On Computational Aspects of the BK-Subproduct Inference
Mechanism
Martin Štěpnička and Balasubramaniam Jayaram
Abstract— The compositional rule of inference (CRI) is
widely used in approximate reasoning schemes using fuzzy sets.
In this work we discuss the suitability of the Bandler-Kohout
subproduct for an alternative inference mechanism from the
computational point of view.
I. I NTRODUCTION
Fuzzy systems are one of the best known applications of
fuzzy logic. These are usually based on a set of fuzzy rules.
Systems using fuzzy rules have been applied in a wide variety
of applications, viz., automatic control, decision making, risk
analysis, etc.
If X is a classical set, we denote the set of all fuzzy sets
on X by F(X). Given two non-empty classical sets X, Y ,
fuzzy rules are usually given in the following form:
IF x is Ai
THEN
y is Bi ,
(1)
where Ai , Bi are membership predicates represented by
fuzzy sets Ai ∈ F(X) and Bi ∈ F(Y ), respectively, for
i = 1, . . . , n. Given a fuzzy observation A0 ∈ F(X) a
corresponding output fuzzy set B 0 ∈ F(Y ) is deduced using
an inference mechanism.
A. Fuzzy Relational Inference
Fuzzy inference systems can be broadly classified as those
inference mechanisms that are fuzzy relation based, i.e., those
that interpret fuzzy rules (1) by a fuzzy relation R ∈ F(X ×
Y ) and the inference mechanism as an image of a fuzzy
set under a fuzzy relation, and those that are not. Similarity
Based Reasoning [1], Inverse Truth-functional Modification
[2] and the Perception-based Logical Deduction [3], [4] are
some representative examples of the latter type of inference
mechanisms which are well established in the literature.
In this work, we focus only on those inference mechanisms that are fuzzy relation based. However, it should
be mentioned that, under certain conditions, an equivalent
fuzzy relation based description of some of these inference
mechanisms can be given, see [5], [6].
There are two main approaches to the construction of a
fuzzy relation interpreting the whole fuzzy rule base (altern.
linguistic description), composed of the n fuzzy rules (1).
The following fuzzy relation Ř ∈ F(X × Y )
Ř(x, y) =
n
_
(Ai (x) ∗ Bi (y)) ,
(2)
i=1
where ∗ is a fuzzy conjunction, typically a t-norm [7].
It represents say Cartesian product approach, which was
motivated by Mamdani and Assilian [8].
Alternatively, to keep the conditional IF-THEN form of
the fuzzy rules (1), fuzzy relation R̂ ∈ F(X × Y ) given as
follows
n
^
(Ai (x) → Bi(y)) ,
(3)
R̂(x, y) =
i=1
can be chosen. It deals with a fuzzy implication [9] →
which is a mathematically correct extension of a classical
implication, basically a residual implication derived from a
left continuous t-norm ∗.
B. Compositional Rule of Inference
The earliest inference scheme, proposed by Zadeh himself
and well established in the literature, is the Compositional
Rule of Inference (CRI) [10]. Given a fuzzy observation A0 ∈
F(X) and a fuzzy relation R interpreting a given fuzzy rule
base, an output fuzzy set B 0 ∈ F(Y ) is deduced as follows:
_
B 0 (y) =
(A0 (x) ∗ R(x, y)), y ∈ Y .
(4)
x∈X
We use the following notation to indicate the CRI scheme:
B 0 = A0 ◦ R .
(5)
C. Motivation for the Bandler-Kohout Subproduct Inference
Generally, the CRI - represented by ◦ - may be replaced
by any appropriate image @ : F(X) × F(X × Y ) → F(Y )
of a fuzzy set under the fuzzy relation R, which fulfills some
required properties.
Fuzzy rules may be viewed as a partial mapping from
F(X) to F(Y ) assigning Bi ∈ F(Y ) to Ai ∈ F(X) for
every i = 1, . . . , n. The inference process then may be
viewed as an extension of this partial mapping to a total
one [11]. For better understanding, let us adopt the notation
from [12] and consider the following structure of fuzzy rules
(1)
S = (X, Y, {Ai , Bi }i=1,...,n , L, @) ,
Martin Štěpnička is with the Institute for Research and Applications of
Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 00 Ostrava, Czech
Republic (e-mail: [email protected]).
Balasubramaniam Jayaram is with the Department of Mathematics,
Indian Institute of Technology Madras, Chennai 600 036, India (e-mail:
[email protected]).
where L is a complete residuated lattice [13] which determines both, the t-norm ∗ as well as its residual implication.
By the choice of the fuzzy relation R interpreting the fuzzy
rule base and by the choice of @, we define a fuzzy function
fR : F(X) → F(Y ) such that fR (A0 ) = A0 @R, for
arbitrary A0 ∈ F(X).
One of the fundamental properties of such a mapping is
its interpolativity, i.e., fR (Ai ) = Bi . In this case, we say
that R is a correct model of given fuzzy rules in the given
structure S.
Therefore we have to deal with a system of fuzzy relation
equations [14]:
Ai @R = Bi ,
i = 1, . . . , n ,
(6)
solved with respect to unknown R ∈ F(X × Y ). If R
is a solution to (6) then the adjoint fuzzy function fulfills
fR (Ai ) = Bi .
Let us recall some main results [15], [16], [14] for the
CRI case:
Ai ◦ R = Bi , i = 1, . . . , n .
(7)
Theorem 1.1: System (7) is solvable if and only if R̂ is
a solution of the system and moreover, R̂ is the greatest
solution of (7).
Theorem 1.2: [16] Let Ai for i = 1, . . . , n be normal.
Then Ř is a solution of (7) if and only if the following
condition
_
^
(Ai (x) ∗ Aj (x)) ≤
(Bi (y) ↔ Bj (y)) ,
(8)
x∈X
y∈Y
holds for arbitrary i, j ∈ {1, . . . , n}.
Theorem 1.2 specifies a sufficient condition under which
the system is solvable and moreover, it ensures that not only
R̂ but also even Ř is a solution of system (7).
Another sufficient condition (with a high practical importance) for the solvability of the system follows.
Theorem 1.3: [17] Let Ai for i = 1, . . . , n be normal and
fulfill the condition
n
X
Ai (x) = 1, x ∈ X.
(9)
i=1
Then the system (7) is solvable.
Besides the most often used CRI inference scheme, another scheme of inference is the Bandler-Kohout subproduct
C (BK-Subproduct, for short) [18], [19]. For a given input
A0 ∈ F(X) the inference obtained using the BK-Subproduct
is given as follows:
^
B 0 (y) =
(A0 (x) → R(x, y)), y ∈ Y .
(10)
x∈X
We use the following notation to indicate the BK-Subproduct
scheme:
B 0 = A0 C R .
(11)
In the case of BK-Subproduct, the system of equations (6),
reduces to the following one
Ai C R = Bi ,
i = 1, . . . , n .
(12)
Concerning system (12), let us recall the following two
basic theorems .
Theorem 1.4: [20] System (12) is solvable if and only if
Ř is a solution of the system and moreover, Ř is the least
solution of system (12).
Theorem 1.5: [21] Let Ai for i = 1, . . . , n be normal.
Then R̂ is a solution of (12) if and only if the condition (8)
holds for arbitrary i, j ∈ {1, . . . , n}.
Again, condition (8) to which Theorem 1.5 refers to, is
on one hand very transparent but not very convenient from a
practical point of view. Fortunately, the sufficient condition
for solvability of the system stated in Theorem 1.3 is valid
even for system (12).
Theorem 1.6: [17] Let Ai for i = 1, . . . , n be normal and
fulfill the Ruspini condition (9). Then the system (12) is
solvable.
As we may see from the main results, the popular CRI
itself does not keep any advantage in comparison with
the BK-Subproduct, with respect to the interpolativity. It
is always a combination of the inference and the fuzzy
relation interpreting fuzzy rules which have to fulfill the
desired requirements. Furthermore, in [12] the authors prove
an equivalence of the solvability by a fuzzy relation R with
a special kind of continuity of the derived fuzzy function fR
in a given structure S = (X, Y, {Ai , Bi }i=1,...,n , L, ◦) and
analogous results are showed in [22] even for the structure
S = (X, Y, {Ai , Bi }i=1,...,n , L, C).
Finally, the preservation of the indistinguishability that
may be inherent in the input fuzzy sets which is a kind of
desired robustness and which was proved in [23] for the CRI
is valid even in case of the BK-Subproduct, see [22].
II. C OMPUTATIONAL ASPECTS OF F UZZY R ELATIONAL
I NFERENCES
In this section, we introduce new results aiming at the
computational aspects of CRI and BK-Subproduct inferences. Firstly, we show that all the advantages enjoyed by
CRI are also available with the BK-Subproduct too. However,
both CRI and BK-Subproduct - being fuzzy relational inferences - possess some drawbacks. Recently JAYARAM [24]
had proposed a modified form of CRI - Hierarchical CRI
scheme. We show that a similar hierarchical inferencing is
possible even in the case of BK-Subproduct and hence is a
computationally viable alternative for the CRI.
Let us recall, that by a fuzzy singleton we mean a fuzzy
set A : X → [0, 1] with the following representation:
(
1, if x = x0 ,
A(x) =
(13)
0, if x 6= x0 ,
for some x0 ∈ X and we say that A attains normality at x0 .
A. Inferencing in the case of singleton fuzzy inputs
It is not uncommon in some contexts to deal with fuzzy
singleton inputs. Typically in fuzzy control, the singleton
fuzzifier which converts a crisp input x0 ∈ X into a singleton
A0 ∈ F(X) which attains normality at x0 is used. Note, that
this seeming formal conversion is crucial since it allows us
to apply any fuzzy relational inference which, in principal,
deals only with fuzzy inputs.
However, from the computational point of view, it is highly
desirable to deal with such an inference mechanisms which
in case of a crisp input x0 ∈ X and the singleton fuzzifier
infers output fR (A0 ) = B 0 ∈ F(Y ) where B 0 (y) = R(x0 , y).
This property, which saves computational costs, holds for
CRI. From the following equalities, we see that the discussed
property is valid even for the BK-Subproduct. Let the given
singleton fuzzy input A0 attains normality at some x0 ∈ X.
Then the inferred output using the BK-Subproduct is given
by
^
B 0 (y) =
(A0 (x) → R(x, y))
x∈X
= (A0 (x0 ) → R(x0 , y)) ∧
^
(A0 (x) → R(x, y))
x∈X
x6=x0
= (1 → R(x0 , y)) ∧
^
(0 → R(x, y))
x∈X
x6=x0
C. Drawbacks of a Fuzzy Relational Inference
= R(x0 , y) ∧ 1 = R(x0 , y), y ∈ Y.
It should be emphasised that the above property is not
generally valid for any image of a fuzzy set under a fuzzy
relation, e.g., in cases of the Bandler-Kohout superproduct
or the Bandler-Kohout square product [18], [25].
B. Equivalence between FITA and FATI
It is a well known fact that one of the reasons to use the
Mamdani-Assilian approach to interpret a fuzzy rule base is
the possible saving of computational efforts. In other words,
combination of ◦ with Ř requires fewer computations than
the combination with the fuzzy relation R̂.
This is due to the fact that we do not have to compose
all rules to a fuzzy relation, we just find the highest degree
of intersection of a given input A0 and a particular rule
antecedent and multiply it by the corresponding consequent.
This approach is then applied rule per rule and the results
are composed together by the maximum operation. Such an
effect of reducing the computational costs is nothing but
the effect of the equivalence of FITA and FATI inference
strategies [26].
So, it may be again generally considered even for other
inference mechanisms. Due to the following sequence of
equalities:
!
n
^
^
0
B(y) =
A (x) →
(Ai (x) → Bi (y))
i=1
x∈X
=
=
=
n
^ ^
x∈X i=1
n ^
^
(A0 (x) → (Ai (x) → Bi (y)))
((A0 (x) ∗ Ai (x)) → Bi (y))
i=1 x∈X
n
^
_
i=1
by R̂, the FATI inference strategy is equivalent to the FITA
inference strategy.
The difference lies in the propriety of the chosen fuzzy
relation interpreting fuzzy rules (1) with respect to a chosen
inference mechanism. In the case of CRI, there is no other
choice but Ř if we want to reduce the computational efforts
by an equivalent FITA strategy. While in case of the BKSubproduct, the same holds for R̂. In other words, there
should be other reasons leading to the use of Ř then the
computational costs because if this is the only reason, we still
may reduce the computational efforts by the use of the BKSubproduct as an inference mechanism while still keeping the
conditional nature of rules (1) by the choice of R̂. Then the
correctness of the model (interpolativity) should be ensured
by the conditions given in Theorem 1.5. On the other hand,
this is also the case of the use of Ř where the same conditions
have to be imposed, see Theorem 1.2.
!
0
(A (x) ∗ Ai (x)) → Bi (y) , y ∈ Y,
x∈X
(14)
we may state that even in the case of the BK-Subproduct
C and an appropriate interpretation of fuzzy rules (1) given
So far, we have considered only Single-Input-SingleOutput (SISO) fuzzy rules. However, in practice, one often
encounters situations which demand that inputs from multiple
sources / dimensions be considered and hence the need arises
for dealing with Multiple-Input-Single-Output (MISO) fuzzy
rules of the following type - for the sake of notational
simplicity and ease of understanding we only deal with a
2-input-1-output MISO fuzzy rules, which can be extend in
an obvious way to more than 2 input dimensions:
IF x is A1
IF x is An
AND
...
AND
...
AND
y is B1 THEN
z is C1
(15)
y is Bn THEN
z is Cn
where Ai , Bi , Ci are represented by fuzzy sets Ai ∈ F(X),
Bi ∈ F(Y ) and Ci ∈ F(Z), respectively.
Note that both in the case of MISO and SISO rules the
input fuzzy set(s) can be seen to be a fuzzy set on either a
single domain or a Cartesian product of the domains, all the
results presented so far, although discussed in the framework
of SISO rules, are valid even when dealing with MISO fuzzy
rules.
However, fuzzy relational inference schemes are not without their drawbacks because of the computational and space
complexities involved, see [27], [28], [29]. These are compounded greatly, especially, while dealing with MISO fuzzy
rules. Since CRI and the BK-Subproduct both belong to the
class of fuzzy relational inferences they are not immune to
these drawbacks.
The complexity of an inference algorithm stems mainly
from two factors:
(i) The process of inference itself. The fuzzy inferencing schemes are generally resource consuming (both
memory and time). Many of the inference schemes discretize the underlying domains and hence the process
becomes computationally intensive.
(ii) The structure, complexity and the number of rules.
Depending on the shape of the underlying fuzzy sets
the number of parameters stored and processed varies.
Similarly, the manner in which multiple antecedents
are combined affects the processing complexity. Also
an increase in the number of rules only exacerbates the
problem. As the number of input variables and/or input
fuzzy sets increases, there is a combinatorial explosion
of rules in multiple fuzzy rule based systems.
We illustrate the above through the following example.
Example 2.1: Let A = [0.9 0.8 0.7 0.7] , B =
[1 0.6 0.8] , C = [0.1 0.1 0.2] , denote fuzzy sets defined
on, respectively, the following classical sets
X = {x1 , x2 , x3 , x4 }, Y = {y1 , y2 , y3 }, Z = {z1 , z2 , z3 }.
Let S = (X, Y, Z, {A, B, C}, L, C) be the structure considered for the single fuzzy rule
IF x is A AND y is B THEN z is C .
Let L be the Łukasiewicz complete residuated lattice, i.e.,
L = ([0, 1], ∧, ∨, ⊗, →⊗ , 0, 1) where ⊗ stands for the
Łukasiewicz t-norm x⊗y = max(0, x+y−1) and →⊗ stands
for the Łukasiewicz implication x →⊗ y = min(1, 1−x+y).
Now, taking the Cartesian product of A and B with respect
to ⊗, we have


0.9
0.5
0.7
 0.8
0.4
0.6 
 .
A⊗B =
 0.7
0.3
0.5 
0.7
0.3
0.5
Now, we have R̂(A, B; C) = [R̂(z1 ) R̂(z2 ) R̂(z3 )], where
(A ⊗ B) →⊗ C = (A ⊗ B) →⊗ [0.1 0.1 0.2] and R̂(zi ) =
(A ⊗ B) →⊗ zi . Thus


0.2
0.6
0.4
 0.3
0.7
0.5 
 ;
R̂(z1 ) = R̂(z2 ) = 
 0.4
0.8
0.6 
0.4
0.8
0.6

0.3
 0.4
R̂(z3 ) = 
 0.5
0.5
0.7
0.8
0.9
0.9

0.5
0.6 
 .
0.7 
0.7
Let A0 = [0 0 1 0], B 0 = [0 1 0] be the given fuzzy singleton
inputs. Then


0 0 0
 0 0 0 

A0 ⊗ B 0 = 
(16)
 0 1 0  .
0 0 0
The output obtained from the BK-Subproduct is
C 0 = (A0 ⊗ B 0 ) C ((A ⊗ B) →⊗ C) = [0.8 0.8 0.9] . (17)
Remark 2.2: With the help of Example 2.1 above the
following observations can be made:
(i) Computational complexity: Though the computational
complexity largely depends on the choice of operators
employed, let us consider the following general case of
a p-input 1-output system where i-th fuzzy rule is modelled by the fuzzy relation Ri ∈ F(X1 , . . . , Xp , Y )
where Ri = (A1i ∗ · · · ∗ Api ) → Bi . Let the universe
of discourse Xj be discretized into pj points for each
j = 1, . . . , p. Then the
complexity
Q
of a single inference
p
is proportional to O
j=1 pj . If pj = m, then it is
O(mp ).
(ii) Space complexity: Again, for a p-input 1-output
Qp system we have a p-dimensional matrix having j=1 pj
entries. Hence we need to store p-dimensional matrices
for every fuzzy if-then rule.
(iii) Run-time Space Requirements: For example, consider
inferencing with the BK-Subproduct inference scheme
(10) in the case of a 2-input fuzzy if-then rule. Let the
universes of discourse X1 , X2 and Y be discretized
into m, k, l points, respectively. Then the memory
requirements of the algorithm are as follows (see also
[29]): m · k · l for (A ∗ B) → C, m · k for combining
the given facts A0 ∗ B 0 and l for the consequent, where
by ∗ we denote the t-norm used for the Cartesian
product. Overall, it is m · k · l + m · k + l. In the case
m = k = l, the memory requirements of the algorithm
become m3 + m2 + m. Generalizing this, in the case
of p-inputs we have that the memory requirements of
the algorithm is O(m(p+1) ).
There are many works proposing modifications to the classical CRI in an attempt to enhancing the efficiency in its
inferencing, see for example [30], [31] or [32], [33], [34]. In
the case when there are more than two antecedents involved
in fuzzy inference, RUAN and K ERRE [35] have proposed
an extension to the classical CRI, wherein starting from a
finite number of fuzzy relations of an arbitrary number of
variables but having some variables in common, one can infer
fuzzy relations among the variables of interest. D EMIRLI
and T URKSEN [29] proposed a Rule Break-up method and
showed that rules with two or more independent variables
in their premise can be simplified to a number of inferences
of rule bases with simple rules (only one variable in their
premise).
However, no such works exist for the BK-Subproduct to
the best of the authors’ knowledge. In the following we
propose a modified form of BK-Subproduct that alleviates
some of the concerns noted above, along the lines of the
Hierarchical CRI proposed by JAYARAM [24].
D. Hierarchical BK-Subproduct
In [24], JAYARAM proposed a hierarchical variant of
the CRI where observation on particular axes are taken
independently and hierarchically and the overall output is
deduced after all observations were used in this step-by-step
chain procedure. On the contrary to the usual case where a
Cartesian product of all observations is computed and such a
product serves as the only fuzzy input with a vector variable.
We follow this idea with the BK-Subproduct as well.
Procedure for Hierarchical BK-Subproduct
Step 1 FOR i = 1 TO n DO
(i) Calculate Ri0 ∈ F(Y × Z): Ri0 = Bi → Ci
(ii) Calculate Ci0 ∈ F(Z): Ci0 = B C Ri0
(iii) Calculate Ri00 ∈ F(X × Z): Ri00 = Ai → Ci0
(iv) Calculate Ci00 ∈ F(Z): Ci00 = A C Ri00
Step 2 AGGREGATE ALL Ci00 BY MINIMUM
We also remark that, although in [24] the author has given
the algorithm for the Hierarchical CRI only in the case of
inferencing with a single MISO rule, it can be extended in
a straight-forward manner to the case of multiple MISO
rules, as done here. However, for the sake of simplicity,
the following example, which demonstrates the reduction in
computational efforts and memory savings is given in the
context of a single MISO rule.
Example 2.3: Let the fuzzy sets A, B, C, A0 , B 0 be as in
Example 2.1 with the same structure S for the given fuzzy
rule. Inferencing with the Hierarchical BK-Subproduct, given
the input (A0 , B 0 ) we have


0.1
0.1
0.2
0.5
0.6 
Step 1 (i) B →⊗ C =  0.5
0.3
0.3
0.4
0
0
Step 1 (ii)
C
= B C (B →⊗ C)
= [0 1 0] C (B →⊗ C)
= [0.5 0.5 0.6]


0.6
0.6
0.7
 0.7
0.7
0.8 

Step 1 (iii) A →⊗ C 0 = 
 0.8
0.8
0.9 
0.8
0.8
0.9
Step 1 (iv)
C 00
= A0 C (A →⊗ C 0 )
= [0 0 1 0] C (A →⊗ C 0 )
= [0.8 0.8 0.9] .
(18)
Remark 2.4: From the above example it is clear that we
can convert a multi-input system to a single-input hierarchical system, both employing the BK-Subproduct inference.
The effect becomes more pronounced when we have more
than two input variables. From the above Example 2.3 it
can be noticed that the most memory intensive step in the
inference is the calculation of the ‘current’ output fuzzy
set (Steps 1 (ii) & (iv)). Once again, considering the case
of a p-input fuzzy rule, if the input universe of discourse
Xj , j = 1, 2, . . . , p are discretized into pj points and the
output universe of discourse Z into l points, respectively,
then the memory requirements of this step, and hence of the
algorithm itself, can easily be seen to be p∗ · l + l + p∗ , where
p∗ = maxpj=1 pj . In the case m = p∗ = l we have the overall
memory requirements to be 2m + m2 . It should also be
emphasized that the memory requirements are independent
of the number of input variables, as can be expected in any
hierarchical setting.
Not only does the Example 2.3 illustrate the computational
efficiency of Hierarchical BK-Subproduct inference, it also
shows that the inference obtained from the original BKSubproduct is identical to the one obtained from the proposed
Hierarchical BK-Subproduct, i.e., (17) = C 0 = C 00 = (18).
The following result shows that this equivalence is always
guaranteed under the structure S considered in this work.
Theorem 2.5: Let
S = (X, Y, Z, {Ai , Bi , Ci }i=1,...,n , L, C)
be a structure for fuzzy rules (15) and let R̂ ∈ F(X ×Y ×Z)
be given by
n
^
((Ai (x) ∗ Bi (y)) → Ci (z)), x ∈ X, y ∈ Y, z ∈ Z.
i=1
Then for any A ∈ F(X) and B ∈ F(Y ) all i = 1, . . . , n it
is true that
n
^
(A ∗ B) C R̂ ≡
(A C (Ai → (B C (Bi → Ci )))).
i=1
Theorem 2.5 states the equivalence of outputs obtained
from the proposed algorithm of the hierarchical BK inference
mechanism and the original BK-Subproduct with two dimensional inputs. Indeed, it may be systematically extended into
a case of inputs of an arbitrary finite dimension.
Now, we may state the following corollary of Theorem
1.5 and Theorem 2.5, which claims that if condition (8)
certifying the solvability of (12) holds than even the proposed
Hierarchical BK-Subproduct inference procedure keeps the
fundamental interpolation condition fulfilled.
Corollary 2.6: Let all the assumptions of Theorem 2.5
be valid. Furthermore, let
n
n
^
_
(Ci (z) ↔ Cj (z))
(Ai (x) ∗ Aj (x) ∗ Bi (y) ∗ Bj (y)) ≤
i=1
i=1
holds for arbitrary x ∈ X, y ∈ Y, z ∈ Z and for arbitrary
i, j ∈ {1, . . . , n}. Then
n
^
(Ai C (Ai → (Bi C (Bi → Ci )))) ≡ Ci .
i=1
III. C ONCLUSIONS
We have briefly recalled that some of the properties that
are usually cited in favor of using the Compositional Rule
of Inference, e.g., reasonable conditions for the solvability
of the fuzzy interpolation problem or the preservation of the
indistinguishability that may be inherent in the input fuzzy
sets, does possess even for the Bandler-Kohout subproduct,
see [22]. Hence, it is equally suitable for consideration
when reasoning with a system of fuzzy rules up to the
computational complexity limitations.
This paper investigates the complexity limitations and
shows that the BK-Subproduct does cope fuzzy singleton
inputs as effectively as the CRI. Moreover, under certain conditions the equivalence of FITA and FATI can be shown for
the BK-Subproduct, much like in the case of CRI. Furthermore, we propose a hierarchical inferencing scheme using the
BK-Subproduct that alleviates most of the recalled computational drawbacks of fuzzy relational inferences with multiple
inputs. This method is amply illustrated with numerical
examples. Finally, we have also shown that if the structure for
the considered fuzzy rules is chosen appropriately then the
outputs obtained from the Hierarchical BK-Subproduct and
the original BK-Subproduct are identical, thus addressing the
issues related to computational complexity.
Based on this work, one can see that the BK-Subproduct is
as much advantageous as the classical CRI and hence can be
employed alternatively in applications. The main difference
lies in the fuzzy relation interpreting a fuzzy rule base
which is combined with a particular inference mechanism.
It is shown that some computational advantages of the very
popular Mamdani-Assilian approach are valid only in the
case of the CRI inference mechanism, while if we use the
BK-Subproduct, then advantage in using Ř is no more true
and the implicational approach employing the fuzzy relation
R̂ assumes this privilege.
Therefore, we conclude that there is another added value
of the existence of two inference schemes with the same
appropriate properties - the possibility to freely choose
between two approaches of the interpretation of a fuzzy
rule base. Up to now, the one approach denoted by R̂
employing genuine implication was disadvantaged because
of computational complexity weakness although for some
problems it is much more suitable [36]. This investigation
shows, that from the computational point of view, there is
neither preferable interpretation of a fuzzy rule base nor
preferable inference mechanism, there are only preferable
combinations of them.
ACKNOWLEDGMENT
This work has been supported by projects
MSM6198898701 and DAR 1M0572 of the MŠMT
ČR.
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