6
-21.2
6.0
-15.6
5.3
-11.2
4.6
-7.6
-4.6
-2.1
1.6
-24.0
3.4
-16.4
3.1
-11.7
2.9
-8.3
-5.9
2.2
CA
WI
W2
-57.4
0.8
7.7
-38.9
0.9
3.6
2.5
3.7
2.7
-3.9
1.8
-2.4
1.3
-1.1
0.8
161
Partial absorption function
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
-89.4
52.0
-78.8
47.7
---68.3
43.1
-58.0
38.3
-47.8
33.2
-37.8
27.7
-28.0
21.8
-18.5
15.3
-9.1
8.2
0.2
-88.6
38.8
-77.4
35.9
---66.4
32.8
-55.7
29.5
-45.4
25.9
-35.4
21.9
-25.9
17.5
-16.9
12.6
-8.2
6.9
0.3
-87.7
29.9
-75.6
27.9
---64.0
25.6
-53.0
23.2
-42.6
20.5
-32.8
17.5
-23.7
14.2
-15.2
10.3
-7.3
5.8
0.4
-86.5
23.2
-73.5
--61.2
21.7
20.0
-49.8
18.2
-39.4
16.2
-29.8
14.0
-21.2
11.4
-13.4
8.4
---6.3
4.8
0.5
-85.0
17.7
-57.8
-70.8
16.6
15.4
-46.1
14.1
-35.7
12.6
-26.5
10.9
-18.5
9.0
-11.5
6.7
-5.3
3.9
0.6
-82.9
13.2
---67.2
12.4
-53.4
11.5
-41.4
10.6
-31.3
9.5
-22.7
8.3
-15.5
6.8
-9.5
5.1
-4.3
3.0
0.7
-79.9
9.2
---62.2
8.7
-47.4
8.1
-35.5
7.5
-26.0
6.7
-18.4
5.9
-12.2
4.9
-7.3
3.7
-3.3
2.2
0.8
-74.9
5.8
-54.4
5.5
-39.0
5.1
-27.7
4.7
-19.4
4.3
:-13.3
3.7
-8.6
3.1
-5.0
2.4
-2.2
1.4
0.9
---64.0
2.7
-40.1
2.6
-25.7
2.4
-16.8
2.3
-11.1
2.0
-7.3
1.8
-4.6
1.5
-2.6
1.2
-1.1
0.7
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
c+-
WI
W2
c+
WI
W2
0.1
-89.7
37.0
-79.4
34.7
---69.1
.32.1
-58.8
29.3
-48.6
26.1
-38.5
22.5
-28.5
18.3
:-18.7
13.4
-9.2
7.5
0.2
-89.3
26.9
-78.7
25.4
---68.1
23.7
-57.5
21.9
-47.0
19.7
-36.8
17.2
-26.9
14.3
-17.3
10.7
-8.3
6.2
0.3
-88.9
20.5
-77.9
19.5
-66.9
18.3
-55.9
16.9
-45.2
15.4
-34.9
13.6
-25.0
11.4
-15.8
8.7
-7.5
5.2
0.4
-88.4
15.8
-65.4
-76.9
14.2
15.0
-54.1
13.2
-43.1
12.0
-32.7
10.7
-23.0
9.0
-14.2
7.0
-6.6
4.2
0.5
-87.8
12.0
-75.7
11.5
-63.6
10.8
-51.8
10.1
-40.5
9.3
-30.0
8.3
-20.6
7.0
-12.4
5.5
-5.6
3.4
0.6
-87.0
8.9
-74.1
8.5
-61.3
-48.9
8.1
7.5
-37.3
6.9
-26.9
6.2
-17.9
5.3
-10.5
4.2
-4.6
2.6
0.7
-85.9
6.2
-71.9
6.0
-58.1
5.7
-45.1
5.3
-33.2
4.9
-23.0
4.4
-14.7
3.8
-8.3
3.0
-3.5
1.9
0.8
-84.2
3.9
-68.5
3.7
-53.3
3.6
-39.4
3.4
-27.4
3.1
-17.9
2.8
-10.9
2.4
-5.9
1.9
-2.4
1.2
0.9
-80.7
1.8
-61.8
-44.2
1.8
1.7
-29.4
1.6
-18.4
1.5
-10.9
1.3
-6.2
1.2
-3.1
0.9
-1.2
0.6
11 Pulllikacije
Elektr<>tebni<!kol
falrulteta
162
J. J. Du.imovic
c++
0.8
0.5
0.6
0.7
-59.5
17.1
-49.4
15.9
-39.3
14.4
-29.2
12.5
-19.2
9.9
-9.4
6.2
-69.3
13.0
-59.0
12.4
-48.8
11.6
-38.5
10.6
-28.3
9.4
-18.3
7.6
-8.7
4.9
-79.2
10.2
-68.8
9.9
-58.4
9.4
-48.0
8.9
-37.6
8.2
-27.3
7.3
-17.3
6.0
-8.0
4.0
-89.4
8.1'
-78.8
7.8
-68.3
7.6
-57.7
7.3
-47.1
6.9
-36.6
6.4
-26.2
5.7
-16.1
4.7
-7.2 .
3.2
0.5
-89.2
6.1
-78.4
6.0
-67.6
5.8
-56.8
5.5
-46.0
5.3
-35.3
4.9
-24.8
4.4
-14.8
3.7
-6.3
2.5
0.6
-88.9
4.5
-77.9
4.4
-66.8
4.3
-55.7
4.1
-44.7
3.9
--:-33.7
3.6
-23.0
3.3
-13.2
2.8
-5.3
1.9
0.7
-88.6
-77.2 -65.7 -54.3
-42.9 -31.6 -20.8 -11.3
3.2
3.1
3.0
2.9
2.7
2.6
2.3
2.0
-4.2
1.4
0.8
-88.1
2.0
-76.1
1.9
-64.2
1.9
-52.2
1.8
-40.3
1.7
-28.6
1.6
-17.7
1.5
-8.8
1.3
-3.0
0.9
0.9
-87.1
0.9
-74.2
0.9
-61.3
0.9
-48.4
0.9
-35.7
0.8
-23.4
0.8
-12.7
0.7
-5.4
0.6
-1.6
0.4
0.1
0.2
0.3
0.4
0.5
0.7
0.8
0.9
0.1
-60.0
-50.0
-40.0
-90.0
-80.0 -70.0
-30.0 -20.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-10.0
0.0
0.2
-40.0 -30.0 -20.0
-90.0 -80.0 -70.0 -60.0 -50.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-10.0
0.0
0.3
-20.0
-50.0 -40.0
-30.0
-90.0 -80.0
-70.0 -60.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-10.0
0.0
0.4
-50.0 -40.0
-90.0 -80.0 -70.0 -60.0
-30.0 -20.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-100.
0.0
0.5
-40.0 -30.0
-90.0 -80.0 -70.0 -60.0 -50.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-20.0
0.0
-10.0
0.0
0.6
-80.0 -70.0 -60.0
-50.0 -40.0
-90.0
-30.0 -20.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-10.0
0.0
0.7
-90.0 -80.0
-70.0 -60.0 -50.0
0.0
0.0
0.0
0.0
0.0
-40.0 -30.0 -20.0
0.0
0.0
0.0
-10.0
0.0
0.8
-90.0
0.0
-80.0 -70.0 -60.0
-20.0
-50.0
-40.0 -30.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-10.0
0.0
0.9
-20.0
-90.0
-80.0 -70.0
-60.0
-50.0 -40.0
-30.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-10.0
0.0
0.1
0.2
0.3
0.1
-89.9
19.6
-79.8
18.9
-69.6
18.1
0.2
-89.8
14.0
-79.5
13.5
0.3
-89.6
10.5
0.4
wI
.0.4
W2
C
WI
W2
0.6
0.9
Partial absorption function
163
Extended continuous logic, introduced in [1], represents a specific area connecting means and logic. Means are considered to be logic functions and vice-versa.
Consequently, the partial absorption function, as well as any other extended continuous logic function, also represents a mean. It is important to point out that
the partial
absorption
function
has found
many
applications
-
it is extensively
used in practically all complex criteria for system evaluation.
REFERENCES
1. J. J. DUJMOVIC:Extended continuous logic and the theory of complex criteria. These
Publications N.! 498-N.! 541 (1975), 197 - 216.
2. D. S. MmuNOYIC,P. M. VASIC:Sredine. Matemati~ka biblioteka,sv. 40, Beograd 1969.