Int. J. Appl. Math. Comput. Sci., 2002, Vol.12, No.3, 403–410
ON GRANULAR DERIVATIVES AND THE SOLUTION
OF A GRANULAR INITIAL VALUE PROBLEM
ILDAR BATYRSHIN∗
∗
Institute of Problems of Informatics, Academy of Sciences of Tatarstan
and Kazan State Technological University, K. Marx Str., 68, Kazan, 420015, Russia
e-mail: batyr@emntu.kcn.ru
Perceptions about function changes are represented by rules like ��
If X
is SMALL then Y
is QUICKLY INCREASING.��
The consequent part of a rule describes a granule of directions of the function change when X is increasing on the fuzzy
interval given in the antecedent part of the rule. Each rule defines a granular differential and a rule base defines a granular
derivative. A reconstruction of a fuzzy function given by the granular derivative and the initial value given by the rule is
similar to Euler��s piecewise linear solution of an initial value problem. The solution method is based on a granulation of the
directions of the function change, on an extension of the initial value in directions and on a propagation of fuzzy constraints
given in antecedent parts of rules on possible function values. The proposed method is illustrated with an example.
Keywords: fuzzy differential, fuzzy granule, initial value problem, cylindrical extension
1. Introduction
In spite of the great success of crisp mathematics in the de-
scription and modeling of quantitative processes, the nat-
ural language has often been used for these purposes till
now. The uncertainty in the understanding of described
processes, the complexity of processes and the absence
of resources for a detailed description are only some of
the reasons behind using such rough and qualitative tool
as the language, instead of exact and fine mathematical
methods developed during the last centuries. Besides, in
many real tasks it is sufficient to have a qualitative descrip-
tion of a system and a qualitative solution instead of some
crisp mathematical result. In such situations the method-
ology of computing with words tolerant for imprecision
to achieve tractability, robustness, a low solution cost and
better rapport with reality may be considered as an alter-
native or additional tool with respect to traditional mathe-
matical methods of modeling (Zadeh, 1997; 1999).
Computing with words is based on a translation of
propositions expressed in a natural language into propo-
sitions expressed as a generalized constraint, and a fuzzy
graph constraint is often used for these purposes (Zadeh,
1999). The most important step in the explicitation of
generalized constraints is a fuzzy information granula-
tion which involves a decomposition of the whole into
parts such that the resulting granules are clumps of physi-
cal or mental objects drawn together by indistinguishabil-
ity, similarity, proximity or functionality (Zadeh, 1997).
This approach was used in (Batyrshin and Panova, 2001),
where the new type of rules describing the shapes of
dependencies between variables was introduced and the
methods of representation of such rules by granular direc-
tions were discussed. The rules are often represented as
follows:
If X
is A
then Y
is B,
(1)
where X and Y are variables and A, B are constrain-
ing fuzzy relations. The following are examples of rules
discussed in (Batyrshin and Panova, 2001):
R1 :
If TEMPERATURE is LOW then DENSITY
is SLOWLY INCREASING,
(2)
R2 :
If TEMPERATURE is HIGH then DENSITY
is QUICKLY DECREASING.
(3)
The rules (2) and (3) are considered as linguis-
tic expressions of dependencies between variables Y =
DENSITY and X = TEMPERATURE, such that Y
is a
SLOWLY INCREASING function of X on the fuzzy
interval
LOW and Y is a
QUICKLY DECREASING func-
tion of X on the fuzzy interval
HIGH. In this paper the
set of rules such as (2) and (3) is translated into rule-based
derivatives.
Differential equations play an important role in math-
ematical modeling. But often the values of the variables
used in the problem considered are uncertain. Moreover,
the functional dependencies between variables may be un-
known. In the first case the model of the process may
I. Batyrshin
404
be based on fuzzy differential equations, i.e. on differ-
ential equations with fuzzy parameters (Ma
et al., 1999;
Nieto, 1999; Park and Han, 2000; Song and Wu, 2000;
Vorobiev and Seikkala, 2002). In the second case the
model of the process may be based on a qualitative de-
scription which uses the signs of derivatives instead of
the derivatives or, equivalently, the labels ��increasing��,
��steady�� and ��decreasing�� (De Kleer and Brawn 1984,
Forbus 1984, Kuipers 1984). If the first approach requires
a crisp description of quantitative dependencies between
variables, the second approach uses very poor informa-
tion about the dependencies. Fuzzy differentiation based
on the extension principle was considered by Dubois and
Prade (1982).
The rule-based approach to representation of deriva-
tives considered here occupies an intermediate position
between the two approaches considered above. The prob-
lem of the reconstruction of a function based on the set
of rules considered and on the initial value given by a
rule such as ��If X is
APPROXIMATELY 5 then Y is
APPROXIMATELY 10�� is considered here as a granular
initial-value problem. The method of solving the prob-
lem discussed in this paper may be considered as a gran-
ular generalization of Euler��s method of solving an initial
value problem for an ordinary differential equation.
In Section 2, we translate the consequent parts of
rules (2) and (3) into linguistic values of derivatives.
These values are also considered as evaluations of slopes
of the tangent line to the curve of the function. The meth-
ods of the fuzzy granulation of such slopes are discussed
and granular differentials defined by these slopes are con-
sidered. The solution of the initial-value problem based
on the examined type of rules is discussed in Section 3.
This procedure is based on the reconstruction of a func-
tion from rule to rule starting from an initial value similar
to Euler��s method. The procedures considered are illus-
trated with an example. In conclusions, we discuss possi-
ble applications and extensions of the proposed approach
to the modeling of complex processes.
2. Granular Differentials
The linguistic label
SLOWLY INCREASING in the conse-
quent part of rule (2) may be interpreted as a linguistic
evaluation of the speed of the change of the variable Y =
DENSITY when the variable X = TEMPERATURE
is increasing within the fuzzy interval
LOW. Since the
speed of the function change is related to the derivative
of the function, the consequent part of this rule may be
also considered as a linguistic evaluation of the derivative
dY/dX on this interval. In terms of derivatives the rules
(2) and (3) may be translated in the following form:
R1 :
If X
is LOW then dY/dX
is POSITIVE SMALL,
(4)
R2 :
If X
is HIGH then dY/dX
is NEGATIVE LARGE.
(5)
Since the value of the derivative is equal to the slope
of the tangent line to the curve of a function, the linguistic
labels in the consequent parts of rules may be considered
also as linguistic evaluations of this slope or parameter p
in the equation of the tangent line y = px + q. A gran-
ular direction of the function change defined by the tan-
gent will be represented by a fuzzy clump of directions.
From another point of view, the granule of directions de-
fines fuzzy sets of differential values dY corresponding
to given crisp values of increment ∆x as dY = P∆x,
where P is a granular slope value defined by a rule. We
will suppose that the range of crisp values of increment
∆x (or differential dx) is defined by the antecedent part
of the corresponding rule. As a result, the granular differ-
ential dY may be considered as a fuzzy function of the
crisp argument ∆x. For example, the rule (5) will define a
fuzzy differential as a fuzzy function dY = P∆x, where
P is a fuzzy set corresponding to the linguistic term
NEG-
ATIVE LARGE and ∆x takes values in the fuzzy interval
defined by the term
HIGH.
For explicitation of rules it is necessary to define lin-
guistic scales for linguistic variables used in the rules, to
define a granulation of possible slope values and to estab-
lish a correspondence between the grades of scales and
slope values.
The explicitation of consequent parts of rules can be
based on perceptions about the graphical representation of
dependencies between linguistic variables (Batyrshin and
Panova, 2001). Such perceptions may arise as a result of a
visual analysis of graphics representing the dependencies
between the variables, and may denote the directions of
the change of the variable Y with the change of the vari-
able X. In this case, instead of the granulation of slope
values, granulation of angles of the directions of func-
tion changes or granulation of arctangent of slopes may
be used.
Suppose that the domain of slope values is equal to
the interval [−10, 10], and seven granules of slopes are
defined by fuzzy sets with central modal values pi, i =
1,..., 7. The possible linguistic scales and centers of
membership functions corresponding to linguistic grades
of the scales are shown in Table 1. Each grade of the scale
represents some fuzzy granule of directions that is a fuzzy
clump of similar directions.
We consider two methods of construction of granu-
lar directions. The first method is called the
proportional
On granular derivatives and the solution of a granular initial value problem
405
Table 1. Linguistic scales of slope values.
li
Linguistic description of the speed Linguistic value of
pi
of the function change
the derivative (slope)
7
QUICKLY INCREASING
POSITIVE LARGE
9
6
INCREASING
POSITIVE MIDDLE 6
5
SLOWLY INCREASING
POSITIVE SMALL
3
4
CONSTANT
ZERO
0
3
SLOWLY DECREASING
NEGATIVE SMALL −3
2
DECREASING
NEGATIVE MIDDLE −6
1
QUICKLY DECREASING
NEGATIVE LARGE −9
extension in direction. Suppose that Pi is a fuzzy slope
value, e.g. a fuzzy set defined on the domain of slope val-
ues p. For each value ∆x > 0 from the domain of in-
crements Dom(∆x), the corresponding fuzzy set dYi of
differential values dy associated with the direction li is
defined by the extension principle of fuzzy logic from the
equation dY = Pi∆x as follows:
µ
prop
dYi
(dy) = µPi (p) ,
(6)
where p = dy/∆x. The corresponding fuzzy relation is
defined as follows:
µ
prop
Di
(∆x, dy) = µPi
( dy
∆x
)
.
(7)
If fuzzy sets are defined by generalized bell member-
ship functions (GBMF) (Jang
et al., 1997), then from (6)
we obtain the following definition of the granular differ-
ential:
µ
prop
dYi
(dy) =
1
1 + ∣∣p−pi
ai
∣
∣
2bi
,
(8)
where ai is the width of the fuzzy set on the level 0.5 and
bi is the steepness of the membership function. Examples
of fuzzy clumps of proportional extensions of directions
based on GBMF and trapezoidal membership functions
are shown in Figs. 1(a), (b). The parameters (a, b, c, d)
of the trapezoidal membership function (Jang
et al., 1997)
are defined by means of the central slope values pi as
follows: a = pi − w1, b = pi − w2, c = pi + w2, d =
pi + w1, where w1 > w2 > 0.
The corresponding fuzzy relations are considered as
granular differentials which define for a given value of in-
crement ∆x a fuzzy set of differential values dY . Such
a fuzzy relation may be considered as an extending fuzzy
linear function representing granular differential values.
A granular differential obtained by (6) will also be called
a
proportional differential. Since ∆x > 0, the fuzzy
set of differentials for increment ∆x = 0 is not de-
fined. Nevertheless, we can define fuzzy sets Di0 at the
point ∆x = 0 as singletons, such that Di0(dy)=1 for
dy = 0 and Di0(dy) = 0 for all other values of dy.
These fuzzy sets Di0 defined for ∆x = 0 will be called
starting sets for proportional extensions of the direction li.
The ��width�� of proportional differentials dY is an
extending value with the increasing of the increment value
∆x. If the extending ��width�� of the fuzzy differential dY
is not desirable, then we can use a
cylindrical extension in
direction (Zadeh, 1966; 1997) and, correspondingly, the
cylindrical differential:
µ
cyl
Di
(∆x, dy) = µdYi (dy) ,
(9)
where dy = p∆x (for all ∆x > 0) and dYi is a given
fuzzy set of differential values in the direction li. For ex-
ample, the cylindrical extension of generalized bell mem-
bership functions for each value ∆x > 0 will be defined
as
µ
cyl
dYi
(dy) =
1
1 + ∣∣dy−dyi
ai
∣
∣
2bi
,
(10)
where dyi = pi∆x. The fuzzy value of the cylindri-
cal differential will have a constant cross-section. Exam-
ples of cylindrical differentials constructed by means of
GBMF and trapezoidal membership functions are shown
in Figs. 1(c), (d).
For ∆x = 0 we define D
cyl
i0
by (9) with dyi =
0, which will be called a starting set for the cylindrical
extension of the direction li.
3. Solution of the Granular Initial-Value
Problem
The total set of rules with granular derivatives in the con-
sequent parts of rules may be considered as a granular de-
scription of the derivative dY/dX = F(X) of a function
Y piecewise defined on the domain of the variable X.
Each rule defines some piece of the derivative on the fuzzy
interval corresponding to the value of X in the antecedent
part of a rule. The use of linguistic values of X in the an-
tecedent parts of rules implies that the set of terms of the
linguistic variable X is defined (Zadeh, 1975). This set of
terms can include the labels
VERY SMALL, SMALL, MID-
DLE, LARGE, VERY LARGE, APPROXIMATELY N
, BE-
TWEEN N
AND M
, GREATER THAN N, etc., where
N and M are some real values or fuzzy numbers. The
meaning of these terms may be explicitated by the defini-
tion of the corresponding fuzzy sets defined on X.
Generally, for the same rule base there may exist sev-
eral different explicitations of linguistic values of X de-
pendent on some parameter or context. The role of such a
parameter or context may be played by another variable.
The explicitation of granular slopes may also depend on
the value of this parameter. In this case the rule base de-
scribes the parametric family of granular derivatives with
I. Batyrshin
406
(a)
(b)
(c)
(d)
Fig. 1. Proportional ((a) and (b)) and cylindrical ((c) and (d)) differentials in directions 3 (��
SLOWLY DECREASING��) and 4 (��
CON-
STANT��) based on generalized bell membership functions ((a) and (c)) and trapezoidal membership functions ((b) and (d)).
explicitation dependent on the value of this parameter. For
example, the rules (4) and (5) may describe the derivative
dDENSITY /dTEMPERATURE for different values
of the third parameter Z =
PRESSURE, but the explic-
itation of this derivative may be different and will depend
on the explicitation of linguistic values of X and the ex-
plicitation of slopes defined by the value of parameter Z.
Let us consider the way of solving the granular ordi-
nary differential equation
dY/dX = F(X)
(11)
satisfying the initial condition
��
If X
is X0
then Y
is Y0
��,
(12)
where X0 and Y0 are fuzzy sets defined on X and Y ,
respectively, and (11) is given by the rule base
Ri :
If X
is Ai
then dY/dX
is Pi, i = 1,...,m,
(13)
with piecewise description of the derivative of Y . Sup-
pose that all Ai��s in (13) are normal and convex fuzzy
sets defined on the domain Dom(X) of X, and the set of
intervals Ai, i = 1,...,m defines some fuzzy partition
of Dom(X), i.e., the following conditions are fulfilled:
supx(Aj �� Ak)(x) = s1, infx��X((��m
i=1Ai)(x)) = s2,
where s1 and s2 belong to [0, 1] such that s1 < 1 and
s2 > 0. Since the cores of the fuzzy intervals in a fuzzy
partition do not intersect, these fuzzy intervals may be lin-
early ordered in such a way that Aj < Ak iff xj < xk
for some points xj and xk from the cores of Aj and Ak,
respectively. We will suppose that this ordering coincides
with the numbering of rules such that Ai < Ai+1 for all
i = 1,...,m.
The problem of solving the granular differential
equation (11) with initial condition (12) will be called a
granular initial-value problem.
With no loss of generality we will suppose that the
intersection of the initial value X0 with the fuzzy interval
A1 from the first rule is a normal fuzzy set. The solution
of the granular initial-value problem based on cylindrical
On granular derivatives and the solution of a granular initial value problem
407
extension in directions defined by the slope values will
include the following steps:
1. Find the core [x11,x12] of the fuzzy set X0 �� A1,
k = 1.
2. Select a starting point x0 in [x11,x12], e.g., as fol-
lows: x0 = (x11 + x12)/2.
3. Construct a fuzzy set Y0 in x0.
4. Choose a fuzzy set Y0 as a starting fuzzy set Di0
for the direction li determined by the slope P1.
5. Construct a granular extension D1 in the direction
li based on the initial fuzzy set Di0. Set k = 2.
6. Select a starting point xk−1
in the interval
[xk1,xk2] maximizing the intersection of fuzzy sets
Ak−1 and Ak.
7. Cut the granular extension in the direction Dk−1
at the point xk−1. The result will give a fuzzy set
DY k−1(y) = Dk−1(y, xk−1).
8. Construct a granular extension Dk based on a fuzzy
set DY k−1 and on a slope value defined by Pk. Set
k = k + 1.
9. Repeat Steps 6–8 while k �� m.
10. Construct cylindrical extensions of constraints
Ak, k = 1,...,m along the Y axis, i.e.
CY (Ak)(x, y) = Ak(x).
11. Propagate the cylindrical extensions of constraints
Ak, k = 1,...,m on the corresponding granular
directions Dk.
12. Aggregate in overall fuzzy graph the constrained di-
rections obtained in Step 11.
As a result of the above procedure, a fuzzy relation
R on X �� Y which will give a solution YR(X) to the
granular initial-value problem will be constructed. The
calculation of the function value for a given fuzzy value
X∗ of the input variable X represented by a fuzzy set
A∗ can be performed as a result of the following steps:
13. Construct a cylindrical extension CY (A∗) of A∗
along the Y axis.
14. Calculate a granular solution YR(X∗) = CY (A∗) ��
R.
15. Find a projection B∗ = PY (YR(X∗)) on the Y
axis.
16. Find a linguistic retranslation of the fuzzy set
Y (X∗) = B∗.
17. Find a numerical solution y∗ = Defuz(Y (X∗)) as a
result of the defuzzification procedure.
The linguistic value of the function Y obtained as a
result of the retranslation of the fuzzy set Y (X∗) = B∗
may be considered as a reply to the query ��
What is the
value of Y
if X
is A∗
?��.
Let us discuss some steps of the procedure consid-
ered. If we use fuzzy intervals Ak with strict monotonic
membership functions from both the sides of the cores,
then in Step 6 each interval [xk1,xk2] will contain only
one point.
Steps 11 and 12 can be realized by several meth-
ods. A max-min aggregation of rules is based on the in-
tersection of each granular direction with the correspond-
ing cylindrical extension of the fuzzy constraint Ak in
Step 11 and the aggregation of results obtained for each
rule with the union operation in Step 12 as follows:
R =
m
⋃
k=1
(Dk �� CY (Ak)).
For this method Steps 11 and 12 are reduced to
R(x, y) = max
k=1,...,m
(min (Dk(x, y), Ak(x))) .
Another method is based on the weighting of gran-
ular directions by the corresponding membership values
of the cylindrical extensions of fuzzy constraints Ak in
Step 11 and on the averaging of results in Step 12:
R(x, y) =
m
��
k=1
(Dk(x, y)Ak(x))
m
��
k=1
Ak(x)
.
This method will be called a weighted-average ag-
gregation. It gives a smoother overall graph than the first
method and is illustrated in Fig. 2 with the example con-
sidered below. As it follows from the last formulas, for
both the methods it is not necessary to calculate cylindri-
cal extensions of constraints Ak.
The procedure of linguistic retranslation in Step 16
can be based on linguistic approximation procedures
(Zadeh, 1975) and will not be discussed here.
If it is necessary, a defuzzification procedure in
Step 17 can be applied. Different types of such procedures
are described in (Jang
et al., 1997).
I. Batyrshin
408
Fig. 2. Construction of the overall graph defined by the fuzzy
rule set from example: (a) The fuzzy constraints on the
variable X; (b)–(e) the propagation of constraints from
X on the corresponding granular directions defined by
fuzzy rules: (b) R1; (c) R2; (d) R3; (e) R4; (f) the
overall fuzzy graph is obtained as weighted-average ag-
gregation of the constrained directions.
4. Example
Consider the following rule base describing the depen-
dency between the variables Y and X:
R1 :
If X
is SMALL then Y
is SLOWLY INCREASING,
R2 :
If X
is MIDDLE then Y
is QUICKLY INCREASING,
R3 :
If X
is SLIGHTLY GREATER THAN MIDDLE
then Y
is CONSTANT,
R4 :
If X
is LARGE then Y
is QUICKLY DECREASING.
This rule base gives a context-insensitive initial data
set (IDS). For the explicitation of this IDS we should
define the corresponding fuzzy sets for the variable X
and granular directions for the variable Y . Moreover, we
should also define an initial fuzzy point (X0,Y0) that will
define the starting point for the process of the reconstruc-
tion of the function Y . Suppose that the initial point is
given by the rule
R5 :
If X
is APPROXIMATELY 0
then Y
is APPROXIMATELY 10,
with an appropriate definition of the fuzzy sets
APPROX-
IMATELY 0 and
APPROXIMATELY 10. Figure 2 illus-
trates the solution to the initial-value problem given by
this example based on some explicitation of the member-
ship functions used in the model. In all constructions gen-
eralized bell membership functions were used. In the rule
R5 the parameters ci are equal to 0 and 10 for GBMF
X0 and Y0, respectively. The point (x1,y1) = (0, 10) is
used as the starting point for construction of a granular di-
rection in the rule R1. The parameters of Y0 are used as
parameters of the cylindrical extension of granular direc-
tions. The weighted average method of aggregation was
used. For each rule the corresponding granular directions
weighted by fuzzy sets given in the antecedent parts of
rules are shown in Figs. 2(b)–(e). The overall fuzzy graph
is shown in Fig. 2(f).
The calculation of a reply to the query
Q :
What is the value of Y
if X
is VERY LARGE?
is based on Steps 13–17 described above and illustrated in
Fig. 3.
Fig. 3. Calculation of a reply to a query: (a) fuzzy graph de-
fined by the fuzzy rule base; (c) constraint on X in a
query: ��X is
VERY LARGE��; (b) result of intersection
of a fuzzy graph with cylindrical extension of a con-
straint on X; (d) projection of the result onto the Y
axis with possible retranslation ��Y is
MIDDLE�� or with
defuzzification result y = 16.38.
On granular derivatives and the solution of a granular initial value problem
409
The retranslation of the resulting fuzzy set Y (X∗)
on Y in the linguistic form �� Y
is MIDDLE�� was ob-
tained as a result of linguistic approximation. The real
value y from the fuzzy set Y (X∗) was obtained as the
mean of the modal values of Y (X∗).
The answer to the query
Q2 :
What is a maximal value of Y
?
is considered for a general case in (Zadeh, 1997). For our
rule base, as the maximal value of Y the projection of the
constrained granular direction corresponding to the rule
R3 (see Fig. 2(d)) on the Y axis can be used.
5. Conclusions
The methods of translating perceptions about function
changes into rule based derivatives, and the methods of
constructing granular derivatives and granular directions
of function changes have been discussed. The problem of
function reconstruction from a rule-based derivative and
the initial value given by the rule is called the granular
initial-value problem. The solution of this problem is
based on a sequential rule by rule reconstruction of the
function starting from a given initial value. The method
can be considered as a generalization of Euler��s method
of piecewise linear solution to the crisp initial-value prob-
lem for ordinary differential equations. The method uses
the basic procedures of computing with words introduced
by Zadeh (1966; 1997; 1999), such as information granu-
lation, extension principles, cylindrical extension along an
axis and along a direction, projections etc. Additionally,
new methods of aggregation of fuzzy rules were proposed.
When solving a granular initial-value problem, the
initial data set (IDS) given by linguistic expressions about
dependencies between variables is transformed into a ter-
minal data set (TDS) which gives a linguistic reply to a
query about the value of the function for a linguistically
given value of the independent variable. This procedure
of inference of the linguistic reply from propositions ex-
pressed in a natural language uses the steps of computing
with words such as the explicitation of IDS into an initial
constraint set (ICS) given by fuzzy relations, constraint
propagation from ICS to a derived constraint set (DCS)
which gives a fuzzy reply to the query, and a retranslation
of DCS into TDS as a result of linguistic approximation
(Zadeh, 1997).
This work may be considered as an initial step in the
modeling and solution of granular differential equations
based on the new types of rules. The approach consid-
ered can be extended to systems of differential equations,
higher-order differential equations and partial differential
equations. The proposed approach can be applied to the
modeling and control of complex technological processes
with several inputs and outputs when the description of
dependencies between input and output parameters may
be given in the form of rules considered in this paper. An
example of fuzzy expert system modeling such a process
with another type of rules describing similar dependen-
cies between variables was considered in (Batyrshin
et al.,
1994). It would be interesting to combine the proposed
approach with the methods developed in qualitative rea-
soning about processes and systems based on the use of
the signs of derivatives (De Kleer and Brawn, 1984; For-
bus, 1984; Kuipers, 1984).
The rules containing some specific features of the
shape of the output variable in consequent parts can be
used for granular shape analysis. The granular models
with the rules of type (2) and (3) can be also more suit-
able for representing the knowledge base invariant to the
change of parameters of some problem area or insensitive
to the context (Batyrshin and Fatkullina, 1995). For ex-
ample, such models may describe the parametric family
of the fuzzy graphs Y (X) when the value of some vari-
able Z is considered as a (perhaps hidden) parameter.
Acknowledgements
The author would like to thank A. Panova and M. Wa-
genknecht for useful discussions, and the referees for their
valuable comments. The research was partly supported by
the RFBR Grant 02–01–00092.
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