Although in the literature, we can find different versions of fuzzy inference systems, all of them can be divided, in general, in four steps: fuzzification, knowledge database, inference engine and defuzzification. The first step is the fuzzification process, where crisp data are transformed into fuzzy information. Usually, this step is performed by means of a fuzzy partition. The knowledge database comprises some knowledge by rules of the type If–Then. The inference engine is the central stage, where the fuzzy inference system takes the fuzzified input and returns an output by combining the input with the rules in the knowledge database. Finally, the defuzzification process takes the output of the inference engine (usually a fuzzy set) and returns a crisp output according to the applied context of the FIS. Next, we specify how these four phases are formalized in our inference system based on the f-index of inclusion. We start with the knowledge database, to show how the information is comprised in If–Then-type rules. Then, under the semantics of the rules in the knowledge database, the inference engine is introduced using the Description Logic as support. Finally, the fuzzification and defuziffication are analyzed according to the previous consideration.
4.1. The Knowledge Database
Definition 14. A frame is a tuple where X and Y are sets and and are two fuzzy partitions over the universes X and Y, where and .
Definition 15. A knowledge database () on the frame is a set of rules of the formwhere , and . To provide an interpretation to the connection between the implication rule and its associated function , we use the f-inclusion relation between fuzzy sets, as presented in the following definition.
Definition 16. Let Γ be a on the frame . A subset of pairs is a model of Γ, if, for every rule , we have that for all . The set of all models of Γ is denoted by .
The use of the
f-inclusion to model implication rules in our frame
is justified by the ability of this operator to serve as fuzzy implication, as explained in [
15].
Note that a model of a is a crisp set, i.e., a subset of is either a model or not. Consequently, with a fixed , we can consider in the set of models of , , the standard order between sets and then analyze the structure of . The next result shows that is a complete lattice:
Theorem 5. Let Γ be a on a frame , and let be the set of models of Γ. Then, has a complete lattice structure with the standard order between sets.
Proof. Let us prove that both intersection and union of an arbitrary number of models are also models. Let
be a subset of models of
and let us begin by showing that
is a model. Let
, then, necessarily, there exists a model
such that
. As a result, for any rule
in
, we have
. In other words,
is a model.
The proof that shows that
is a model of
is similar. □
Corollary 1. Let Γ be a on a frame , and let be the set of models of Γ. Then, the greatest element of , denoted by , is the join of every model of Γ, and the least model of is the empty set.
The previous result highlights a substantial difference with respect to Description Logic and logic programming. In the aforementioned formal theories, the emphasis is placed on minimal models (the least model in logic programming or canonical models in DL). In our work, we focus on the maximal model, as it determines the possible pairs of points in
that are consistent with its associated
. Later, in
Section 4.2, it is shown that we can reduce our analysis to the greatest model of
to check correct inferences.
Below, we provide an illustrative example describing a knowledge database where the step-by-step construction of its greatest model is described. For the sake of showing the potential application of this inference system, it has been contextualized in pediatrics despite the synthetic nature of the example.
Example 3. Let us assume that we have conducted a study on heart rate in infants and children between the ages of one and nine. In order to establish a relation between these two properties, we consider the frame consisting of
The interval to represent the ages between 1 and 9, e.g., the age of an infant who is 18 months old would be ;
The interval to cover a wide range of heart rates (in bpm).
The fuzzy partition consists of four fuzzy sets which distinguish four fuzzy age ranges. The membership functions corresponding to each element of are The fuzzy partition only consists of three elements, which can be classified into = “low heart rate”, = “standard heart rate”, and = “high heart rate”, respectively. These fuzzy sets have the following respective associated membership functions:
Both of these fuzzy partitions are represented in Figure 1. Once the frame has been defined, it is time to set the rules that constitute the knowledge base Γ. This contains different rules, each one corresponding to a different combination of elements and . As explained before, the mapping associated with each rule determines a relation of f-inclusion . In other words, when the relationship between and is stronger (i.e., patients with ages corresponding to tend to have a heart rate within ), the mapping is closer to the identity (i.e., the f-inclusion relation is more restrictive). Conversely, when the elements and are not related at all, the inclusion function is null, i.e., for all .
Since, in general, the average heart rate of infants and children decreases with age, the knowledge base associated with this frame is represented as in Table 3. Whereand for all . Now that the has been defined (denoted by Γ), it is time to compute its greatest model . Let us recall from Definition 16 that a model M of Γ is a set of pairs in which satisfy all the following f-inclusion relations:for every rule . Thus, in order to obtain the greatest model, we have to compute which pairs satisfy each of these f-inclusions. This calculation can be performed graphically in the plane through the representation of all the restrictions imposed by each f-inclusion associated with each rule . This plotting process is performed step by step in order to facilitate its understanding. Let us start by focusing on the second element of , , which covers the fuzzy age range from 2 to 6 years. According to Table 3, the three rules associated with this fuzzy set are First of all, note that the former rule, , does not impose any restriction on the set of models, since all pairs satisfy null f-inclusions, i.e., . Therefore, the step-by-step graphical representation is only performed for the next two rules (since the graph associated to the first rule does not impose any constraint on the maximal model).
The graph in Figure 2 illustrates the pairs in that satisfy the condition imposed by the rule . This is carried out in two simple steps: First, the fuzzy set is computed, and its graph is represented over in the partition .
Then, the pairs with positive membership degree in are selected. Among these, the pairs whose y-coordinate has a membership degree in lower than are discarded from .
Note that the mapping associated with the rule implies the existence of a patient whose age belongs to with the highest possible membership degree, that is, 4 years old, and with a heart rate with a membership degree in equal to , e.g., 85 or 115 beats per minute.
A similar process is followed to plot in Figure 3 the set of pairs that satisfy the rule . The apparition of this mapping implies the existence of two different patients: one of them who is 4 years old and whose heart rate is approximately 110 and another one whose age has a membership degree in higher than (i.e., their age is between 30 and 66 months old) and whose heart rage is lesser or equal than 100 bpm, so it does not belong to partition . The final step consists in computing the intersection of the shaded regions obtained in Figure 2 and Figure 3, as the resulting region contains the set of pairs that satisfy all the rules associated with the fuzzy set . Figure 4 shows the result of this computation. By performing an analogous process with the remaining elements of , we obtain the graphical representation in the plane of the greatest model , which corresponds to the shaded region in the plane shown in Figure 5. Note that the pairs that lie on the “boundary” of the shaded section also belong to . The computation of the greatest model provides an easy way to calculate other models of the , since the search for a model is reduced to selecting a set of pairs that lie within the shaded region. In some applied environments, some models different from the greatest one may be of interest. For example, Figure 6 shows (in bold) two different models. The left graph presents a finite model, , consisting of seven elements : On the other hand, the right graph in Figure 6 shows a model consisting of a functional dataset, namely, a set of pairs . The following result shows that with this semantics, we can identify each model of a with one particular T-Box of the Description Logic defined in the previous section. As shown in a subsequent section, this link allows the application of inference rules of in our .
Proposition 3. Let Γ be a on a frame , and let be the T-Box of constructed as follows:
For each in (resp., in ) consider the concept (resp., );
For each appearing in Γ, consider the concept modifier ;
For each rule , consider in the specialization concept
Then, given a model M of Γ, the interpretation given by
;
If , then ;
If , then ;
;
is a model of .
Proof. Let . Then, by definition of a model, for every rule in , which implies that . In conclusion, satisfies for every concept specialization in . In other words, is a model of . □
Obviously, the converse of the previous result is not true in general, since the models in may be very general. Actually, a with only empty models may be connected to a T-Box with non-empty models via the translation described in Proposition 3.
Example 4. Let us consider, on the universes and , the fuzzy partitions and given by the fuzzy sets defined in Table 4. On this frame , we define the composed by the next four rules:
Consider each element , and let us check that there is at least one rule in Γ that is not satisfied by :
and do not satisfy , since .
does not satisfy , because .
and do not satisfy , since .
does not satisfy , because .
and do not satisfy , since .
Finally, does not satisfy , since .
In conclusion, there is no pair that belongs to a model of ; in other words, the only model is the empty set, .
Now, let us define in the set of concepts associated with each one of the elements of and , respectively, and consider the T-Box defined as in Proposition 3: Given the interpretation where
it can be easily checked that satisfies every concept specialization in . Therefore, is a model of .
4.2. Adding Inputs to s: Defining Consequences
In the previous subsection, we described the structure of ; now, let us explain how to reason with it. The purpose of this fuzzy inference system is to draw conclusions from a and a certain input. Let us begin by defining what is a -input on a frame.
Definition 17. A -inputon the frame is a fuzzy set defined either on X or on Y.
Thanks to Proposition 3, it looks natural to incorporate inputs to s by giving a similar semantics to assertions in . Since assertions in resemble the so-called -cuts in fuzzy set theory, our semantics considers -models that represent the -cuts of a (fuzzy) model.
Definition 18. Let A (resp., B) be a -input on a frame such that (resp., ), and let . A subset of pairs is an -model of A (resp., B) if and only if if and only if ).
Under the previous definition of an -model, -inputs can be interpreted as constraints that bound the pairs of elements in . Note that the greater the -input (as a fuzzy set), the weaker the restriction, since it admits more -models. On the other hand, the value also determines a restriction in the sense that the greater the value , the stronger the restriction imposed to pairs in to be in an -model.
Note that from a theoretical point of view, we can assume theoretically that only two inputs can be considered: one fuzzy set on the universe X and another on Y. In other words, if we are interested in reasoning with two different fuzzy sets and on X, this is equivalent to considering the fuzzy set as a -input.
Proposition 4. Let and be two -inputs on a frame , both of them defined on the same universe. A subset of pairs is an α-model of and if and only if M is an α-model of .
Proof. Let
and let
M be an
-model of
and
. By definition of an
-model, we have that for all
,
and
. That is equivalent to saying
Therefore, M is a model of . □
As a consequence of the previous result, we can combine different fuzzy sets defined on the same universe into one -input and keep the semantics. Thus, as mentioned before, from a theoretical point of view, only two inputs are possible, one fuzzy set on the universe X and another on the universe Y. For the sake of simplicity in this approach, we only consider one -input for reasoning.
The following result shows that the set of -models has a complete lattice structure.
Proposition 5. Let A be a -input on a frame . Fixing , the set of α-models of A has the structure of a complete lattice.
It is also convenient for the reader to keep the following meaning of a -input A: the fuzzy set A determines the set of possible values of X (or Y) a system may consider for reasoning with. In this way, we focus on determining the greater set of possible values in that satisfy the constraints represented by the rules in both a and in a -input. The following definition determines the semantics for the fusion of the knowledge represented by a and the restriction imposed by a -input as a fuzzy set on .
Definition 19. Let Γ be a , and let A be a -input on a frame . A model of is a fuzzy set M on the universe such that for each , the α-cut is a model of Γ and an α-model of A.
Note that a model of the pair of a and a -input A is a fuzzy set constructed by -cuts that intersect models of and -models of A. The following result shows an equivalent definition of a model that is used in some proofs of further results.
Theorem 6. Let Γ be a , and let A be a -input on a frame . M is a model of if and only if is a model of Γ and .
Proof. Let us start by proving the backward implication, as it is more straightforward:
- ⇐
Let , and let us assume that is a model of and that for all . Let us prove that M is a model of . Consider the -cut with fixed :
- –
First, since and is a model of , is also a model of .
- –
Second, for each , we have that by assumption. Then, for all . In consequence, , namely, is an -model of A.
- ⇒
Assume that is a model of . Then,
- –
On the one hand, by Definition 19, every
with
is a model of
. Moreover, the support of a fuzzy set can be characterized by
Since, by Theorem 5, is a complete lattice, is also a model of .
- –
On the other hand, by Definitions 18 and 19, every
with
is also an
-model of
A, which means that
. Let us prove by reductio ad absurdum that
for all
. Suppose that there exists an element
such that
for certain
. Then, by definition,
belongs to the
-cut of
M, i.e.,
. By assumption,
is a
-model of
A, which implies
which implies that
. This leads us to a contradiction since
which completes the proof.
□
In Theorem 5, it was already demonstrated that for every , there exists a greatest model associated with it. The following theorem presents an analogous result for models s with -inputs.
Theorem 7. Let Γ be a , and let A be a -input on a frame . The set of models of , denoted by , has a complete lattice structure with the standard Zadeh’s ordering between fuzzy sets (i.e., if and only if .
Proof. Let us begin by proving that is closed by the union of arbitrary models. Let , and let us show that both fuzzy sets and are models of , i.e., they are in . By Theorem 6:
We know that for all and ;
The result is reduced to prove that prove that and for all .
Let
; then,
This is what we wanted to prove. □
It it straightforward to check that the empty set is the least model of . The following theorem shows the expression of the greatest model of .
Theorem 8. Let Γ be a and A be a -input on a frame . The greatest model of , denoted by , is the fuzzy set given by the restriction of the support of A to the greatest model of Γ: Proof. Let us prove that is a model of and that every model M of satisfies . Then, necessarily, is the greatest model of .
By Theorem 6, is a model of , since by definition, for all .
On the other hand, let M be a model of and let us assume by reductio ad absurdum that there exists such that . By definition, we have that , which contradicts the fact that M is a model according to the characterization of Theorem 6. □
We can now introduce the notion of logical consequence as usual in formal logic theories.
Definition 20. Let Γ be a , and let A be a -input on a frame . We say that a fuzzy set (resp., ) is a consequence of , denoted by , if for every model M of , we have for all (resp., for all ).
Thanks to the complete lattice structure of the set of models of (see Theorem 7), we can reduce the validation of consequences to checking the greatest model of .
Theorem 9. Let Γ be a , and let A be a -input on a frame . A fuzzy set (resp., ) is a consequence of if and only if we have for all (resp., ), where is the greatest model of .
Proof. Suppose that (the proof for the case is similar) and that . Then, every model M of satisfies for all . In particular, that inequality holds for the greatest model . That is, for all .
Conversely, let us assume that
for all
. Given a model
M of
, we have that
for all
, since
is the greatest model of
. Therefore,
for all
and
. □
From the previous characterization, it is obvious that in order to determine consequences, we can focus only on the greatest model of . The following result shows that when considering a with a simple condition of coherence, determining whether a fuzzy set defined on the same universe as the -input is a consequence is trivial.
Theorem 10. Let Γ be a on a frame . Let us assume that for each and , there exists and such that is a model of Γ. Then:
Given , if and only if ;
Given , if and only if .
Proof. Let us prove the first item, since the second is proved similarly. Let
be the greatest model of
. From Theorem 8, we have that
for all
and
if
. Let
such that
. Then, by Theorem 9, that is equivalent to say that
. Then, for each
, and choosing
such that
, we have
which is equivalent to say that
. □
From the previous result we have the following: given a on a frame and a -input , the only non-trivial consequences of are those fuzzy sets defined on Y. The following proposition shows that there is a monotonicity on the set of consequences with respect to the ordering of fuzzy sets with respect to Zadeh’s ordering.
Proposition 6. Let Γ be a and be a -input on a frame . Let such that and ; then, .
Proof. Consider a model M of . If , then (Theorem 9) for all . Since for all , then for all . Again, by Theorem 9, we can conclude that . □
It can be also proved that the set of consequences of inherits the complete lattice structure from the set of models of .
Proposition 7. Let Γ be a and be a -input on a frame . The set of consequences of , i.e., the set , has a complete lattice structure with greatest element Y.
Proof. Let be the set of consequences of , and let . By Proposition 6, we have directly that is a consequence of .
Let us prove now that
is a consequence of
as well. By definition of a consequence, we have that for all
’s and all models
M of
, we have
for all
. As a result,
for all models
M and all
. That is,
is a consequence of
. □
Corollary 2. Let Γ be a and be a -input on a frame . Let be two fuzzy sets such that and ; then,
;
.
4.3. Inference Engine: Links with
In the previous section, we showed that given a and a -input , the set of consequences was determined by the greatest model of or by the least consequence on . Determining the greatest model and checking whether a fuzzy set is a consequence by comparison with it may be complex from a computational point of view. In this section, we propose a simple inference process to obtain correct consequences that upper-bound the least consequence of .
In order to keep the support of our fuzzy inference system on the Description Logic (and its power of reasoning), it would be desirable to link the models of both constructions, as linked in the previous section by Proposition 3. The following theorem presents an analogous result for a and a -input by fixing elements in -cuts of models.
Theorem 11. Let Γ be a and let be a -input on a frame . Let be the A-Box of constructed as follows:
For each in , consider the concept ;
For A in , consider the concept ;
For each concept , consider the pair of concept modifiers ;
For each concept , consider in the assertion where .
Consider a non-empty model M of , its α-cut , and . Then, the interpretation given in Proposition 3 plus
is a model of .
Proof. By definition of the
f-index of inclusion,
for all
and for every element
of
. Then, the interpretation
satisfies
for all
. In other words, it satisfies the concept specialization
. Then, by Theorem 2,
must also satisfy
.
At the same time, since the instance
a of
satisfies
and
is an
-model of
A,
. Therefore, given the assertion
:
In conclusion, satisfies the assertion for all , and therefore, is a model of . □
Note that -inputs in a and a frame are translated as a series of assertions in Description Logic through concept modifiers for each one of the elements of the partition . That is to say, for each -input, is considered an A-Box consisting of as many assertions as there are elements in the partition . This identification is carried out individually for each pair in the support of the model , taking into account the -cuts of M due to the specific characteristics of Description Logic, i.e., DL only allows for the inclusion of a countable set of instances, whereas the universes X and Y can be of a fundamentally different nature, such as . However, this identification of the instance a (in ) with its corresponding is fixed but arbitrary within the -cut , as is the choice of the value . This makes it possible to translate the assertions defined in into a fuzzy set A defined on a universe X or Y.
The next result justifies the use of reasoning tools from Description Logic within our framework, and in particular, the application of the Generalized Modus Ponens given in Theorem 4.
Corollary 3. Let Γ be a and let A be a -input on a frame . Let be the T-Box of and the A-Box of constructed as in Proposition 3 and Theorem 11, respectively. Then, all fuzzy assertions entailed by the set of axioms (on the context of ) form a correct inference from Γ and A (interpreted as and -input).
Proof. Let be a fuzzy assertion entailed by , which means that every model of satisfies as well. Then, given a model M of and given the interpretation , with , defined in Proposition 3 and Theorem 11, it is guaranteed that is a model of ; therefore, it also satisfies . As and , it can be concluded that is a -model of . □
As a consequence of the previous corollary, we can apply tools of inference from the Description Logic
on
and
-inputs. Among all of them, we are interested here on the Generalized Modus Ponens (GMP) described on
Section 3.3. Specifically, we can join all the possible GMP applicable on the T-Box and A-Box constructed according to Proposition 3 and Theorem 11 and obtain the following inference on
, which is the core of the inference engine considered in our approach.
Theorem 12. Let Γ be a and let A be a -input on a frame , where . Then, for all , we havewhere is the fuzzy set given bywhere is the only mapping such that forms an adjoint pair. Proof. Consider in
the T-Box
and the A-Box
constructed as in Proposition 3 and Theorem 11, respectively, by a fixed model
M of
and a fixed
-cut of
M,
, with
. Given the construction of both sets, one can rearrange them in pairs of elements
related through the concept specialization antecedent
. Then, by applying the Generalized Modus Ponens given in Theorem 4, we have
where the concept
obtained as an output is interpreted as
for all
. If we denote this fuzzy set as
, it is concluded that
is an
-model of
. □
The previous result leads to the application of the following inference engine for a frame
and a
. Given a
-input
, for each pair
, there exists a rule
from which we can obtain the following inference:
where
is the only mapping such that
is an adjoint pair, as is
with respect to
, i.e., the
f-index of inclusion of
A into
is restricted to the set
(Definition 6).
Thanks to Theorem 12, we obtain the consequence
for each pair
, where
and
. By Corollary 2, we can conclude that the intersections and unions of the
are also consequences. Since our goal is to find the smallest set that is a consequence of
and
A, the inference engine takes the infimum of all of them. This leads to our inference engine:
which is called the
-output of
. The next result states that the fuzzy set obtained by our inference engine is actually a consequence of a
and a
-input.
Corollary 4. Let Γ be a and let be a -input on a frame . Then, the fuzzy set defined by Equation (4) is a consequence of , that is, . Proof. As a consequence of Theorem 12, given a model of
, every
-model of
A with
is also an
-model if
, which means that every
satisfies
. Then, the fuzzy set defined in Equation (
4) also satisfies
for every
. In consequence,
is an
-model of the
-output
B. □
Let us look at a pair of examples applying this inference engine to the defined previously in Example 3. First, we study the case where the -input introduced is a singleton. Second, the case where the -input is a more general fuzzy set is studied.
Example 5. Let be the frame and Γ the defined in Example 3, where a study on the heart rate (in bpm) (Y) in children between 1 and 9 years old (X) is conducted. In this example, the greatest model of Γ, , was plotted in Figure 6. Suppose that we want to study the possible heart rate of a child who is 30 months old, that is, two and a half years old. This case is translated on the frame as a crisp singleton -input , whose membership function is given below in Figure 7: The inference engine consists of the application of the GMP based on the f-index of inclusion for each of the elements in Γ. Since has 4 elements and has 3 elements, we apply a total of GMP as described in Theorem 12:where is described in terms of Equation (3): Let us recall that is the only mapping such that forms an adjoint pair, as is with respect to the f-index of inclusion restricted to of A in . Therefore, in order to apply the GMP, it is necessary to compute each of the mappings and . The following formula [30] allows us to compute the the adjoint pair associated with each that appears in Γ by a straightforward manual calculation:which results in the Table 5. Whereand for all . Secondly, in order to apply the inference engine, the f-index of inclusion restricted to of the -input A in each of the elements of the partition must be computed. The results obtained are as follows: Nevertheless, the GMP defined in Theorem 12 employs the adjoint pair associated with each of these f-inclusions. By using the formula described in Equation (5), adapted for each , we obtain each , with : Now, we can proceed to apply the GMP based on the f-index of inclusion restricted to . Let us first show an example of computation of this inference rule, considering the rule . The f-index of inclusion of the -input A into is ; then, according to Theorem 12, the result of applying GMP is Note that this composition of mappings is applied to ; in other words, it does not depend on the -input A. Recall that the membership function of is: Let us compute the output step by step. The result of applying this fuzzy set to the mapping is: Finally, the output of this GMP is: In other words, applying the mapping does not affect the final result.
In the vast majority of cases, we obtain the output due to the appearance of the top element as one of the components of the function composition. All these cases are consequences of two possibilities: either the respective rule in Γ determines no restriction or the inclusion of A into is null. Recall that the final inference is the intersection of all the outputs ; hence, all these cases are degenerate and do not produce any restriction. Accordingly, only those outputs where the mapping is not involved in the application of the GMP are shown below: Once each of the inferences is obtained, the -output is given by the fuzzy set B obtained by computing the intersection of each . Its membership function is given by: Figure 8 shows the membership function of this fuzzy set. Finally, let us study the greatest model of . Unlike the greatest model of Γ, is a fuzzy set, and therefore, it may be difficult to represent its membership function on the plane ; instead, we can consider their α-cuts to study the relation between the -output and the greatest model. Since the -input A is a crisp singleton (i.e., A only takes values in ), let us consider the core (i.e., the 1-cut) of the -output B, which is represented in Figure 9. Its algebraic expression is: Observe that for every element , the X-component is contained in the core of A, and the Y-component is contained in the core of B, which is the interval , that is, Later on, we demonstrate that this phenomenon is not a coincidence. This result is used to justify the defuzzification procedure of our inference system, as can be seen in the following section.
Let us see now another example of application of the inference engine, this time involving a fuzzy set as -input with a continuous membership function.
Example 6. Consider the same frame and the same Γ defined in Examples 3 and 5. Suppose now that we want to study the heart rate of a child who is “approximately seven years old”. This statement is represented by the -input defined in Figure 10. Just like in the previous example, 12 GMP based on the f-index of inclusion have to be carried out. Each one generates an output given by the following expression:where and . Let us now compute each f-index of inclusion of the -input A in each element of the partition : Let us now compute the mapping such that forms an adjoint pair by using Equation (5): Only two outputs different from Y are obtained, and they are given by those GMP obtained by applying rules and : The final inference is obtained through the intersection of each output of the 12 GMP, which results in . This -output is represented in Figure 11, and it is given by the following expression: Finally, let us study the greatest model of , as in the previous example. In Figure 12, we have represented the -input A, the -output B, and the core of the greatest model of . In this case, we can also check that this set can be rewritten in terms of the cores of both A and B, namely, . Moreover, there is an additional property satisfied by the greatest model , which is related to the support of (represented in Figure 12 by the shaded region within ), the support of A, and the support of B: is contained in the X-component of , and is contained in its Y-component. Nevertheless, this last property is not always satisfied by the -output of a Γ and a -input; e.g., the -output obtained in Example 5 has a greater support than the Y-component of (see Figure 9). 4.4. On Fuzzification and Defuzzification Procedures
Last but not least, we discuss the procedures of fuzzification and defuzzification. The reason is mainly because firstly, they depend strongly on the particular context of application of the inference system, and secondly, the scope of this paper is to show the correctness of the inference system. Nevertheless, here, we present some discussions and some issues to be considered in further approaches.
Obviously, the definition of rules in the knowledge database requires a pair of fuzzy partitions and on the universes X and Y, respectively. That is a fuzzification procedure, but there is another that transforms the input of the system into a -input. In most approaches, the same fuzzy partitions used in the construction of the knowledge database are used to fuzzify the input of the system. However, in our approach, the -input may be any fuzzy set defined independently from the fuzzy partition considered in X for the construction of the . That opens an endless number of possibilities for transforming the input of the fuzzy inference system into the -input. For instance, consider that for the sake of accuracy, we use two fuzzy partitions and on the universes X and Y with hundreds of fuzzy sets each. On the other hand, in order to keep an interpretable input and output of the fuzzy inference system, we use linguistic labels in the form of fuzzy sets on X. These linguistic labels may have more or less elements, may be defined differently for each user, and may even have no relation with the partition used in X for the . In all cases, given a linguistic label as a fuzzy set on X, we can reason with the .
Another possibility for a fuzzification is to iterate fuzzy inference systems, and then, the -output of one fuzzy inference system becomes the -input of the next one, so the fuzzification is then the result of a fuzzy inference system. Note that we can even not consider any fuzzification. That is, given a value or an interval of values in X (i.e., a crisp input), we can consider it as a set (or a singleton as a set as in Example 5) and then reason with it. As mentioned above, the fuzzy partitions considered in the construction of knowledge databases do not limit the fuzzification procedure for the -input; it is open to any possible fuzzification.
On the other hand, the defuzzification also depends on the scope of the application rather than the fuzzification. Therefore, it is hard to talk about it in this approach. Nevertheless, it is convenient to bear in mind the following characteristic of our fuzzy inference systems. The consideration of crisp models for the semantics of allows us to focus on models as a defuzzification procedure. In this respect, we have two options, to either take into consideration all the possible values that are coherent with the modeling of a and the -input (which results in choosing the greatest model as the (crisp) output of our system) or to focus on selecting a specific kind of model. This latter case is certainly more interesting, and the characteristics of these models depend on the scope of the application; for example, a regression model focuses on a functional model, whereas a classification model may focus on measure models with a certain in the second component.
It is worth finishing this section by focusing on the simplest case of fuzzification and defuzzification procedures: a singleton crisp input and the search for a set of possible values for the variable Y in the output. This is exactly the case illustrated in Example 5, were the crisp value is directly considered as a crisp-singleton fuzzy set as -input. The following result shows that in that case, the core of the -output determines the greatest model. Note that given , the singleton can be interpreted as the fuzzy set A such that and if .
Corollary 5. Let Γ be a on a frame , let , let M be the greatest model of and let B be the fuzzy set obtained in Equation (4) as the -output of . Then, M is a crisp set (i.e., ) and Proof. Proving that M is crisp comes directly from Definitions 18 and 19, since all -models of coincide with the one-model of .
Let us prove the “if and only if” part. By definition of a consequence and the correctness of Corollary 4,
implies
. In other words, to prove the other implication, let us assume that
. By definition of the inference engine in terms of intersections, we have that necessarily, for all rules
,
Moreover, since
is a singleton, it can be proved that
Then, we have the following chain of inequalities:
In other words, satisfies all the rules in , and is a model of . Moreover, since the -input is crisp, is a one-model of . By maximality of M, necessarily, we have that , which is what we wanted to prove. □
In other words, in the case of working in the simplest setting of fuzzification (a crisp entry) and defuzzification (interval of plausible values in the greatest model), the values in the core of the -output directly determine the greatest model of the and the -input. That fact is illustrated in the following example.
Example 7. Let us reconsider Example 5, where the -input was the crisp singleton set . The result of the fuzzy engine iswhose core is the interval . From Corollary 5, we can conclude that the greatest model of is The reader can check in Figure 9 that effectively, all pairs of values in the greatest model of Γ with are exactly the interval .