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GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY NGUYEN THU ANH Study of real-world semantics-based interpretability of fuzzy system Major: MATHEMATICAL BASIS FOR INFORMATICS Code: 62.46.01.10 SUMMARY OF MATHEMATICS DOCTORAL THESIS SCIENTIFIC INSTRUCTOR: Ph.D Tran Thai Son Hanoi 2019 INTRODUCTION In some areas, we expect machinery to be able to simulate behavior, reasoning ability like human and give human reliable suggestions in the decision-making process A prominent feature of human is the ability to reason on the basis of knowledge formed from life and expressed in natural language Because the language characteristic is fuzzy, the first problem that needs to be solved is how to mathematically formalize the problems of linguistic semantic and handle semantic language that human often uses in daily life In response to those requirements, in 1965, Lotfi A Zadeh was the first person to lay the foundation for fuzzy set theory Based on fuzzy set theory, Fuzzy Rule Based System (FRBS) has been developed and become one of the tools of simulating reasoning method and making decisions of human in the most closely manner FRBS has been successfully applied in solving practical problems such as control problem, classification problem, regression problem, language extraction problem, etc When building FRBSs, we need to achieve two goals: accuracy and interpretability The thesis will focus on the study of interpretability In [1]1 Gacto finds that there are currently two main approaches to interpretability The first approach is based on complexity and the second approach is based on semantics Another approach proposed by Mencar et al in [2]2, called similar measure function-based approach to assess the interpretability of semantics-based fuzzy rules The interpretability of fuzzy rules is measured by the similarity between knowledge represented by fuzzy set expression and linguistic expression in natural language In 2017, a new approach to the interpretability of fuzzy system is Realworld-semantics-based approach – RWS-approach, has been first-time proposed and initially surveyed in [3]3 This approach is based on real-world semantics of words and relations between semantics of fuzzy system components and corresponding component structures in the real world Derived from the recognition that fuzzy set expressions, especially fuzzy rules of fuzzy systems have no relationship based on methodology with real world semantics and, therefore, there are no formal basis to study the nature of interpretability, his thesis chooses the real-world-semantics-based approach proposed in [3] to study the interpretability of fuzzy systems M.J Gacto, R Alcalá, F Herrera (2011), Interpretability of Linguistic Fuzzy Rule-Based Systems: An Overview of Interpretability Measures Inform Sci., 181:20 pp 4340–4360 C Mencar, C Castiello, R Cannone, A.M Fanelli (2011), Interpretability assessment of fuzzy knowledge bases: a cointension based approach, Int J Approx Reason 52 pp 501–518 Cat Ho Nguyen, Jose M Alonso (2017), “Looking for a real-world-semantics-based approach to the interpretability of fuzzy systems” FUZZ-IEEE 2017 Technical Program Committee and Technical Chairs, Italy, July 9-12 At the same time, at present, methods of building FRBS from data in fuzzy set theory-based approach lack a full formal link between fuzzy sets representing the assumed semantics of a word and its inherent semantics The words used in FRBS are only considered as labels or symbols assigned to corresponding fuzzy sets, are very difficult to fully convey underlying semantics compared with natural linguistic words Therefore, this thesis wishes to further study the interpretability of linguistic fuzzy systems in the semantic approach based on the hedge algebra proposed by Nguyen and Wechler [4] [5]5 In this approach, the computational semantics of words shall be defined based on the inherent order semantics of the words and word domains of the variables that establish an order-based structure that are rich enough to solve the problems in fact This thesis has achieved some following results:  Research and analysis of interpretability are as a study of the relationship between RWS of linguistic expressions and computational semantics of computational expressions assigned to linguistic expressions The schema proposal solves the problem of interpretability of the computational representation of liguistic frame of cognitive (LFoC)  The study proposing constraints on interpretation operations is built to convey, preserve the desired semantic aspects of the LFoC for fuzzy systems  Application of HA approach solves the problem of interpretability of computional representation of LFoC by establishing a granular polymorphism structure of triangular fuzzy sets or trapezoidal fuzzy sets  Further clarify RWS interpretation of human natural languages and word domains of variables and its basic role in checking RWS interpretability of components of fuzzy system, at the same time, prove that the standard fuzzy set algebras are not RWS interpretability  Propose formalization method to solve RWS interpretation of fuzzy systems in the second case and n input variable CHAPTER I : BASIC KNOWLEDGE 1.1 Fuzzy set Definition 1.1 [6]6 Let U be the universe of objects The fuzzy set A on U is the set of ordered pairs (x, A(x)), with A(x) being the function from U to [0,1] C.H Nguyen and W Wechler (1990), “Hedge algebras: an algebraic approach to structures of sets of linguistic domains of linguistic truth variables”, Fuzzy Sets and Systems, vol 35, no.3, pp 281293 Cat-Ho Nguyen and W Wechler (1992),” Extended hedge algebras and their application to Fuzzy logic”, Fuzzy Sets and Systems, 52, 259-281 L A Zadeh, Fuzzy set, Information and control, 8, (1965), pp 338-353 assigned to each element x of U value A(x) reflects the degree of x belong to fuzzy set A If A(x) = 0, then we say x does not belong to A, otherwise if A(x) = 1, then we say x belongs to A In Definition 1.1, function  is also called is a membership function 1.2 Linguistic variable Simply as said by Zadeh, a linguistic variable is a variable in which "its values are words or sentences in natural language or artificial language" 1.3 Fuzzy rule based system 1.3.1 The components of the fuzzy system A fuzzy rule based system consists of the following main components: Database, Fuzzy Rule-based - FRB and Inference System - Database is sets of 𝔏j including linguistic label Tj corresponding to fuzzy sets used to reference domain fuzzy partition UjR (real number set) of variable 𝔛j, (j=1, ,n+1) of problem n input output - Fuzzy rule base is a set of fuzzy rules if-then - Reasoning system performs an approximate reasoning based on rules and input values to produce the predicted output value Some approximate reasoning directions are as follows: + Approximate reasoning based on fuzzy relationship + Approximate reasoning by linear interpolation on fuzzy set + Reasoning based on the rule burning 1.3.2 Objectives upon building FRBS  Evaluation of the effectiveness (accuracy) of FRBS For the objective of the effectiveness of FRBS, we have mathematical formulas to evaluate how an FRBS is effective  Problem of interpretability of FRBS Interpretability is a complex and abstract problem, it involves many factors In [1] Gacto finds that there are currently two main approaches to the interpretability: - Interpretability is based on complexity:  Rule basis level: The less the number of rules of the rule system is, the shorter the length of the rule is  Fuzzy partition level: number of attributes or number of variables, number of variables used less will increase the interpretability of the rule system The number of functions is used in the fuzzy partition, the number of functions should not be exceeded 7±2 [6] - Interpretability is based on semantics:  Semantics at the rule basis level: The rule basis must be consistent, ie it does not contain contradictory rules, the rules with the same premise must have the same conclusion, the number of rules burned by an input data is as little as possible  Semantics at fuzzy partition level (word level): The defined domain of variables must be completely covered by the function of fuzzy sets 1.4 Hedge algebra 1.4.1 The concept of hedge algebra Definition 1.2 [7]7: A hedge algebra is denoted as a set of components denoted by AX = (X, G, H, ) where G is a set of generator, H is a set of hedges, and “” is a partial ordering relation on X The assumption in G contains constants 0, 1, W with the meaning of the smallest element, the largest element and the neutral element in X We call each language value xX a term in HA If X and H be linearly ordered sets, then AX = (X, G, H, ) is sais a linear hedge algebra And if two critical hedges are fitted  and  with semantics being the right upper bound and right lower bound of the set H(x) when acting on x, then we get the complete linear HA, denoted by AX* = (X, G, H, , , ) Note that hn h1u is called a canonical representation of a term x for u if x = hn h1u and hi h1uhi-1 h1u for i is integer and in We call the length of a term x is the number of hedges in its canonical representation for the generated element plus 1, denoted by l(x) 1.4.2 Some properties of linear hedge algebra Theorem 1.1: [7] Let the sets H- H+ of a hedge algebra AX = (X, G, H, ) be linearly ordered Then, the following statements hold: i) For every uX, H(u) is a linearly ordered set ii) If X is a primarily generated hedge algebra and the set G of the primary generators of X is linearly ordered, then so is the set H(G) Furthermore, if u 0} 3.2.1.1 Try to discover structural relationships between words of 𝔉 being considered as an 𝕃E: As previously discussed, 𝔉 can be regarded as a RW-counterpart (or, its words can properly convey their RW-semantics) and we try to find out key structural features of 𝔉 It can be seen that on this set there exist two relations, denoted by ≤ and GS(x, y), which are still not considered in the fuzzy set framework ∘ 𝔉 is linearly orderedset induced by the word meaning Its structure is denoted as (𝔉, ≤) ∘ GS(x, y) is a generality-specificity relation on 𝔉 For example, for the variable AGE, “old” is more general than “very_old”,and “rather very young” is more specific than“young”, if they all are in 𝔉 It can be verified that G(x, y) have the following properties: - Anti-symmetry: GS(x, y) &GS(y, x) =>x = y - Transitivity: GS(x, y) &GS(y, z) =>GS(x, z) These relations are taken as constraints imposed on two respective interpretation mappings of sound computational representations of LFoCs, where a sound computational representation of an LFoC Definition 3.2 A fuzzy set presentation of a given LFoC 𝔉, FR(𝔉)={F(x): x∈𝔉}, where F(x) denotes the fuzzy set assigned to x, is said to be RWSinterpretable if the following conditions hold: (i) On FR(𝔉) can be defined two relations: the first one is denoted by ≤*, which is reflexive, anti-symmetric and transitive, and the second one is denoted by GS*, which is anti-symmetric and transitive (ii) There exists an interpretation assignment I and IGS that both map 𝔉 into FR(𝔉), such that they respectively preserve the relations ≤ and GS on 𝔉, i.e., for all x, y in 𝔉, we have x ≤ y =>I(x) ≤*I(y) and GS(x, y) =>GS*(IGS(x), IGS(y)) 3.2.1.2 Try to construct a computational space that can properly represent the semantics of 𝔉: The study [3] argued that the topology of multi-granularity structure of fuzzy sets, as shown in Fig 3.3 can meet the both above discovered constraints, where the fuzzy sets are arranged in three level of their generality Here, the fuzzy setson the k th-level represent the semantics of the words of the generality of degree k For k = 0, the words on this level are the most general and the fuzzy setsare either all triangles or all trapezoids, whore core are therefore represented by bolded points in the figure and whose supports are uniquely defined by the fuzziness intervals of the respective words Representing triangles/trapezoids by 16 triples of the form (a, b, d), where b’s are the cores of triangles/trapezoids, the order between the fuzzy setsare defined as follows: (a, b, d) ≤* (a’, b’, d’) {b ≤ b’& there is at least one of the remaining components of the triples, say the first one, satisfying the W 00 10 inequality a ≤ a’} The relation GS* between the triples are defined as 01 young 11 old follows: GS*((a, b, d), (a’, b’, d’))  [a, d] [a’, d’].Obviously,the triples on 02 Vyoung Ryoung Rold Vold 12 the top level are of the most g generality which is compatible with the Figure 3.3 RWS-Interpretable triangle/trapezoid semantics of the words on multi-granular representation of XAGE,(2) this level 3.2.2 The RWS-interpretability ofcomputational representation of LRBs and ARMs In [3], it is argued that one may acquire a piece of knowledge about a numeric dependentrelationship between two variables only when it is observed that they are monotonically dependent oneach other on a certain interval of each variable, otherwise their dependence is chaotic Compatibly with this, as discussed in that study, it is interesting that the semantics of a multi-variable linguistic rule does also expressmonotonic dependent relationships betweenits unique output variable and one of its input variables Consider a linguistic rule with one output and m input variables written in the following form: (r) IF 𝒳1L is x1 & … &𝒳mL is xm, THEN 𝒳m+1,L is xm+1 (1) in which each expression “𝒳jL is xj” is a linguistic predicate, for j = to m + Similar as for analyzing a classical multi-variable function, for every rule r, one may consider m dependent relations ‘IF 𝒳jL is xj, THEN 𝒳m + 1,L is xm+ 1’, j = to m + 1, and therefore, r denotes m monotonic dependences between variables 𝒳m + 1, L and 𝒳jL on certain interval of each respective RW-variable of the RWcounterpart To further analyze, we assume that a LRB ℛℬ is full of conditions, that is, all m variables 𝒳jL, j = 1, …,m, are explicitly present in each rule (as it happens with the well-known Wang and Mendel [11]11) In this case, it is simply called a full LRB It is natural to require ℛℬto beconsistent, i.e if the 11 L.-X Wang and J M Mendel (1992), “Generating fuzzy rules by learning from examples,” IEEE Transactions on Systems, Man and Cybernetics, vol 22, no 6, pp 1414–1427 17 antecedents of any two rules of ℛℬ are identical, then so are their conclusions Then, a full and consistent LRB ℛℬ may be considered as representing a linguistic functional dependence of 𝒳m+1,L on 𝒳jL’s, j = to m By the RWSinterpretability of natural languages and the RW-semantics of linguistic rules and LRBs, ℛℬ represents a real-world functional dependence of 𝒳m+1 on 𝒳j’s, for j = to m Definition 3.3 Given comput-space 𝒮 = (𝒞, ≤𝒮) defined on Cartesian product oforder-based structures CSj’s A CRep-method ℳ with its interpretations I𝒳j’s, I𝒳j: Dom(𝒳j) → CSj, is said to be RWS-interpretable in 𝒮 provided that 1) The interpretations I𝒳j’s are order isomorphisms 2) For a given LRB 𝔹, ℳ preserves the discovered monotonicity, if any, of 𝔹 That is, if 𝔹 is increasing (or decreasing) and for a = (xi1, …,xim) ≤a’ = (xi’1, …,xi’m), where a and a’are any two linguistic vectors formed by m words appearing in, respectively, some two rules and ra’of 𝔹, then ℳ(ra)|𝒳m+1 ≤ ℳ(ra’)|𝒳m+1 (or, ℳ(ra)|𝒳m+1 ≥ ℳ(ra’)|𝒳m+1) It can be observed in general that any method that represents any LRB as an (m+1)-dimensional fuzzy relation cannot be RWS-interpretable, as the order of words and their fuzzy sets is ignored ARMds eveloped to solve application problems plays an important role to build FSysts and therefore, its interpretability is essential to ensure their performance in solving application problems, due to in the opposite case we have no formal basis to ensure that the outputs of their ARMd are compatible with the results expected by human designer This question strongly depends on the RWS-interpretability of the constructed computational representation method, ℳ, to produce computational representations of LRBs as well as of ARMds running on Any ARMd, say ℝ, needs to be developed to be able to work on the computational representation of ℛℬ and this implies that its realworld-semantics interpretability depends heavily on ℳ Therefore, the RWSinterpretability of an ARMds should be defined based on the computational representation method associated with it In [13], the authors introduced the following definition, in which a = (a1, , am) is the input vector and ℝ(a) denotes the numerical output of the vector a produced by ℝ Definition 3.4 Assume that an ARMd ℝ is developed to work on computational representations of LRBs produced by a computational representation method ℳ Then, ℝ is said to be RWS-interpretable if for any give LRB ℛℬ being increasingly monotonic to all individual input variables of ℛℬ, ℝ must satisfy the following condition: (a, a’){[a ≼ a’  ℝℳ(𝔹)(a)  ℝℳ(𝔹)(a’)] and [a  a’  ℝℳ(𝔹)(a)  ℝℳ(𝔹)(a’)]} (2) 18 3.3 The RWS-interpretability concept of linguistic fuzzy expressions/theories As discussed in previous sections, the novel RWS-interpretability of any fuzzy theories, in general, and of any FSysts, in particular, seems to be very essential and practical So, a natural question that arises is that whether the fuzzy sets theory or its expressions are RWS-interpretable and if it is not, whether there exist methodologies to develop RWS-interpretable FSysts? 3.3.1 Examination of the RWS-interpretability of some fuzzy expressions of the fuzzy set theory The RWS-interpretability of the fuzzy set theory is a too big problem and, therefore, in this section it is restricted to examine the RWS-interpretability of the standard fuzzy set algebras 3.3.1.1 An analysis of the RWS-interpretability of the standard fuzzy set algebra Let us consider a universe U and denote by F(U) the set of all fuzzy sets of U, F(U) = { : ∈ [0, 1]U}, where [0, 1]U is the set of all (membership) functions from U into [0, 1] and the fuzzy sets and their membership functions can be viewed as to be identical It is well-known that the union (), intersection (), complementation () can be defined in F(U) as a generalization of the respective operations on the crisp sets of U They are pointwise defined on the membership functions of the fuzzy sets in the whole F(U) Then, we have a standard fuzzy set algebra that can be denoted by F𝔸= (F(U), , , ) To examine the RWS-interpretability of F𝔸, we have to find out which is the RW-semantics of individual fuzzy sets and the RW-semantics of the operations ,  and  when they act on fuzzy sets To answer these questions, we should come back to the ultimate aim of the development of the fuzzy sets theory: to simulate human capabilities in handing words This is why in applications the operations ,  and  are usually interpreted as representing the computational semantics of the respective logical connectives in natural languages, AND, OR and NOT So, we will examine the RWS-interpretability of the operations of the standard fuzzy set algebra based on the real-world semantics of the connectives AND, OR and NOT There are two main reasons that show that F𝔸 is not be able RWSinterpretable 1) A methodological reason present in the fuzzy set framework Let us consider the variable HIGH of people of a community and the meaning of the sentence “he is ‘Tall OR Rather_tall’” The semantics of this sentence must be considered in the context of the word-expressions of the word-domain of HIGH, LDom(HIGH) Assume that the standard fuzzy set algebra defined on the universe U of HIGH is F𝔸= (F(U), , , ) Proposition 3.2 As LDom(HIGH) is finite, while F(U) is innumerable and even of continuum power, there does not exists an interpretation mapping ℑ 19 from LDom(HIGH) into F(U) that can maintains the relationships that characterize the structure of LDom(HIGH), noting that the operations of F𝔸 are defined in the whole F(U) The above clause is demonstrated and illustrated by Figure 3.4 2) A methodological reason on standpoint of the RWS-approach First, we adopt an assumption AB that we deal withonly variables with numeric linear universe and A B hence the word-domains of their linguistic variables are linearly Figure 3.4 The union of the two given fuzzy ordered So, their respective HAs sets of variable HIGH are also linear In Section 1, it is shown by Proposition that the HA AXHIGH is RWS-interpretable That is there exists an interpretation mapping ℑHIGH from LDom(HIGH) into the underlying set of AXHIGH, which implies that ℑHIGH(wA AND wB)=ℑHIGH(wA) ℑHIGH(wB), where  is join operation defined in the order-based structure AXHIGH Since AXHIGH is RWS-interpretable, ℑ preserves the relationships between words of LDom(HIGH), the expression ℑHIGH(wA) ℑHIGH(wB) represents the semantics of the word-expression “wA AND wB” As AXHIGH is linear, ℑHIGH(wA)ℑHIGH(wB) = max{ℑHIGH(wA), ℑHIGH(wB)} and it represents the RW-semantics of the expression “wA AND wB” Since as mentioned above, the fuzzy sets A and B associated with respectively the word wA and wB, but AB{A, B}, which is not compatible with ℑHIGH(wA) ℑHIGH(wB) = max{ℑHIGH(wA), ℑHIGH(wB)} which represents the RW-semantics of “wA AND wB” This asserts that the standard fuzzy set algebra F𝔸 is not RWS-interpretable 3.3.1.2 A discussion of the RWS-interpretability of Mamdani fuzzy reasoning method Table 3.1 Simplified FRB for In Mamdani fuzzy reasoning the actuator on the 1th-storey method which is denotedby ARMMmd, ẋ2 NS Z PS its fuzzy rule base (FRB), 𝔹, consists of x2 n rules of a similar form as in (1), but at NS NM NS Z the positions of the words xjk’s are Z NS Z PS fuzzy setf(xjk)’s assigned to the words denoted also by xjk’s: (𝔹) IF 𝒳1L is f(x1,k)& … &𝒳mL is xmi,k, THEN 𝒳(m+1)L is x(m+1),k, for k = to n Z PS (3) PS PM 20 This thesis has proved with monotonous language rule base (LRB) 𝔅, given in Table 3.1, simplified from the LRB given in the study [12]12 to include only fuzzy rule, a Mamdani fuzzy approximation method ARMMmd based on rule burning is not RWS-interpretable because it does not satisfy condition (2) of Definition 3.4 3.3.2 Discuss the RWS-interpretability of graphic representation of LRBs and of the interpolation reasoning HA-based method A question arising is whether there exists an RWS-interpretable ARM? In this section, we will follow the HA approach to the inherent order-based semantics of words and the inherent semantic structures of word-domains of variables Since this approach establishes a formalism to immediately handle the variable words, we use the terminology linguistic rules (or,linguistic rule bases (LRBs)) instead of fuzzy rules (or, fuzzy rule bases (FRBs) to emphasize this linguistic characteristic There are three basic quantitative semantics of the words of each variable 𝒳, defined in close relation to each other: fuzzy measure, fuzzy interval (considered as interval semantics) and semantically quantifying mapping (SQM) of the words of variables They are uniquely defined when the numerical values of the independent fuzzy parameters of variables are provided The SQM-values of words are called the numerical semantics of words In this section, however, we utilize only SQMs which are characterized by two properties that they are order isomorphisms, i.e they must preserve the order relations among words and the images of linguistic domains of variables under these isomorphisms are dense in the reference domains of the corresponding variables (similar as the countable set of the rational numbers is dense in the real line) On the mathematical point of view, when word-domains are formalized to become math-structures, everylinguistic rule of the form given in (3) can be considered as a point in the respective linguistic(m+1)-dimensional Cartesian space So, every given LRB of the form (3) can be considered as modeling a linguistic function of m variables going through these n points, called graphic representation of (3) Construction method a graphic representation of 𝔅: Given a linguistic rule base 𝔅 describes the function relationship of XL,(m+1) into variables XL,j, j = 1, , m Then, the steps to build the function fN represent the language law base calculation 𝔅 including the following steps Step 1) Determine the qualitative and quantitative semantics of linguistic variables: 12 R Guclu, H Yazici (2008), Vibration control of a structure with ATMD against earthquake using fuzzy logic controllers Journal of Sound and Vibration, 318, 36–49 21 (1.1) Building hedge algebra of domain from Dom(Xj), j = 1, …, m +1 (1.2) Selecting values of fuzzy parameters of Xj’s variables This greatly affects quantitative semantics of the words of the variable Step 2) Represent the graph of 𝔅 by defining the grid of points in space [0, 1]m+1 where the function fN,𝔅 , qua goes through: (2.1) For each variable Xj, j = 1, …, m +1, list all the words of the variable present in 𝔅, denoted by xjk, k = 1, …, Kj SQMj notation is a quantitative mapping of Xj determined by the values of the fuzzy parameters of Xj and the numerical semantic values of the words xjk, SQMj(xjk), k = 1, …, Kj (2.2) Set the approximate graph grid of function fN,𝔅 as a calculation representation of the linguistic rule base 𝔅 as follows: - For each language rule ri in 𝔅 there is the form (1), which is of the form: ri : IF X1L is x1,i & … &XmL is xm,i, THEN Xm+1,L is xm+1,i, i = 1, …, n, We denote ri|Xj = xj,i, j = 1, …, m + 1, and set the following points: (SQM1(ri|X1), …, SQMm+1(ri|Xm+1)) ∈ [0, 1]m+1 - Set grid in space [0, 1]m+1: Grid(𝔅) = {( SQM1(ri|X1), …, SQMm+1(ri|Xm+1) : i = 1, …, n } Because the quantitative maps SQMj are all isomorphisms that preserve the order of language words of the variables Xj, it is easy to verify the correctness of the following theorem: Theorem 3.2 The graphic representation of LRBs is RWS-interpretable Proof: Suppose for 𝔅 is monotonous, meaning that if we use the above notation and we have ri|Xj ≤ ri’|Xj with all j = 1, …, m, then we also have ri|Xm+1 ≤ ri’|Xm+1 Because the quantitative mappings SQMj are all isomorphic preserving the order of the linguistic words of Dom(Xj), Grid (𝔅) defines a function fN on the domain of Grid(𝔅) which is monotonous , that is something to prove 3.3.3 Approximate reasoning method on graphical representations of LRBs 3.3.3.1 Interpolative approximate reasoning method Approximate reasoning problem: Give a numerical vector ain = (ain,1, …, ain,m) ∈ U𝒳1 …  U𝒳m and a linguistic rule base ℛℬ, calculate a numerical semantic of the output corresponding to the input ain, denoted by Outℛℬ(ain), based on the knowledge given by ℛℬ This problem can be solved in this study by an interpolative method in Euclide an space as follows: Interpolative method on LRB ℛℬ: Let be given values of the fuzzy parameters of the variables present in ℛℬ and a graphical representation method 𝕄Graph Then, ℳGraph(ℛℬ) defines a grid of a surface Sℛℬ in Euclidean space [0, 1]m+1 So, every (numerical) interpolative method INTMd on the surface Sℛℬ can be apply to define a ARMd to solve the approximate reasoning problem for the given linguistic rule knowledge base ℛℬ 22 For a give an INTMd ℳInter, it is clear that, for each input vector ain, Outℛℬ(ain) can be calculated by applying ℳInter on the surface Sℛℬ, denored by ℳInter(Sℛℬ), and obtain Outℛℬ(ain) = ℳInter(Sℛℬ)(ain), i.e it is the value calculated by ℳInter on Sℛℬ in the Euclidean space [0, 1]m+1 3.3.3.2.RWS-interpretability of interpolative approximate reasoning methods 1) Linear interpolative approximate reasoning methods: In case that the LRB has two inputs, we have a linear interpolative approximate reasoning method on surface in [0, 1]3 For example, the LRB ℛℬ given in Table 3.1 with linguistic rules defines a surface Sℛℬ as represented in Figure 3.6  Interpolative approximate reasoning method Li This interpolative method is called the Li-method, which is extended from the method studied in the work [13]13, but it RWS-interpretability is still not examined, and is described as follows: - For each input vector ain = (a1, a2), define the smallest rectangle, whose three vertices are denoted by Pk, k = 1, 2, 3, in the coordinate plane x  y containing point (a1, a2) - Establish the section whose projection l (0.73) on the coordinate plane x Ll(0.67  y is the above defined ) triangle: Denote by Sℛℬ(Pk), k = 1, 2, 3, the W(0.40) points in [0,1]3 lying on Ls(0.30) the surface Sℛℬ whose s(0.18) projections on the plane x W (0.40) s (0.18) l (0.73)  y are the points Pk, k = s(0.18) 1, 2, and establish the plane equation going W through these points, ( denoted by z = EQ(Sℛℬ(P1), l (0.73) x (x, y) Sℛℬ(P2), Sℛℬ(P3)) graphical representation of LRB - Calculate the Figure 3.6 numerical passing through points output by equality ) Out(ain)=EQ(Sℛℬ(P1), Sℛℬ(P2), Sℛℬ(P3))(a1, a2) We can easily demonstrate the correctness of the following theorem: 13 M Antonelli, P Ducange, B Lazzerini, F Marcelloni (2011), Learning concurrently data and rule bases of Mamdani fuzzy rule-based systems by exploiting a novel interpretability index Soft Comput., 15 pp 1981–1998 23 Theorem 3.3 The linearly interpolative Li-method, denoted by -ℳ, is RWS-interpretable Proof: Assuming that LRB ℛℬ describes an increasing linguistic function, as this equation is linear it is easy to prove that the inequality (a1, b1) ≤ (a2, b2) implies that -ℳ(Sℛℬ)(a1, b1) ≤ -ℳ(Sℛℬ)(a2, b2) 2) In case m > There are many interpolative methods with the number of dimensions n > but they are in general very complicated when n is large In this case, we can use an aggregation operator usually used in fuzzy set theory to convert approximate reasoning problems in m + dimensional space to two-dimensional one Theorem 3.4 Let be given a LRB ℛℬ and assume that the aggregation operator used is a weighted average with weight vector w = (w1, …, wm) corresponding to m antecedent variables of ℛℬ, denoted by 𝔤w Then, the linear interpolation using 𝔤w, denoted by Li_IntM2,w is RWS-interpretable Proof: Assume that ℛℬ is a LRB represented by the graph 𝒩Gph𝕀∘𝕗 (ℛℬ) with the grid Grid2(ℛℬ) = {(𝔤w[SQM1(x1,i),…,SQMm(xm,i)], SQMm+1(xm+1,i)): i =1,…, n } Due to ℛℬ is increasingly monotonic and assume that there are two rules ri and ri’ in form (*) whose linguistic vectors created by the words in their antecedent parts, denoted by x(ri) = (x1,i, …, xm,i) and x(ri’) = (x1,i’, …, xm,i’), satisfy the condition that x(ri) ≤ x(ri’), i.e xj,i ≤ xj,i’, for j = 1, …, m, implies ri|𝒳m+1 = xi,m+1 ≤ ri’|𝒳m+1 = xi’,m+1 As SQMj are order isomorphisms, we have SQMj(xj,i) ≤ SQMj(xj,i’), j = 1, …, m+1, and therefore we obtain 𝔤w(x(ri)) ≤ 𝔤w(x(ri’)) Consider two input vectors ain = (ain,1, …, ain,m) ≤ bin = (bin,1, …, bin,m) Then, similarly as above, we have 𝔤w(ain,1, …, ain,m) ≤ 𝔤w(bin,1, …, bin,m) There are two cases: Case 1: There exists a smallest interval [𝔤w(x(rj1)), 𝔤w(x(rj2))] containing the both values 𝔤w(ain,1, …, ain,m) and 𝔤w(bin,1, …, bin,m) computed from the two given inputs Case 2: The two values 𝔤w(ain,1, …, ain,m) and 𝔤w(bin,1, …, bin,m) lie on different intrvals I1 = [𝔤w(x(rj1)), 𝔤w(x(rj1*))] and I2 = [𝔤w(x(rj2)), 𝔤w(x(rj2*))] created by the adjacent horizon coordinates of the grid Grid 2(ℛℬ) in [0, 1]2 We have proved that the method of approximating linear interpolation Li_IntM2, also preserves the monotonicity of the linguistic law base 𝔅 in these cases The theorem is proven 3.4 Conclusion of chapter The purpose of this chapter is to discuss and analyze clearly that the interpretability of RWS is an essential general concept and not only for FSysts but also for languages, theories, signing methods The interpretability of RWS 24 of two basic concepts, LRB-Rep method and approximate reasoning are formalized more accurately The fuzzy set theory is proved initially including basic parts failing to interpret based on RWS Meanwhile, graph representation of the linguistic rule basis and approximate reasoning method are developed based on linear interpolation proved to be interpretable based on RWS CONCLUSION OF THE THESIS The thesis is conducted with the desire to study in depth the interpretability of the fuzzy linguistic systems in the semantic approach based on the hedge algebra proposed by Nguyen and Wechler In this approach, the computational semantics of a word must be defined based on the inherent order semantics of the words and word domains of the variables that establish an order-based structure that is rich enough to solve the problems in fact The thesis has focused on studying the problem of interpretability of FRBS in HA-based approach and proposed some constraints, definitions and theorems in this approach At the same time, the thesis also studies the approach based on real-world interpretability for the problem of interpretability of fuzzy system and deeper and more practical analysis of the interpretability of formal theories includes natural human languages in general and formalized fuzzy systems From the results achieved in the thesis, we can draw some following conclusions: • The interpretability of a computional expression depends on the idea that the interpretability needs to preserve discovered relationships based on the semantics of linguistic expression, and the proposal of constraints for interpretation operations are different from the concept of interpretability of fuzzy systems studied in the fuzzy set theory • The HA-based approach is similar to the idea of interpretability of formal programming languages In which, the semantics of syntactic expressions defined by interpretation operations assign them certain mathematical objects that are considered to be their semantics This interpretation aims to preserve the validity of their semantics on the basis of preserving axioms for the mathematical structure assigned to syntactic expressions • As shown in this study, the fuzzy set theory, more specifically the standard fuzzy set algebra may not be interpretable of real world semantics Methodologically, it is an essential shortcoming of fuzzy set theory based on the perspective of a real-world semantic approach However, despite this, fuzzy set theory is still one of the great theories because it is very flexible in applications, this omission can be overcome by testing experimental studies until realizing the objective 25 LIST OF WORKS HAS BEEN PUBLISHED Thu Anh Nguyen, Cat Ho Nguyen, On the real-world-semantics interpretability of fuzzy rule based systems under fuzzy set approach and hedge algebra approach, Journal of Computer Science and Cybernetics, V.33, N.1 (2017), 86–109 Thu Anh Nguyen, Cat Ho Nguyen, On the computational interpretability of linguistic fuzzy rules based systems at the low level, Proceeding of XIX National Conference: Some Selected Issues of Information Technology and Communication – Hanoi, Oct 1-2/2016 , 274-281 Thu Anh Nguyen, Thai Son Tran, The real-world-semantics interpretability of linguistic rule bases and the approximate reasoning method of fuzzy systems, Vietnam Journal of Science and Technology, Vietnam Academy of Science and Technology (Accepted) Cat Ho Nguyen, Thi Lan Pham, Cam Ha Ho, Thu Anh Nguyen, A real-world-semantics approach to the interpretability problem of linguistic summaries, IEEE Transactions on Fuzzy Systems (Submited) ... real-world Real world models of formal theories Applications/algorithms designed based on certain formal theories interacting with their RW-counterparts Formal theories developed based on their axioms... development the theory of the real numbers The successful applications of this theory in reality demonstrate that it is RWSinterpretable So, methodologically, the RWS-interpretability of the theory of... are independent, i.e uH(v) vH(u), H(u) H(v) The theorem below looks at the comparison of two terms in the linguistic domain of variable X Theorem 1.2: [7] Let x = hn…h1u and y = km…k1u be two