Mamdani-Assilian rules: with or without continuity? Antonín Dvořák1 , Martin Štěpnička1 1 CE IT4Innovations – IRAFM, University of Ostrava, 30. dubna 22, Ostrava, 70103, Czech Republic Abstract 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 different from continuity. Keywords Fuzzy rules, Mamdani-Assilian rules, inference mechanism, modus ponens, functionality 1. Preliminaries 1.1. Basic concepts Since we assume that all readers are familiar with the basic concepts of fuzzy sets, we only briefly recall them. A fuzzy set 𝐴 is defined as a mapping from non-empty universe 𝑋 to the unit interval, i.e., 𝐴 : 𝑋 → [0, 1]. Let us also recall the denotation of the set of all fuzzy sets on a given universe: ℱ(𝑋) = {𝐴 | 𝐴 : 𝑋 → [0, 1]}. A fuzzy relation is a fuzzy set on a Cartesian product of universes, e.g., a binary fuzzy relation 𝑅 can be an element of ℱ(𝑋 × 𝑌 ). Let us recall the definition of two fundamental properties of a fuzzy set. Definition 1. Fuzzy set 𝐴 ∈ ℱ(𝑋) is called • normal if there exists 𝑥 ∈ 𝑋 such that 𝐴(𝑥) = 1 ; • bounded if there exists 𝑥 ∈ 𝑋 such that 𝐴(𝑥) = 0 . The operations on fuzzy sets can be approached from distinct perspectives; however, prob- ably the most often accepted setting stems from a residuated lattice ⟨[0, 1], ∧, ∨, ⊗, →, 0, 1⟩; therefore, we adopt it too. Thus, any operations on fuzzy sets appearing in this article come from the above-mentioned algebraic structure, where ⊗ is a left-continuous t-norm [1] and → is its adjoint fuzzy implication [2]. WILF’21: The 13th International Workshop on Fuzzy Logic and Applications, December 20–22, 2021, Vietri sul Mare, Italy " Antonin.Dvorak@osu.cz (A. Dvořák); Martin.Stepnicka@osu.cz (M. Štěpnička) ~ https://ifm.osu.eu/antonin-dvorak/10001/ (A. Dvořák); https://ifm.osu.eu/martin-stepnicka/366/ (M. Štěpnička)  0000-0002-0611-491X (A. Dvořák); 0000-0002-0285-075X (M. Štěpnička) © 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings http://ceur-ws.org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) 1.2. Fuzzy rules If we omit the difference between the syntactical (linguistic) level and the semantic level, fuzzy rules can be viewed as conditional sentences: IF 𝑥 is 𝐴𝑖 THEN 𝑦 is 𝐵𝑖 , 𝑖 = 1, . . . , 𝑛 (1) where 𝐴𝑖 ∈ ℱ(𝑋) and 𝐵𝑖 ∈ ℱ(𝑌 ) 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 𝑋 × 𝑌 . These fuzzy relations are either formed as the conjunction of implications (so-called implicative model): 𝑛 ⋀︁ ˆ (𝑥, 𝑦) = 𝑅 (𝐴𝑖 (𝑥) → 𝐵𝑖 (𝑦)) , (2) 𝑖=1 or as the disjunction of conjunctions (so-called Mamdani-Assilian model): 𝑛 ⋁︁ ˇ (𝑥, 𝑦) = 𝑅 (𝐴𝑖 (𝑥) ⊗ 𝐵𝑖 (𝑦)) . (3) 𝑖=1 Fuzzy relational models of a fuzzy rule base constitute one of the major blocks in the block structure of fuzzy relational inference systems. Another crucial block is the inference mechanism. It deals with a fuzzy set 𝐴 ∈ ℱ(𝑋) 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 𝐵 ∈ ℱ(𝑌 ). Mathematically, it is defined as an image of a fuzzy set under a fuzzy relation—a concept derived from fuzzy relational compositions/products [3, 4]. The more common one is known under the name Compositional rule of inference (CRI) [5], and for an input 𝐴 ∈ ℱ(𝑋) and a relational model of a fuzzy rule base 𝑅 ∈ ℱ(𝑋 × 𝑌 ) it is defined as follows: ⋁︁ (𝐴 ∘ 𝑅)(𝑦) = (𝐴(𝑥) ⊗ 𝑅(𝑥, 𝑦)) . (4) 𝑥∈𝑋 Another alternative stems from the composition called Bandler-Kohout subproduct [6, 7], and carries the same name. It is defined by the following formula: ⋀︁ (𝐴 ◁ 𝑅)(𝑦) = (𝐴(𝑥) → 𝑅(𝑥, 𝑦)) , (5) 𝑥∈𝑋 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 𝑥′ ∈ 𝑋 represented by a singleton, i.e., by 𝐴′ ∈ ℱ(𝑋) such that 𝐴′ (𝑥′ ) = 1 and 𝐴′ (𝑥) = 0 elsewhere, both inferences degenerate to a simple and practical substitution that makes the choice between inference mechanisms redundant: (𝐴′ ∘ 𝑅)(𝑦) = (𝐴 ◁ 𝑅)(𝑦) = 𝑅(𝑥′ , 𝑦). 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 first case, we consider the input 𝐴 to be equal to one of the antecedents 𝐴𝑖 , and investigate whether the inferred output 𝐵 is equal to 𝐵𝑖 , 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 𝑅 ˆ ∈ ℱ(𝑋 × 𝑌 ) is called coherent if for any 𝑥 ∈ 𝑋 there ˆ (𝑥, 𝑦) = 1. exists 𝑦 ∈ 𝑌 such that 𝑅 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 𝑅 ˆ of fuzzy rules; however, we avoid ˆ doing so by purpose as the definition was intended for 𝑅. Using it for other fuzzy relations, ˇ , would be meaningless. An analogous approach for the Mamdani-Assilian model e.g., for 𝑅 stemming from the fact that conflicting rules in this model do not lower membership degrees, but generate non-convex results, was investigated in [18]. 2. What are the desirable properties? 2.1. Preservation of modus ponens The preservation of modus ponens leads to the solvability of fuzzy relational equations. We recall the most fundamental results that can be found in the literature cited above. Theorem 1. Let us consider the following systems of fuzzy relational equations 𝐴𝑖 ∘ (◁)𝑅 = 𝐵𝑖 , 𝑖 = 1, . . . , 𝑛 . (6) ˆ (𝑅 Then the system is solvable if and only if 𝑅 ˇ ) is its solution. Theorem 1 states that 𝑅 ˆ has the primary position for the CRI inference while 𝑅 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 𝑅 ˆ to be the predetermined model for the inference ∘ and vice-versa, and analogously we assume that 𝑅 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]. 2.2. Preservation of modus ponens as a sort of functionality or continuity In this section, we show that both the preservation of modus ponens and consistency are closely related to each other and also to the set-theoretic definition of a function, and, consequently, also to (the Lipschitz-type of) the continuity. If there are inconsistent rules in (1), then there does not exist any fuzzy relation that would solve the related system of fuzzy relational equations. Thus, modus ponens cannot be preserved. The equivalence of the solvability with (a sort of Lipschitz-like) continuity of the related fuzzy mapping has been demonstrated in [24]. In principle, for two identical inputs (antecedents), considering two different outputs (conse- quents) 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 different 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, 𝑎 → 𝑏 = 1 if and only if 𝑎 ≤ 𝑏 for any residual implication. Hence, the requirement of the existence of 𝑦 ∈ 𝑌 such that 𝑅 ˆ (𝑥, 𝑦) = 1 actually means that for all rules and for arbitrary 𝑥 ∈ 𝑋, there has to be a 𝑦 such that 𝐴𝑖 (𝑥) ≤ 𝐵𝑖 (𝑦). 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 𝐴1 and 𝐴2 but completely different consequents. Then, obviously, for an input 𝑥 such that 𝐴1 (𝑥) = 𝐴2 (𝑥) = 1, such 𝑦 cannot be found. 2.3. Another view 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 𝑅ˆ given by (2). However, ˇ the Mamdani-Assilian model 𝑅 actually expresses rather the semantics “𝑥 is 𝐴𝑖 AND 𝑦 is 𝐵𝑖 ”, 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 different 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. Consider, for example, the following rules for going around an obstacle: IF 𝑜𝑏𝑠𝑡𝑎𝑐𝑙𝑒 is 𝑓 𝑟𝑜𝑛𝑡 THEN 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 is 𝑙𝑒𝑓 𝑡, IF 𝑜𝑏𝑠𝑡𝑎𝑐𝑙𝑒 is 𝑓 𝑟𝑜𝑛𝑡 THEN 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 is 𝑟𝑖𝑔ℎ𝑡. Clearly, if we consider the implicative interpretation of these rules, that is, if they are supposed 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: 𝑜𝑏𝑠𝑡𝑎𝑐𝑙𝑒 is 𝑓 𝑟𝑜𝑛𝑡 AND 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 is 𝑙𝑒𝑓 𝑡, OR 𝑜𝑏𝑠𝑡𝑎𝑐𝑙𝑒 is 𝑓 𝑟𝑜𝑛𝑡 AND 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 is 𝑟𝑖𝑔ℎ𝑡, 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 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 “𝑜𝑏𝑠𝑡𝑎𝑐𝑙𝑒 is 𝑓 𝑟𝑜𝑛𝑡 AND 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 is 𝑓 𝑜𝑟𝑤𝑎𝑟𝑑”, these three rules together would bear no information, since all possible directions would be equally represented. From the logical inference point of view, it is not necessary to expect the equality in (6), but an inclusion 𝐴𝑖 ∘ 𝑅ˆ ⊆ 𝐵𝑖 would be sufficient. 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 information, fulfils the inclusion. Taking this into account and adding the coherence as an additional property to the assump- tions, we may obtain the following proposition ensuring that we do not get a meaningless or even empty fuzzy set on the output. ˆ be coherent. Then (𝐴 ∘ 𝑅 Proposition 1. Let 𝐴 be normal and let 𝑅 ˆ ) ∈ ℱ(𝑌 ) is normal. Proof: Let 𝑥′ ∈ 𝑋 be a point of normality of 𝐴. Then, ⋁︁ (︁ )︁ ˆ )(𝑦) = (𝐴 ∘ 𝑅 ˆ (𝑥, 𝑦) ≥ 𝐴(𝑥′ ) ⊗ 𝑅 𝐴(𝑥) ⊗ 𝑅 ˆ (𝑥′ , 𝑦) = 𝑅 ˆ (𝑥′ , 𝑦) 𝑥∈𝑋 and, as 𝑅ˆ is coherent, for the given 𝑥′ there has to exist some 𝑦 ′ such that 𝑅 ˆ (𝑥′ , 𝑦 ′ ) = 1. □ Consequently, if some antecedent 𝐴𝑖 is the input, we automatically obtain the desirable inclusion 𝐴𝑖 ∘ 𝑅ˆ ⊆ 𝐵𝑖 , and, jointly with the assumption on the normality of 𝐴𝑖 and coherence ˆ , we have ensured that the output will be meaningful. Thus, the inclusion 𝐴𝑖 ∘ 𝑅 of 𝑅 ˆ ⊆ 𝐵𝑖 will not be satisfied by trivial outputs. The remarks above as well as Proposition 1 are still closely related to the functionality/continuity. However, as we know, the situation between ∘ and 𝑅 ˆ on the one side and ◁ and 𝑅 ˇ on the other side is dual [26]. We obtain, for example, an automatic preservation of the following inclusion: 𝐴𝑖 ◁ 𝑅 ˇ ⊇ 𝐵𝑖 , and, with respect to the inference ◁ that is based on an implication, the meaningless output would be a fuzzy set equal to 1 on the whole universe 𝑌 . This naturally brings us to defining a concept for 𝑅 ˇ that would be dual to the coherence for 𝑅 ˆ. Definition 3. Fuzzy relation 𝑅ˇ ∈ ℱ(𝑋 × 𝑌 ) is called concise if for any 𝑥 ∈ 𝑋 there exists ˇ (𝑥, 𝑦) = 0. 𝑦 ∈ 𝑌 such that 𝑅 As the concept of coherence is related to the concept of normality, the concept of a concise ˇ is closely related to boundedness. This immediately leads to the following fuzzy relation 𝑅 proposition. ˇ be concise. Then (𝐴 ◁ 𝑅 Proposition 2. Let 𝐴 be normal and let 𝑅 ˇ ) ∈ ℱ(𝑌 ) is bounded. Proof: Let 𝑥′ ∈ 𝑋 be a point of normality of 𝐴. Then ⋀︁ (︀ ˇ )(𝑦) = ˇ (𝑥, 𝑦) ≤ 𝐴(𝑥′ ) → 𝑅 ˇ (𝑥′ , 𝑦) = 𝑅 ˇ (𝑥′ , 𝑦) )︀ (𝐴 ◁ 𝑅 𝐴(𝑥) → 𝑅 𝑥∈𝑋 ˇ is concise, for the given 𝑥′ there has to exist some 𝑦 ′ such that 𝑅 and, as 𝑅 ˇ (𝑥′ , 𝑦 ′ ) = 0. □ And again, if some antecedent 𝐴𝑖 is the input, we automatically obtain the desirable inclusion 𝐴𝑖 ◁ 𝑅 ˇ ⊇ 𝐵𝑖 , and, jointly with the assumption on the normality of 𝐴𝑖 and conciseness of 𝑅 ˇ, we have ensured that the output will be meaningful. Thus, the inclusion will not be satisfied by trivial outputs. Propositions 1 and 2 state an analogous knowledge that could be expressed as follows: “if 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 different models and inferences, which influence what is a ‘significant information’. 2.4. Is functionality always desirable? Let us move a bit forward in our thoughts on the intuitively expected properties of fuzzy rules, especially the Mamdani-Assilian ones. Consider their crisp variant. Thus, let all antecedents and consequents be classical sets 𝐴𝑖 ⊂ 𝑋, 𝐵𝑖 ⊂ 𝑌 , e.g., intervals. Let the antecedents meet the finitary condition, that is, let for each antecedent 𝐴𝑖 there is a point 𝑥𝑖 such that 𝑥𝑖 ∈ 𝐴𝑖 but 𝑥𝑖 ̸∈ 𝐴𝑗 for any 𝑗 ̸= 𝑖. Then it is easy to prove that 𝐴𝑖 ∘ 𝑅ˆ = 𝐵𝑖 and 𝐴𝑖 ◁ 𝑅 ˇ = 𝐵𝑖 for all 𝑖. 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 𝐴𝑖 = 𝐴𝑗 and 𝐵𝑖 ̸= 𝐵𝑗 , 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 𝐴𝑖 = 𝐴𝑗 and disjoint 𝐵𝑖 ̸= 𝐵𝑗 , we obtain 𝐴𝑖 ◁ 𝑅 ˇ = 𝐵𝑖 ∪ 𝐵𝑗 . This result is not contradicting anything, nor our intuition. If we have two rules, one of them 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 situ- ations 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 𝐼 = {1, . . . , 𝑛} be the index set and let 𝐴𝑖 be normal for 𝑖 ∈ 𝐼. Then 𝐴𝑖 are said to meet the finitary condition if for any 𝑖 ∈ 𝐼 there exists an 𝑥 ∈ 𝑋 such that 𝐴𝑖 (𝑥) = 1 and 𝐴𝑗 (𝑥) = 0 for any 𝑗 ̸= 𝑖. The finitary condition has been proved to be sufficient for the solvability of both systems. For distinct proofs and formulations of the problem, we refer to [27, 28, 29]. Let us recall the version devoted to the Bandler-Kohout subproduct. Theorem 2. [28] Let 𝐴𝑖 meet the finitary condition. Then, ˇ = 𝐵𝑖 , 𝐴𝑖 ◁ 𝑅 𝑖 = 1, . . . , 𝑛 . (7) Theorem 2 might be viewed as violating the functionality or continuity idea as it imposes no assumptions on the closeness of the consequents 𝐵𝑖 ; however, it is simply due to the fact that finitarity has to be understood as “sufficient 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. As discussed above, preservation of modus ponens is a natural expectation; however, only 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 𝑅 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 𝐼 = {1, . . . , 𝑛} be the index set and let 𝑖, 𝑗 ∈ 𝐼 be such that 𝐴𝑖 = 𝐴𝑗 . Then, ˇ ⊇ 𝐵𝑖 ∪ 𝐵𝑗 . 𝐴𝑖 ◁ 𝑅 (8) Proof: Using the isotonicity of → in the second argument and the property 𝑎 → (𝑎 ⊗ 𝑏) ≥ 𝑏, we get (︃ )︃ ⋀︁ ⋁︁ ˇ )(𝑦) = (𝐴𝑖 ◁ 𝑅 𝐴𝑖 (𝑥) → (𝐴𝑘 (𝑥) ⊗ 𝐵𝑘 (𝑦)) 𝑥∈𝑋 𝑘∈𝐼 ⋀︁ ≥ ((𝐴𝑖 (𝑥) → (𝐴𝑖 (𝑥) ⊗ 𝐵𝑖 (𝑦))) ∨ (𝐴𝑖 (𝑥) → (𝐴𝑗 (𝑥) ⊗ 𝐵𝑗 (𝑦)))) 𝑥∈𝑋 ≥ 𝐵𝑖 (𝑦) ∨ 𝐵𝑗 (𝑦) . □ Note that Proposition 3 does not assume anything, no finitarity or normality is needed. The result is intuitive, but we still do not know whether not ‘too much’ would be produced by the inference system and what is needed to prevent that the output is trivial, that is, the universal fuzzy set. Therefore, let us add the finitarity. Proposition 4. Let 𝐼 = {1, . . . , 𝑛} be the index set and let 𝑖, 𝑗 ∈ 𝐼 be such that 𝐴𝑖 = 𝐴𝑗 . Let the set {𝐴𝑘 | 𝑘 ∈ 𝐼 ∖ {𝑗}} meet the finitary condition. Then, ˇ = 𝐵𝑖 ∪ 𝐵𝑗 . 𝐴𝑖 ◁ 𝑅 (9) Proof: Let 𝑥′ ∈ 𝑋 be such that 𝐴𝑖 (𝑥′ ) = 1 and 𝐴𝑘 (𝑥′ ) = 0 for any 𝑘 ̸= 𝑖, 𝑗. Then we get (︃ )︃ (︃ )︃ ⋀︁ ⋁︁ ⋁︁ ˇ )(𝑦) = (𝐴𝑖 ◁ 𝑅 𝐴𝑖 (𝑥) → (𝐴𝑘 (𝑥) ⊗ 𝐵𝑘 (𝑦)) ≤ 𝐴𝑖 (𝑥′ ) → (𝐴𝑘 (𝑥′ ) ⊗ 𝐵𝑘 (𝑦)) 𝑥∈𝑋 𝑘∈𝐼 𝑘∈𝐼 ⎛ ⎞ ⋁︁ = 1 → ⎝(1 ⊗ 𝐵𝑖 (𝑦)) ∨ (1 ⊗ 𝐵𝑗 (𝑦)) ∨ (𝐴𝑘 (𝑥′ ) ⊗ 𝐵𝑘 (𝑦))⎠ 𝑘∈𝐼∖{𝑖,𝑗} ⋁︁ = 𝐵𝑖 (𝑦) ∨ 𝐵𝑗 (𝑦) ∨ (0 ⊗ 𝐵𝑘 (𝑦)) = 𝐵𝑖 (𝑦) ∨ 𝐵𝑗 (𝑦) . 𝑘∈𝐼∖{𝑖,𝑗} □ Proposition 4 provides us with a valuable result, i.e., that the output of the system is the desirable union of both consequents of the fully fired rules. Thus, the system does not build a confusing fog of information around the necessary one we want to be given. 3. Conclusions 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. Acknowledgments The authors announce the support of the Czech Science Foundation through the grant 20-07851S. References [1] E. P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer, Dordrecht, 2000. [2] M. Baczyński, B. Jayaram, Fuzzy Implications, Springer-Verlag, Heidelberg, 2008. [3] R. Bělohlávek, Fuzzy relational systems: Foundations and principles, Kluwer, Plenum Press, Dordrecht, New York, 2002. [4] L. Běhounek, M. Daňková, Relational compositions in fuzzy class theory, Fuzzy Sets and 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: Proc. Int. Workshop on Fuzzy Reasoning Theory and Applications, Queen Mary College, London, 1978. [7] W. Bandler, L. J. Kohout, Semantics of implication operators and fuzzy relational products, International Journal of Man-Machine Studies 12 (1980) 89–116. [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, H. Prade (Eds.), The Handbook of Fuzzy Set Series Vol. 1, Kluwer, Boston, 2000, pp. 291–340. [16] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relation Equations and Their Applica- tions 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.), Foundations of Fuzzy Logic and Soft Computing, Springer, Berlin, 2007, pp. 502–514. [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: Complete set of solutions, Fuzzy Sets and Systems 159 (2008) 2256–2271. [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, Infor- mation 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.