Abstract
Fuzzy systems are commonly considered suitable tools to express knowledge in a human comprehensible fashion. This kind of characterization makes them eligible for being applied in several contexts where interpretability is a major issue and humans may profit from a self-explanatory form of automatic computation. However, fuzzy systems are not interpretable per se and the simple adoption of a natural language representation does not guarantee full acceptance and plain confidence among human experts. A number of constraints and criteria have been proposed in literature to drive the design and the construction of fuzzy systems so that they can be deemed interpretable. The aim of this chapter is to provide a thorough exposition of constraints and criteria which have been variously adopted in the research community in this context of investigation. Due to their heterogeneity and multiplicity, we resorted to a particular arrangement of their presentation. By following a hierarchical organization, we start from the basic constituents of a fuzzy system (namely, the fuzzy sets) and we go through the other design levels where compound elements are involved: for each level, an exhaustive review of interpretability constraints and criteria is expounded supplying formal definitions, illustrative examples, and bibliographical references.
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Notes
- 1.
For ease of referencing, both constraints and criteria will be defined as “constraints” when they are introduced.
- 2.
It is possible that \(\left[ A\right] _{1}=\emptyset \) and \({{\,\mathrm{hgt}\,}}(A)=1\) if the membership function \(\mu _{A}\) is asymptotically approaching 1; nevertheless, this is quite a rare case in fuzzy modeling.
- 3.
Natura non facit saltus (Carl von Linné, Philosophia Botanica, 1751).
- 4.
It should be remarked that the notion of convexity of fuzzy sets is different from the notion of convexity of functions. Actually, convex fuzzy sets are often characterized by membership functions that are not convex.
- 5.
A fuzzy vector is a vector of fuzzy numbers.
- 6.
In this book we adopt the unimodality concept as generally defined in Mathematics. Alternative definitions exist in fuzzy set literature, e.g. (Gitman and Levine 1970).
- 7.
i.e. \(\forall x \in U:\min \{A(x),B(x)\}=0\).
- 8.
We are assuming 1-complement as negator operator. This is the simplest definition of negation and is widely used in literature.
- 9.
The simplest definition of the cardinality of a fuzzy set is \(\left| A\right| =\int _{U}\mu _{A}\left( x\right) dx\). A deeper discussion on fuzzy set cardinalities can be found in Wygralak (2000).
- 10.
In a weak sense, i.e. with the smallest prototype.
- 11.
If completeness is satisfied, then for each input \(\mathbf {x}\) there exists a rule R whose antecedent \(\mathbf {A}\) is satisfied with \(\mu _{\mathbf {A}}(\mathbf {x})>0\). Since the antecedent \(\mathbf {A}\) is defined by the conjunction of fuzzy sets (one for each feature), i.e. \( \mathbf {A} = A_1 \times ... \times A_n \) then \(0<\mu _{\mathbf {A}}(\mathbf {x})=\mu _{A_1}(x_1) \otimes ... \otimes \mu _{A_n}(x_n)\) . But this implies that \(\forall i: \mu _{A_i}(x_i)>0\). Therefore, \(\forall x_i, \exists A_i : \mu _{A_i}(x_i)>0\), i.e. coverage is satisfied.
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Alonso Moral, J.M., Castiello, C., Magdalena, L., Mencar, C. (2021). Interpretability Constraints and Criteria for Fuzzy Systems. In: Explainable Fuzzy Systems. Studies in Computational Intelligence, vol 970. Springer, Cham. https://doi.org/10.1007/978-3-030-71098-9_3
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