Semantic-based constraints

The main objective of imposing semantic constraints during the fuzzy rule-based system construction is to preserve the semantics associated with the membership functions. In other words, the fuzzy partitions of a given variable can be interpreted as linguistics labels such as: low, medium and high. In addition, semantic constraints include also some other properties related to the logic side of the rules such as the consistency of the rules (Gacto, Alcalá, & Herrera, 2011). Semantic-based constraints can be divided into

University

49

the following three classes according to the components on which the constraints are applied: constraints for fuzzy sets, fuzzy partition and fuzzy rules.

a) - Semantic-based constraints for fuzzy sets

 Normality

Normality for a fuzzy set means that there exists at least one element or data point in the universe of discourse with full membership, i.e., has a membership value equal to 1 (see Figure 3.1). This can be formally stated with the following expression:

∃ 𝑥 ∈ 𝑋, 𝜇_𝐴(𝑥) = 1 (3.1)

For interpretable fuzzy rule-based systems, the linguistic terms should have a clear semantic meaning; that is, one element of the universe of discourse should exhibit full matching with the linguistic term semantically represented by the fuzzy sets (De Oliveira, 1999).

Normality is a requirement that is implicitly assumed by the overwhelming majority of literature related to the interpretability with few exceptions that were explicitly cited normality as one of the requirements for fuzzy rule-based systems’ interpretability (De Oliveira, 1999; Serge Guillaume, 2001; S. Guillaume & Charnomordic, 2011, 2012; Mencar & Fanelli, 2008; Roubos & Setnes, 2001; Setnes & Roubos, 2000; Zhou & Gan, 2008).

0.0 1.0

Figure 3.1 An example of non-normalized fuzzy set (fuzzy set with dotted points)

University

50

 Convexity

A fuzzy set is a convex if the membership values of elements belonging to any interval are not lower than the membership values at the interval’s extremes. Formally this can be written as follows:

∀ 𝑎, 𝑏, 𝑥 ∈ 𝑋, 𝑎 ≤ 𝑥 ≤ 𝑏 → 𝜇_𝐴(𝑥) ≥ min{𝜇_𝐴(𝑎), 𝜇_𝐴(𝑏)} (3.2) It is semantically considered as a completion of the normality requirement. According to Mencar and Fanelli (2008), convexity assures that the concept represented by the fuzzy set is related to a single specific property of a perceived object. In other words, the concept can be conceived as elementary.

Convexity is very important requirement of interpretability and it is implicitly assumed in interpretable fuzzy rule-based systems except with few articles (De Oliveira, 1999; S. Guillaume & Charnomordic, 2012; Mencar & Fanelli, 2008; Zhou & Gan, 2008).

In fact, the normality and convexity of a fuzzy set can be easily satisfied by selecting the most commonly used membership function types such as triangular and Gaussian and this explains why most of the researchers do not explicitly discussed these semantic constraints in their works.

b) - Semantic-based constraints for fuzzy partition

 Coverage and completeness

The universe of discourse of a variable is complete if every data point of element belongs at least of one of the generated membership functions. Formally, this can be written as follows:

∀ 𝑥 ∈ 𝑋, ∋ 𝐴 ∈ 𝐹, 𝜇_𝐴(𝑥) > 0 (3.3) Where 𝐹 is the set of fuzzy sets defined in the universe of discourse 𝑋.

The expression (3.3) suggests that for every data point, it is required that the membership value should not be zero for at least one of the fuzzy sets. That is, every data point is semantically represented by at least one of the linguistic terms (see Figure

University

51

3.2). This also means that a fuzzy rule-based system should be able to infer a proper conclusion for every input (De Oliveira, 1999). Completeness is justified by the fact that in human reasoning there will never be a gap of description within the range of the variable (Herrmann, 1997).

In interpretable fuzzy models, another definition of the completeness constraint is known as α-completeness can be defined with the following expression:

∀ 𝑥 ∈ 𝑋, ∋ 𝐴 ∈ 𝐹, 𝜇𝐴(𝑥) ≥ 𝛼 (3.4)

α-completeness is preferred because it guaranties that every element in the universe of discourse is well presented by a fuzzy set with a minimum degree 𝛼, given the rise to the concept of strong coverage (De Oliveira, 1999).

 Distinguishability

Distinguishability means that each fuzzy set should be distinct enough from the other fuzzy sets defined on the same universe of discourse so they represent distinct concepts that can be assigned to linguistic terms with clear and different semantic meanings (De Oliveira, 1999; Mencar & Fanelli, 2008; Zhou & Gan, 2008).

Distinguishability is a basic and essential constraint that has been widely adopted in interpretable fuzzy modelling literature (see, for example, (De Oliveira, 1999; Espinosa & Vandewalle, 2000; S. Guillaume & Charnomordic, 2004; Mencar, Castellano, &

0.0 1.0

Figure 3.2 An example of bad coverage of the universe of discourse, some elements (potted points) in the universe of discourse are not covered

University

52

Fanelli, 2007; Mencar & Fanelli, 2008; Setnes, Babuska, Kaymak, & Van Nauta Lemke, 1998). This property offers a number of advantages for interpretable fuzzy modeling including: reduce redundancy, which may be present in the form of similar fuzzy sets that represent compatible concepts (Setnes et al., 1998), and more importantly the ease of the linguistic interpretation of the model since fuzzy sets represent well-separated concepts (Setnes et al., 1998). Thus, when the distinguishability is lost, especially during an accuracy-oriented learning process, it is difficult to assign distinct linguistic terms to fuzzy sets. An example of distinguishable and non-distinguishable fuzzy sets is shown in Figure 3.3. As can be seen in the Figure 3.3, it is easy to assign labels or linguistic terms such as: very low, low, average, large and very large to the distinguishable fuzzy sets while it is difficult to do that for non- distinguishable fuzzy sets as most of them represent almost the same concept. During the design process of fuzzy models, other constraints such as coverage may require overlapping fuzzy sets, and thus distinguishability should be carefully balanced with other constraints (Mencar & Fanelli, 2008).

Figure 3.3 Example of non-distinguishable fuzzy sets (right) and distinguishable fuzzy sets (left)

University

53

c) - Interpretability constraints for the rule base

 Consistency

Consistency means the absence of contradictory in the rule base, i.e., if two or more rules have similar antecedents, they should have different consequents (Dubois, Prade, & Ughetto, 1997; Serge Guillaume, 2001; Y Jin, von Seelen, & Sendhoff, 1999). In a knowledge base of expert systems, inconsistency in the rule base occurs when there are two rules in the form 𝐴 → 𝐵 and 𝐴 → 𝐶, where 𝐵 and 𝐶 are mutually exclusive concepts. Inconsistency means that a statement and its negation can be derived from the same knowledge base which makes it useless. In fuzzy logic, consistency is a matter of degree since the statement 𝐴⋀¬𝐴 can be true with a certain degree greater than zero. Thus, in the fuzzy rule base, partial inconsistency can be tolerated if it is acceptably small (Mencar & Fanelli, 2008). Checking the fuzzy rule base for consistency remains an important constraint for its validation and interpretation (Dubois et al., 1997; Mencar & Fanelli, 2008).

 Type of fuzzy rules

Actually, there are two well-known types of fuzzy rule-based systems, namely, Mamdani and Takagi systems. The only difference between these two models lies in their consequent part; as Mamdani model uses fuzzy set in its consequent part while Takagi model uses a linear real function. Actually, Mamdani model is more interpretable because a fuzzy set is suitable to express human perception knowledge while the linear function in the consequent of Takagi model does not represent any physical meaning (Zhou & Gan, 2008). For this reason, most of the researchers whose interpretability is their main objective have been using Mamdani model to build their fuzzy rule-based systems (Cordon, 2011).

University

54