Fuzzy rules and fuzzy inference systems have become the central point of fuzzy modeling since the early beginnings of fuzzy systems. Hence, distinct desirable properties of the rules, their models, and the whole systems are studied. The non-conflictness of rules and/or the preservation of modus ponens seem to be considered the most crucial one(s). However, under the standard setting, such properties are semantically equivalent to the continuity of the modeled dependency. A natural question arises whether such a requirement is consistent with semantics of fuzzy rules. While the answer is positive in the case of implicative rules, in the case of the more often used Mamdani-Assilian rules, we may consider another perspective. This article foreshadows another perspective that could lead to the investigation of a desirable property of the Mamdani-Assilian model that is diferent from continuity.
briefly recall them. A fuzzy set is defined as a mapping from non-empty universe to the
unit interval, i.e., : → [
Definition 1.
Fuzzy set ∈ ℱ ( ) is called • normal if there exists ∈ such that () = 1 ; • bounded if there exists ∈ such that () = 0 .
ably the most often accepted setting stems from a residuated lattice ⟨[
IF is THEN is , = 1, . . . , (1) where ∈ ℱ () and ∈ ℱ ( ) are antecedent and consequent fuzzy sets, respectively.
If a fuzzy relational inference is considered, fuzzy rule base (1) is modeled by a single fuzzy relation on the Cartesian product of both universes × . These fuzzy relations are either formed as the conjunction of implications (so-called implicative model): or as the disjunction of conjunctions (so-called Mamdani-Assilian model):
ˆ(, ) = ⋀︁ (() → ()) , =1
ˇ(, ) = ⋁︁ (() ⊗ ()) .
=1
structure of fuzzy relational inference systems. Another crucial block is the inference mechanism.
It deals with a fuzzy set ∈ ℱ () as an input, and with a direct use of a fuzzy relational
model of the given fuzzy rule base, maps it to the output fuzzy set ∈ ℱ ( ). Mathematically,
it is defined as an image of a fuzzy set under a fuzzy relation—a concept derived from fuzzy
relational compositions/products [
The more common one is known under the name Compositional rule of inference (CRI) [5], and for an input ∈ ℱ () and a relational model of a fuzzy rule base ∈ ℱ ( × ) it is defined as follows: ( ∘ )() = ⋁︁ (() ⊗ (, )) . (4)
∈
Another alternative stems from the composition called Bandler-Kohout subproduct [6, 7], and carries the same name. It is defined by the following formula: ( ◁ )() = ⋀︁ (() → (, )) ,
∈ and it is worth mentioning that it has been firstly proposed as an alternative to CRI already in [8] and later on it has been shown to be an equally appropriate inference mechanism [9].
Note that if we consider a crisp input ′ ∈ represented by a singleton, i.e., by ′ ∈ ℱ () such that ′(′) = 1 and ′() = 0 elsewhere, both inferences degenerate to a simple and practical substitution that makes the choice between inference mechanisms redundant: (′ ∘ )() = ( ◁ )() = (′, ).
There are numerous research studies focusing on the preservation of desirable properties of fuzzy rules or whole inference systems [10, 11, 12, 13]. Vast majority of them include the preservation of modus ponens or consistency of fuzzy rules. In the rfist case, we consider the (2) (3) (5) input to be equal to one of the antecedents , and investigate whether the inferred output is equal to , which leads to the solvability of fuzzy relational equations [14, 15, 16]. In the latter case, most works consider concepts that define the conflict as the existence of two rules with equal or very similar antecedents but dissimilar consequents. Let us recall, e.g., the coherence.
Definition 2. [17] Fuzzy relation ˆ ∈ ℱ ( × ) is called coherent if for any ∈ there exists ∈ such that ˆ(, ) = 1.
Clearly, the coherence of a fuzzy relation can be defined for an arbitrary fuzzy relation, not only restrictively as a property of the implicative model ˆ of fuzzy rules; however, we avoid doing so by purpose as the definition was intended for ˆ. Using it for other fuzzy relations, e.g., for ˇ, would be meaningless. An analogous approach for the Mamdani-Assilian model stemming from the fact that conflicting rules in this model do not lower membership degrees, but generate non-convex results, was investigated in [18].
Theorem 1. Let us consider the following systems of fuzzy relational equations ∘ (◁) = , = 1, . . . , . (6) Then the system is solvable if and only if ˆ ( ˇ) is its solution.
Theorem 1 states that ˆ has the primary position for the CRI inference while ˇ keeps the same position for the Bandler-Kohout subproduct inference. Although we may find conditions under which the opposite combinations also preserve modus ponens [19, 20], they usually bring other disadvantages as long as we do not accept additional restrictions, e.g., on the choice of the algebra, see [21]. Therefore, we avoid going into a deeper discussion on assumptions for these combinations, and we simply consider ˆ to be the predetermined model for the inference ∘ and vice-versa, and analogously we assume that ˇ and ◁ constitute such a pair too. Note that the latter pair does not constitute a logical inference; however, Mamdani-Assilian rules have their meaningful logical models that have been successfully studied by logical tools [22, 23].
mapping has been demonstrated in [24].
In principle, for two identical inputs (antecedents), considering two diferent outputs (consequents) is in contradiction with the definition of a function in set theory. And if we consider it in a bit weaker form, i.e., two close inputs cannot lead to two outputs located far from each other, we come to the Lipschitz continuity. And taking into account that the consistency means the nonexistence of conflicting rules, i.e, rules that at the same time impose diferent outputs for the same inputs, the connection of the preservation of modus ponens and the continuity is straightforward.
The same view is mirrored in the definition of the coherence. Indeed, → = 1 if and only if ≤ for any residual implication. Hence, the requirement of the existence of ∈ such that ˆ(, ) = 1 actually means that for all rules and for arbitrary ∈ , there has to be a such that () ≤ (). Therefore, for any input, there is an element in the output universe that belongs to any consequent in a degree higher than or equal to the degree enforced by the respective antecedent. Suppose that there are two rules with identical and normal antecedents 1 and 2 but completely diferent consequents. Then, obviously, for an input such that 1() = 2() = 1, such cannot be found.
If we consider the argumentation mentioned above, it corresponds to the semantics expressed in the conditional form of (1) that is mirrored in the implicative model ˆ given by (2). However, the Mamdani-Assilian model ˇ actually expresses rather the semantics “ is AND is ”, with the disjunctive aggregation by the connective OR, see [11, 25]. And then, one can doubt what is wrong with having two identical or close inputs and two diferent outputs. If rules specify “options” or possibilities (see [11]) a decision-maker has, then there seems to be nothing wrong with rules with similar or even equal antecedents but contradictory consequents.
IF is THEN is , IF is THEN is ℎ.
to hold simultaneously (connected by the AND connective), we observe that it is impossible to fulfill their requirement to change the direction of our vehicle to the left and to the right at the same time, and we obtain inconsistency. However, if we consider the disjunctive interpretation of these rules: is AND is , OR is AND is ℎ, we see that we are given two options for going around, and it is up to us which one we choose. However, what is important, these rules implicitly exclude the direction forward causing the equally represented. an inclusion ∘ ⊆
ˆ information, fulfils the inclusion. crash with the obstacle. Note also that, provided that there are three possible directions (left, right and forward), if there were also the third rule “ is AND is ”, these three rules together would bear no information, since all possible directions would be
would be suficient. However, this inclusion is ensured automatically, but the problem is that even an empty fuzzy set on the output of the system, which carries no
Proposition 1. Let be normal and let ˆ be coherent. Then ( ∘ ˆ) ∈ ℱ ( ) is normal.
Proof: Let ′ ∈ be a point of normality of . Then, ( ∘ ˆ)() = ⋁︁ (︁ ∈
︁) () ⊗ ˆ(, ) ≥ (′) ⊗ ˆ(′, ) = ˆ(′, ) not be satisfied by trivial outputs. and, as ˆ is coherent, for the given ′ there has to exist some ′ such that ˆ(′, ′) = 1.
Consequently, if some antecedent is the input, we automatically obtain the desirable inclusion ∘ ⊆
ˆ of ˆ, we have ensured that the output will be meaningful. Thus, the inclusion ∘ ⊆ will ˆ , and, jointly with the assumption on the normality of and coherence
However, as we know, the situation between ∘ and ˆ on the one side and ◁ and ˇ on the other side is dual [26]. We obtain, for example, an automatic preservation of the following inclusion: ◁ ˇ ⊇
, and, with respect to the inference ◁ that is based on an implication, brings us to defining a concept for ˇ that would be dual to the coherence for ˆ. the meaningless output would be a fuzzy set equal to 1 on the whole universe . This naturally Definition 3. ∈ such that ˇ(, ) = 0.
Fuzzy relation ˇ ∈ ℱ ( × ) is called concise if for any ∈ there exists □ □
fuzzy relation ˇ is closely related to boundedness. This immediately leads to the following proposition.
Proposition 2. Let be normal and let ˇ be concise. Then ( ◁ ˇ) ∈ ℱ ( ) is bounded.
Proof: Let ′ ∈ be a point of normality of . Then ( ◁ ˇ)() = ⋀︁ (︀ () → ˇ(, ))︀ ≤ (′) → ˇ(′, ) = ˇ(′, )
∈ and, as ˇ is concise, for the given ′ there has to exist some ′ such that ˇ(′, ′) = 0.
And again, if some antecedent is the input, we automatically obtain the desirable inclusion ◁ ˇ ⊇ , and, jointly with the assumption on the normality of and conciseness of ˇ, we have ensured that the output will be meaningful. Thus, the inclusion will not be satisfied by trivial outputs.
the input is significant and the fuzzy rule base model is correct (coherent or concise), then the output also brings a significant information”. We only have to carefully distinguish between two diferent models and inferences, which influence what is a ‘significant information’ .
especially the Mamdani-Assilian ones. Consider their crisp variant. Thus, let all antecedents and consequents be classical sets ⊂ , ⊂ , e.g., intervals. Let the antecedents meet the ifnitary condition, that is, let for each antecedent there is a point such that ∈ but ̸∈ for any ̸= . Then it is easy to prove that ∘ ˆ = and ◁ ˇ = for all .
Now, let us consider that the rule base (still in the crisp case) describes the driving example of avoiding the obstacle introduced above. There, we have two rules such that = and ̸= , and they are even disjoint. In the case of implicative rules, this amounts to a clear conflict, incoherence, or inconsistency. Indeed, if the rules were viewed as special axioms of some theory, such a theory would be contradictory. However, if we do not view the rules as special axioms in the conditional form but consider the Mamdani-Assilian form that determines possibilities, we do not observe any contradiction. Then, for the rule base containing two rules with = and disjoint ̸= , we obtain ◁ ˇ = ∪ .
giving us an option to avoid an obstacle located in front of us by going to the left, the other one giving an option to go to the right, and the observation is that there is an obstacle in front of us, we should deduce the conclusion that we may go either to the left or to the right.
The principal problem is that this view rather fits decision-making situations, not control situations with expected functional dependency, where a defuzzification is employed. Consequently, most of the defuzzifications such as COG or COA would “average” the output which would lead to a frontal collision with the obstacle. However, the problem does not lie in the rules nor in the union of disjoint intervals on the output but in an inappropriately chosen combination of the tool (Mamdani-Assilian interpretation of rules) and the modeled functional dependency. Let us shortly come back to the solvability of systems of fuzzy relational equations. In [27] and then independently in [28], so called finitary (originally boundary) condition has been defined. Definition 4. Let = {1, . . . , } be the index set and let be normal for ∈ . Then are said to meet the finitary condition if for any ∈ there exists an ∈ such that () = 1 and () = 0 for any ̸= .
Theorem 2. [28] Let meet the finitary condition. Then, ◁ ˇ = , = 1, . . . , .
◁ ˇ ⊇ ∪ .
∈
no assumptions on the closeness of the consequents ; however, it is simply due to the fact that finitarity has to be understood as “suficient disjointness” of the input (fuzzy) nodes. And if input nodes are far from each other, the respective output nodes can be arbitrary, and none of the above-mentioned properties is harmed.
when a functional relationship is assumed, which is often not the case in decision-making situations, where two or more (disjoint) choices are possible and natural. For such cases, Mamdani-Assilian rules seem to perfectly fit the goal with their semantics. However, it does not mean that we should not expect any reasonable behavior of the system or no reasonable properties should be preserved. Analogously to the case of crisp inputs, where the conciseness of the fuzzy relation ˇ played the “good property” role, we should be willing to obtain not too general outputs of the system. The following sequence of propositions will provide us with a knowledge showing that Mamdani-Assilian systems can give us natural and reasonable outputs no matter the fact that we are harming the modus ponens.
Proposition 3. Let = {1, . . . , } be the index set and let , ∈ be such that = . Then, (7) (8) □ (9)
the set { | ∈ ∖ {}} meet the finitary condition. Then, Proposition 4. Let = {1, . . . , } be the index set and let , ∈ be such that = . Let ◁ ˇ = ∪ . we get
Proof: Using the isotonicity of → in the second argument and the property → ( ⊗ ) ≥ , ( ◁ ˇ)() = ⋀︁ ⋁︁ (() ⊗ ())
)︃ ∈ ∈
︃( ≥ () ∨ () . ≥ ⋀︁ ((() → (() ⊗ ())) ∨ (() → ( () ⊗ ()))) )︃ (︃
⋁︁ ∈∖{,} ∈ ⎞ ︃( ⎛ ∈
∈
Proof: Let ′ ∈ be such that (′) = 1 and (′) = 0 for any ̸= , . Then we get ⋁︁ (() ⊗ ()) ≤ (′) →
⋁︁ ((′) ⊗ ()) ⋁︁ ∈∖{,} = 1 → ⎝(1 ⊗ ()) ∨ (1 ⊗ ()) ∨ ((′) ⊗ ())⎠ = () ∨ () ∨
(0 ⊗ ()) = () ∨ () .
We have recalled the basic components of fuzzy inference systems and the most frequently discussed desirable properties, namely, the consistency of rules that is also mirrored in the preservation of modus ponens. We showed that under the standard setting this property leads to the functionality or continuity of the model. However, taking into account the semantic meaning of the Mamdani-Assilian rules, this requirement does not seem that natural. An alternative approach to investigating the “correct” behavior of Mamdani-Assilian rules is foreshadowed. )︃ □
Systems 160 (2009) 1005–1036. [5] L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems, Man, and Cybernetics 3 (1973) 28–44. [6] W. Bandler, L. J. Kohout, Fuzzy relational products and fuzzy implication operators, in:
London, 1978. [7] W. Bandler, L. J. Kohout, Semantics of implication operators and fuzzy relational products, [8] W. Pedrycz, Applications of fuzzy relational equations for methods of reasoning in presence of fuzzy data, Fuzzy Sets and Systems 16 (1985) 163–175. [9] M. Štěpnička, B. Jayaram, On the suitability of the Bandler-Kohout subproduct as an inference mechanism, IEEE Transactions on Fuzzy Systems 18 (2010) 285–298. [10] M. Mizumoto, H.-J. Zimmermann, Comparison of fuzzy reasoning methods, Fuzzy Sets and Systems 8 (1982) 253–283. [11] D. Dubois, H. Prade, What are fuzzy rules and how to use them, Fuzzy Sets and Systems 84 (1996) 169–185. [12] B. Moser, M. Navara, Fuzzy controllers with conditionally firing rules, IEEE Transactions on Fuzzy Systems 10 (2002) 340–348. [13] M. Štěpnička, S. Mandal, Fuzzy inference systems preserving Moser–Navara axioms, Fuzzy
Sets and Systems 338 (2018) 97–116. [14] E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control 30 (1976) 38–48. [15] B. De Baets, Analytical solution methods for fuzzy relational equations, in: D. Dubois,
[16] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relation Equations and Their Applications to Knowledge Engineering, Kluwer, Boston, 1989. [17] D. Dubois, H. Prade, L. Ughetto, Checking the coherence and redundancy of fuzzy knowledge bases, IEEE Transactions on Fuzzy Systems 5 (1997) 398–417. [18] D. Coufal, Coherence index of radial conjunctive fuzzy systems, in: P. M. et al. (Ed.),
[19] F. Klawonn, Fuzzy points, fuzzy relations and fuzzy functions, in: V. Novák, I. Perfilieva (Eds.), Discovering the World with Fuzzy Logic, Springer, Berlin, 2000, pp. 431–453. [20] I. Perfilieva, L. Nosková, System of fuzzy relation equations with inf- → composition:
[21] M. Štěpnička, B. Jayaram, Y. Su, A short note on fuzzy relational inference systems, Fuzzy
Sets and Systems 338 (2018) 90–96. [22] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998. [23] V. Novák, I. Perfilieva, J. Močkoř, Mathematical Principles of Fuzzy Logic, Kluwer, Boston, 1999. [24] I. Perfilieva, S. Lehmke, Correct models of fuzzy if-then rules are continuous, Fuzzy Sets and Systems 157 (2006) 3188–3197. [25] U. Bodenhofer, M. Daňková, M. Štěpnička, V. Novák, A plea for the usefulness of the deductive interpretation of fuzzy rules in engineering applications, in: Proc. 16th IEEE Int.
Conf. on Fuzzy Systems, London, 2007, pp. 1567–1572. [26] R. Bělohlávek, Sup-t-norm and inf-residuum are one type of relational product: Unifying framework and consequences, Fuzzy Sets and Systems 197 (2012) 45–58. [27] F. Chung, T. Lee, A new look at solving a system of fuzzy relational equations, Fuzzy Sets and Systems 88 (1997) 343–353. [28] I. Perfilieva, Finitary solvability conditions for systems of fuzzy relation equations, Information Sciences 234 (2013) 29–43. [29] M. Štěpnička, B. Jayaram, Interpolativity of at-least and at-most models of monotone fuzzy rule bases with multiple antecedent variables, Fuzzy Sets Systems 297 (2016) 26–45.