=Paper= {{Paper |id=Vol-2648/paper14 |storemode=property |title=Inference Methods for Mamdani-Type Systems Based on Fuzzy Truth Value |pdfUrl=https://ceur-ws.org/Vol-2648/paper14.pdf |volume=Vol-2648 |authors=Vasily G. Sinuk,Sergey V. Kulabukhov }} ==Inference Methods for Mamdani-Type Systems Based on Fuzzy Truth Value== https://ceur-ws.org/Vol-2648/paper14.pdf 
Inference Methods for Mamdani-Type Systems Based
on Fuzzy Truth Value
Vasily G. Sinuka , Sergey V. Kulabukhova
a
 Belgorod State Technological University named after V. G. Shukhov, Department of Software Engineering for
Computers and Computer-Based Systems, Belgorod, Russia


                                         Abstract
                                         The article introduces inference methods for Mamdani-type fuzzy systems, which can be implemented
                                         with polynomial computational complexity for any t-norms and multiple fuzzy inputs. Center average
                                         and center of gravity defuzzification methods were used for case of multiple rules in rule base. Network
                                         architectures of systems corresponding to inference methods introduced in the article are provided.




1. Introduction
Mamdani’s approach addresses the question of interpretation of the expression β€œif 𝑋 is 𝐴 then
π‘Œ is 𝐡”, where 𝑋 and π‘Œ are linguistic variables, 𝐴 and 𝐡 are linguistic values of 𝑋 and π‘Œ
respectively. The source of uncertainty consists in the fact that β€œif 𝑋 is 𝐴 then π‘Œ is 𝐡” can be
interpreted in two different ways. First, the most obvious way is to consider this expression as
β€œπ‘‹ is 𝐴 and π‘Œ is 𝐡”, or as (π‘₯, 𝑦) is 𝐴 Γ— 𝐡, where 𝐴 Γ— 𝐡 is a Cartesian product of fuzzy sets 𝐴 and
𝐡. Hence, with this interpretation β€œif 𝑋 is 𝐴 then π‘Œ is 𝐡” is a joint constraint on 𝑋 and π‘Œ . An
alternative way consists in understanding β€œif 𝑋 is 𝐴 then π‘Œ is 𝐡” as a conditional constraint or,
equivalently, an implication. Many different implications are known. This way was considered
in [Mik18] for systems with multiple inputs. The subject of this article is the development of
Mamdani’s approach.
   For systems with multiple fuzzy inputs, which represent a formalization of terms of lin-
guistic variables or inaccurate measurements, inference methods based on max-min and max-
product composition operations are known [Rut10]. Operators min (taking minimum) and
product (arithmetical product) are t-norms [Als06] that correspond to Mamdani’s [Mam74]
and Larsen’s [Lar80] inference rules respectively. But for other t-norms, replacement of which
can be necessary for learning of fuzzy systems, implementation of inference for multiple fuzzy
inputs with polynomial computational complexity is impossible. In this article, methods that
solve this problem are considered.
   The statement of the problem and estimation of complexity of fuzzy inference is made in
section 2. In section 3, an inference method using a measure of possibility for each input of a
multiple-input system is considered. Section 4 introduces an inference method based on fuzzy
Russian Advances in Artificial Intelligence: selected contributions to the Russian Conference on Artificial intelligence
(RCAI 2020), October 10-16, 2020, Moscow, Russia
" vgsinuk@mail.ru (V.G. Sinuk); qlba@ya.ru (S.V. Kulabukhov)

                                       Β© 2020 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)
truth value. Sections 5 and 6 consider inference for a rule base with use of center average and
center of gravity methods respectively.


2. Statement of the problem
A linguistic model is represented by a fuzzy rule base π‘…π‘˜ , π‘˜ = 1, 𝑁 of the form:
               π‘…π‘˜ ∢ If π‘₯1 is 𝐴1π‘˜ and π‘₯2 is 𝐴2π‘˜ and … and π‘₯𝑛 is π΄π‘›π‘˜ then 𝑦 is π΅π‘˜ ,                   (1)
where 𝑁 is the number of fuzzy rules, π΄π‘–π‘˜ βŠ† 𝑋𝑖 , 𝑖 = 1, 𝑛, π΅π‘˜ βŠ† π‘Œ are fuzzy sets, defined by
membership functions πœ‡π΄π‘–π‘˜ (π‘₯𝑖 ) and πœ‡π΅π‘˜ (𝑦) respectively; π‘₯1 , π‘₯2 , … , π‘₯𝑛 are input variables of the
linguistic model, while
                              [π‘₯1 , π‘₯2 , … , π‘₯𝑛 ]T = 𝒙 ∈ 𝑋1 Γ— 𝑋2 Γ— β‹― Γ— 𝑋𝑛 .
Symbols 𝑋𝑖 , 𝑖 = 1, 𝑛 and π‘Œ stand for input and output variables spaces respectively.
  Let us denote 𝑿 = 𝑋1 Γ— 𝑋2 Γ— β‹― Γ— 𝑋𝑛 and π‘¨π’Œ = 𝐴1π‘˜ Γ— 𝐴2π‘˜ Γ— β‹― Γ— π΄π‘›π‘˜ , whereas
                                        πœ‡π‘¨π’Œ (𝒙) = T1 πœ‡π΄π‘–π‘˜ (π‘₯𝑖 ),                                    (2)
                                                   𝑖=1,𝑛

where T1 is an arbitrary t-norm, then rule (1) can be represented in the form of fuzzy implication
                                    π‘…π‘˜ ∢ π‘¨π’Œ β†’ π΅π‘˜ ,         π‘˜ = 1, 𝑁 .
Rule π‘…π‘˜ can be formalized as a fuzzy relation, defined over set 𝑿 Γ— π‘Œ , i.e. π‘…π‘˜ βŠ† 𝑿 Γ— π‘Œ is a fuzzy
set with membership function
                                      πœ‡π‘…π‘˜ (𝒙, 𝑦) = πœ‡π‘¨π’Œ β†’π΅π‘˜ (𝒙, 𝑦).
Mamdani’s model defines the function πœ‡π‘¨π’Œ β†’π΅π‘˜ (𝒙, 𝑦) based on known membership functions
πœ‡π‘¨π’Œ (𝒙) and πœ‡π΅π‘˜ (𝑦) as follows [Rut10, Peg09]:
                                                                        T
                      πœ‡π‘¨π’Œ β†’π΅π‘˜ (𝒙, 𝑦) = T2 (πœ‡π‘¨π’Œ (𝒙), πœ‡π΅π‘˜ (𝑦)) = πœ‡π‘¨π’Œ (𝒙) βˆ—2 πœ‡π΅π‘˜ (𝑦),
where T2 is an arbitrary t-norm.
   The problem consists in defining fuzzy inference π΅π‘˜β€² βŠ† π‘Œ for a system, represented in the
form (1), if the inputs are assigned fuzzy sets 𝑨′ = 𝐴′ 1 Γ— 𝐴′ 2 Γ— β‹― Γ— 𝐴′ 𝑛 βŠ† 𝑿 or β€œπ‘₯1 is 𝐴′1 and π‘₯2 is
𝐴′2 and … and π‘₯𝑛 is 𝐴′𝑛 ” with the corresponding membership function πœ‡π‘¨β€² (𝒙), which is defined
as
                                       πœ‡π‘¨β€² (𝒙) = T3 πœ‡π΄β€²π‘– (π‘₯𝑖 ),                                     (3)
                                                   𝑖=1,𝑛
where T3 is an arbitrary t-norm.
  According to fuzzy modus ponens rule [Zad73], fuzzy set π΅π‘˜β€² is defined by the composition of
fuzzy set 𝐴′ and relation π‘…π‘˜ , i.e.
                                    π΅π‘˜β€² = 𝑨′ β—¦(π‘¨π’Œ β†’ π΅π‘˜ ),
or, at the level of membership functions,
                                       {        T           T          }
                         πœ‡π΅π‘˜β€² (𝑦) = sup πœ‡π‘¨β€² (𝒙) βˆ—4 (πœ‡π‘¨π’Œ (𝒙) βˆ—2 πœ‡π΅π‘˜ (𝑦)) ,                           (4)
                                     π’™βˆˆπ‘Ώ

where T4 is an arbitrary t-norm. Computational complexity of expression (4) equals 𝑂(|𝑿 |Γ—|π‘Œ |).
3. Inference method based on possibility measure for each
   input
Let us consider the inference (4) when

                                                   T1 = T2 = T3 = T4 = T,                                       (5)

then                                             {                             }
                                                          T          T
                                   πœ‡π΅π‘˜β€² (𝑦) = sup πœ‡π‘¨β€² (𝒙) βˆ— (πœ‡π‘¨π’Œ (𝒙) βˆ— πœ‡π΅π‘˜ (𝑦)) .                               (6)
                                                  π’™βˆˆπ‘Ώ

Due to associativity of t-norms, the expression (6) can be transformed into
                                                   {        T        }T
                                     πœ‡π΅π‘˜β€² (𝑦) = sup πœ‡π‘¨β€² (𝒙) βˆ— πœ‡π‘¨π’Œ (𝒙) βˆ— πœ‡π΅π‘˜ (𝑦).                                (7)
                                                   π’™βˆˆπ‘Ώ

Using (2) and (3) we can further transform (7):
               {                                                                                   }
                           T            T   T            T            T            T   T             T
 πœ‡π΅π‘˜β€² (𝑦) = sup πœ‡π΄β€²1 (π‘₯1 ) βˆ— πœ‡π΄β€²2 (π‘₯2 ) βˆ— … βˆ— πœ‡π΄β€²π‘› (π‘₯𝑛 ) βˆ— πœ‡π΄1π‘˜ (π‘₯1 ) βˆ— πœ‡π΄2π‘˜ (π‘₯2 ) βˆ— … βˆ— πœ‡π΄π‘›π‘˜ (π‘₯𝑛 ) βˆ— πœ‡π΅π‘˜ (𝑦).
            π‘₯1 βˆˆπ‘‹1
            π‘₯2 βˆˆπ‘‹2
              β‹―
            π‘₯𝑛 βˆˆπ‘‹π‘›

Associativity and commutativity of t-norms enables us to rearrange πœ‡π΄β€²π‘– (π‘₯𝑖 ) and πœ‡π΄π‘–π‘˜ (π‘₯𝑖 ), which
allows us to obtain
               {                                                                                  }
                           T             T            T             T   T            T              T
  πœ‡π΅π‘˜ (𝑦) = sup (πœ‡π΄1 (π‘₯1 ) βˆ— πœ‡π΄1π‘˜ (π‘₯1 )) βˆ— (πœ‡π΄2 (π‘₯2 ) βˆ— πœ‡π΄2π‘˜ (π‘₯2 )) βˆ— … βˆ— (πœ‡π΄π‘› (π‘₯𝑛 ) βˆ— πœ‡π΄π‘›π‘˜ (π‘₯𝑛 )) βˆ— πœ‡π΅π‘˜ (𝑦),
    β€²               β€²                         β€²                              β€²
            π‘₯1 βˆˆπ‘‹1
            π‘₯2 βˆˆπ‘‹2
              β‹―
            π‘₯𝑛 βˆˆπ‘‹π‘›

and, since t-norms are non-decreasing,
                          T                   T                     T          T        T            T      T
πœ‡π΅π‘˜β€² (𝑦) = sup {πœ‡π΄β€²1 (π‘₯1 ) βˆ— πœ‡π΄1π‘˜ (π‘₯1 )} βˆ— sup {πœ‡π΄β€²2 (π‘₯2 ) βˆ— πœ‡π΄2π‘˜ (π‘₯2 )} βˆ— … βˆ— sup {πœ‡π΄β€²π‘› (π‘₯𝑛 ) βˆ— πœ‡π΄π‘›π‘˜ (π‘₯𝑛 )} βˆ— πœ‡π΅π‘˜ (𝑦),
          π‘₯1 βˆˆπ‘‹1                                  π‘₯2 βˆˆπ‘‹2                                    π‘₯𝑛 βˆˆπ‘‹π‘›

what can be written as
                               {                      T             }T            {         }T
            πœ‡π΅π‘˜β€² (𝑦) = T           sup {πœ‡π΄β€²π‘– (π‘₯𝑖 ) βˆ— πœ‡π΄π‘–π‘˜ (π‘₯𝑖 )}     βˆ— πœ‡π΅π‘˜ (𝑦) = T Ξ π΄π‘–π‘˜ |𝐴′𝑖 βˆ— πœ‡π΅π‘˜ (𝑦),         (8)
                       𝑖=1,𝑛       π‘₯𝑖 βˆˆπ‘‹π‘–                                          𝑖=1,𝑛

where
                                                                        T
                                            Ξ π΄π‘–π‘˜ |𝐴′𝑖 = sup {πœ‡π΄β€²π‘– (π‘₯𝑖 ) βˆ— πœ‡π΄π‘–π‘˜ (π‘₯𝑖 )}
                                                           π‘₯𝑖 βˆˆπ‘‹π‘–

is a scalar value which, according to its definition in [Dub90], is a measure of possibility for
𝑖-th input, meaning how much 𝐴′𝑖 corresponds to π΄π‘–π‘˜ (or vice versa).
   Thus, we have proved that inference method (8) is possible if all four t-norms are similar (5).
In contrast to [Rut10], this t-norm may be arbitrary.
4. Inference method based on fuzzy truth value
Applying the truth modification rule [Bor82]

                                                    πœ‡π‘¨β€² (𝒙) = πœπ‘¨π’Œ |𝑨′ (πœ‡π‘¨π’Œ (𝒙)),

where πœπ‘¨π’Œ |𝑨′ ( β‹… ) denotes the fuzzy truth value of a fuzzy set π‘¨π’Œ with respect to 𝑨′ , representing
a compatibility membership function 𝐢𝑃(π‘¨π’Œ , 𝑨′ ) of π‘¨π’Œ relatively to 𝑨′ , while 𝑨′ is considered
as true [Zad78, Dub90]:

                            πœπ‘¨π’Œ |𝑨′ (𝑣) = πœ‡πΆπ‘ƒ(π‘¨π’Œ , 𝑨′ ) (𝑣) = sup {πœ‡π‘¨β€² (𝒙)},                            𝑣 ∈ [0; 1],
                                                                       πœ‡π‘¨π’Œ (𝒙)=𝑣
                                                                            π’™βˆˆπ‘Ώ


let us denote 𝑣 = πœ‡π‘¨π’Œ (𝒙). Then we get:

                                           πœ‡π‘¨β€² (𝒙) = πœπ‘¨π’Œ |𝑨′ (πœ‡π‘¨π’Œ (𝒙)) = πœπ‘¨π’Œ |𝑨′ (𝑣).

Hence fuzzy modus ponens rule for systems with 𝑛 inputs can be represented as follows:
                                                            {               T     T          }
                                     πœ‡π΅π‘˜β€² (𝑦) = sup             πœπ‘¨π’Œ |𝑨′ (𝑣) βˆ—4 (𝑣 βˆ—2 πœ‡π΅π‘˜ (𝑦)) .                                           (9)
                                                  π‘£βˆˆ[0;1]

Computational complexity of expression (9) has order of 𝑂(|𝑣| Γ— |π‘Œ |). As proven in [Kut15,
Sin16]:

 πœ‡πΆπ‘ƒ(π‘¨π’Œ , 𝑨′ ) (𝑣) = TΜƒ 1 πœ‡πΆπ‘ƒ(π΄π‘˜π‘– , 𝐴′𝑖 ) (𝑣𝑖 ) =
                       𝑖=1,𝑛

                      =(πœ‡πΆπ‘ƒ(π΄π‘˜1 , 𝐴′1 ) (𝑣1 ) TΜƒ 1 πœ‡πΆπ‘ƒ(π΄π‘˜2 , 𝐴′2 ) (𝑣2 )) TΜƒ 1 πœ‡πΆπ‘ƒ(π΄π‘˜3 , 𝐴′3 ) (𝑣3 ) TΜƒ 1 … TΜƒ 1 πœ‡πΆπ‘ƒ(π΄π‘˜π‘› , 𝐴′𝑛 ) (𝑣𝑛 ),

where TΜƒ 1 is an extended according to the extension principle 𝑛-ary t-norm [Dub90] and

                                            πœ‡πΆπ‘ƒ(π΄π‘˜π‘– , 𝐴′𝑖 ) (𝑣𝑖 ) =         sup {πœ‡π΄β€²π‘– (π‘₯𝑖 )}.
                                                                       πœ‡π΄π‘˜π‘– (π‘₯𝑖 )=𝑣𝑖
                                                                            π‘₯𝑖 βˆˆπ‘‹π‘–


Particularly, if 𝑛 = 2, then

          πœ‡πΆπ‘ƒ(π‘¨π’Œ , 𝑨′ ) (𝑣) = TΜƒ 1 πœ‡πΆπ‘ƒ(π΄π‘˜π‘– , 𝐴′𝑖 ) (𝑣𝑖 ) =            sup            {πœ‡πΆπ‘ƒ(π΄π‘˜1 , 𝐴′1 ) (𝑣1 ) T3 πœ‡πΆπ‘ƒ(π΄π‘˜2 , 𝐴′2 ) (𝑣2 )}.
                                  𝑖=1,2                          𝑣1 T1 𝑣2 = 𝑣
                                                                  (𝑣1 ,𝑣2 )∈[0;1]2


Computational complexity of the latter expression has order of 𝑂(|𝑣|2 ). In case T4 = T2 = T,
then associativity of t-norms allows us to transform (9) into
                      {               T    T         }     {            T }T                    T
πœ‡π΅π‘˜β€² (𝑦) = sup            πœπ‘¨π’Œ |𝑨′ (𝑣) βˆ— (𝑣 βˆ— πœ‡π΅π‘˜ (𝑦)) = sup πœπ‘¨π’Œ |𝑨′ (𝑣) βˆ— 𝑣 βˆ— πœ‡π΅π‘˜ (𝑦) = Ξ π‘¨π’Œ |𝑨′ βˆ— πœ‡π΅π‘˜ (𝑦),
            π‘£βˆˆ[0;1]                                                  π‘£βˆˆ[0;1]
                                                                                                                                         (10)
where π‘˜ = 1, 𝑁 and
                                                                                            T
                                               Ξ π‘¨π’Œ |𝑨′ = sup {πœπ‘¨π’Œ |𝑨′ (𝑣) βˆ— 𝑣}                                                           (11)
                                                            π‘£βˆˆ[0;1]
is a scalar value which represents a generalization of an expression defined in [Yag83] and
means how much terms π‘¨π’Œ of rule π‘˜ correspond to input values 𝑨′ (or vice versa).
   This means that using fuzzy truth values in (9) makes its computational complexity polyno-
mial and does not impose restrictions onto t-norms (5).
   In case π‘¨π’Œ = 𝑨′ , then πœπ‘¨π’Œ |𝑨′ (𝑣) = 𝑣, i.e. 𝐢𝑃(π‘¨π’Œ , 𝑨′ ) is β€œtrue”. Hence
                       {            T }T               { T }T               T
   πœ‡π΅π‘˜β€² (𝑦) = sup       πœπ‘¨π’Œ |𝑨′ (𝑣) βˆ— 𝑣 βˆ— πœ‡π΅π‘˜ (𝑦) = sup 𝑣 βˆ— 𝑣 βˆ— πœ‡π΅π‘˜ (𝑦) = 1 βˆ— πœ‡π΅π‘˜ (𝑦) = πœ‡π΅π‘˜ (𝑦),
             π‘£βˆˆ[0;1]                                π‘£βˆˆ[0;1]

what indicates the fulfillment of the first criterion of correspondence of an inference method
to approximate reasoning [Rut10].
   Let us consider inference based on (10), which belongs to so-called FITA-approaches (First
Inference, Then Aggregate), i.e. when inference for every rule is performed prior to aggregation
of the result. Aggregation for Mamdani model is implemented by means of S-norms [Rut04].
For example, let us use the Lukasiewicz t-norm [Als06] in (10), which could not be used in
inference before due to the computational complexity:
                                      T
                               Ξ π‘¨π’Œ |𝑨′ βˆ— πœ‡π΅π‘˜ (𝑦) = {0, Ξ π‘¨π’Œ |𝑨′ + πœ‡π΅π‘˜ (𝑦) βˆ’ 1}.                  (12)

   FITA-fuzzy process based on (12) is illustrated in figure 1, where three fuzzy sets π΅π‘˜ , π‘˜ = 1, 3
with Gaussian membership functions are depicted subsequently. Here we assume that these
fuzzy sets are normal, i.e. sup𝑦 {πœ‡π΅π‘˜ (𝑦)} = 1. Each of π΅π‘˜β€² is derived from a particular rule
according to formula (11) from fuzzy set π΅π‘˜ by pushing it down. The membership function
obtained as union of fuzzy sets π΅π‘˜β€² , π‘˜ = 1, 3 using maximum operation is depicted at the bottom
of the figure. The maximum operation is an example of S-norms.
   Let us compare the shapes of fuzzy sets π΅π‘˜β€² , derived with the use of Lukasiewicz’s t-norm,
to ones that were obtained using minimum and arithmetical product operations. In the first
case, membership functions are being β€œtruncated”, in the second case they are being β€œscaled”
[Kru01].


5. Fuzzy system based on center average defuzzification
   method
Let us consider the systems introduced in section 4 having fuzzy inputs and using the center
average defuzzification method [Rut04]. In this case, the crisp output value is defined by the
following formula:
                                      βˆ‘π‘˜=1,𝑁 𝑦 π‘˜ β‹… πœ‡π΅π‘˜β€² (𝑦 π‘˜ )
                                  𝑦=                           ,                          (13)
                                           βˆ‘π‘˜=1,𝑁 πœ‡π΅π‘˜β€² (𝑦 π‘˜ )
where 𝑦 is the crisp output of a system, consisting of 𝑁 rules; 𝑦 π‘˜ are centers of membership
functions πœ‡π΅π‘˜ (𝑦), π‘˜ = 1, 𝑁 , i.e. points, for which

                                       πœ‡π΅π‘˜ (𝑦 π‘˜ ) = sup{πœ‡π΅π‘˜ (𝑦)} = 1                            (14)
                                                   π‘¦βˆˆπ‘Œ
          πœ‡π΅1 (𝑦)                                         πœ‡π΅β€² (𝑦)
      1                                               1     1




Π𝐴1 |𝐴′




      0                   𝑦1                  𝑦       0                                    𝑦
          πœ‡π΅2 (𝑦)                                         πœ‡π΅β€² (𝑦)
      1                                               1     2




Π𝐴2 |𝐴′




      0              𝑦2                       𝑦       0                                    𝑦
          πœ‡π΅3 (𝑦)                                         πœ‡π΅β€² (𝑦)
      1                                               1     3


Π𝐴3 |𝐴′




      0                        𝑦3             𝑦       0                                    𝑦
                                                          πœ‡π΅β€² (𝑦)
                                                      1




                                                      0                𝑦2 𝑦1 𝑦3            𝑦

Figure 1: Graphical representation of inference based on (12) and the Lukasiewicz t-norm




is true. According to expressions (9) and (13) we get
                                              {            T4   T2           }
                       βˆ‘π‘˜=1,𝑁 𝑦 π‘˜ β‹… supπ‘£βˆˆ[0;1] πœπ‘¨π’Œ |𝑨′ (𝑣) βˆ— (𝑣 βˆ— πœ‡π΅π‘˜ (𝑦 π‘˜ ))
                    𝑦=                        {            T4   T2           }.            (15)
                           βˆ‘π‘˜=1,𝑁 supπ‘£βˆˆ[0;1] πœπ‘¨π’Œ |𝑨′ (𝑣) βˆ— (𝑣 βˆ— πœ‡π΅π‘˜ (𝑦 π‘˜ ))
From (14) follows
                          {            T     T    }     {            T   }
                 sup       πœπ‘¨π’Œ |𝑨′ (𝑣) βˆ—4 (𝑣 βˆ—2 1) = sup πœπ‘¨π’Œ |𝑨′ (𝑣) βˆ—4 𝑣 = Ξ π‘¨π’Œ |𝑨′ ,        (16)
                π‘£βˆˆ[0;1]                                 π‘£βˆˆ[0;1]

because a t-norm meets boundary condition T(π‘Ž; 1) = π‘Ž by definition. Substituting (16) into
(15), we get
                                   βˆ‘       𝑦 π‘˜ β‹… Ξ π‘¨π’Œ |𝑨′
                                𝑦 = π‘˜=1,𝑁                .                             (17)
                                         βˆ‘π‘˜=1,𝑁 Ξ π‘¨π’Œ |𝑨′
Therefore the result 𝑦 does not depend on the specific t-norm T2 when using the center average
defuzzification method for systems with fuzzy inputs.
  Let us consider the inference with crisp input data, hence
                                                        {
                                                         1, if 𝑣 = π‘£π‘˜ ,
                             πœπ‘¨π’Œ |𝑨′ (𝑣) = 𝛿(𝑣 βˆ’ π‘£π‘˜ ) =
                                                         0, if 𝑣 β‰  π‘£π‘˜ ,

where
                                      π‘£π‘˜ = T1 πœ‡π΄π‘–π‘˜ (π‘₯ 𝑖 ),        π‘˜ = 1, 𝑁 ,
                                            𝑖=1,𝑛

in which π‘₯ 𝑖 , 𝑖 = 1, 𝑛 are crisp input values, and T1 is a t-norm formalizing the conjunction in
π‘˜-th rule’s antecedent. Then
                                               {           T   }
                                Ξ π‘¨π’Œ |𝑨′ = sup 𝛿(𝑣 βˆ’ π‘£π‘˜ ) βˆ—2 𝑣 = π‘£π‘˜ ,
                                           π‘£βˆˆ[0;1]

considering that T2 (1; π‘£π‘˜ ) = π‘£π‘˜ . Therefore, the output value is defined as follows:

                                                βˆ‘π‘˜=1,𝑁 𝑦 π‘˜ β‹… π‘£π‘˜
                                           𝑦=                   ,
                                                    βˆ‘π‘˜=1,𝑁 π‘£π‘˜

what turns out to be the zero order Takagi-Sugeno’s fuzzy inference algorithm [Kru01]. Thus,
system output does not depend on t-norms T2 and T4 in the case of crisp input data and the
center average defuzzification method. The structure of a fuzzy system that is described by
expression (17) is shown in figure 2.


6. Fuzzy system based on the center of gravity defuzzification
   method
Let us consider those systems introduced in section 4 having fuzzy inputs and using a discrete
variant of the center of gravity defuzzification method [Rut04]

                                             βˆ‘π‘˜=1,𝑁 𝑦 π‘˜ β‹… πœ‡π΅π‘˜β€² (𝑦 π‘˜ )
                                        𝑦=                                ,                  (18)
                                                     βˆ‘π‘˜=1,𝑁 πœ‡π΅π‘˜β€² (𝑦 π‘˜ )
                𝐢𝑃(𝐴11 , 𝐴′1 )
                                                                                            𝑦1
                𝐢𝑃(𝐴21 , 𝐴′2 )
                                                TΜƒ                      Π𝐴1 |𝐴′
                                                                                           𝑦2
                                                                                                     Ξ£
                𝐢𝑃(𝐴𝑛1 , 𝐴′𝑛 )


                𝐢𝑃(𝐴12 , 𝐴′1 )
                                                                                            𝑦𝑁
  𝐴′1           𝐢𝑃(𝐴22 , 𝐴′2 )
                                                TΜƒ                      Π𝐴2 |𝐴′
  𝐴′2
                                                                                                         Γ·          𝑦
                𝐢𝑃(𝐴𝑛2 , 𝐴′𝑛 )


                                                                                            1
  𝐴′𝑛

                𝐢𝑃(𝐴1𝑁 , 𝐴′1 )
                                                                                           1
                𝐢𝑃(𝐴2𝑁 , 𝐴′2 )                                                                       Ξ£

                                                TΜƒ                      Π𝐴𝑁 |𝐴′


                𝐢𝑃(𝐴𝑛𝑁 , 𝐴′𝑛 )                                                              1

                        1                        2                         3                     4       5


Figure 2: Network structure of inference process based on (17)




where 𝑦 is the crisp output value, and 𝑦 π‘˜ are the centers of membership functions πœ‡π΅π‘˜ (𝑦), π‘˜ =
1, 𝑁 , defined by expression (14). Fuzzy set 𝐡′ is derived by the union of fuzzy sets π΅π‘˜β€² , π‘˜ = 1, 𝑁
using the maximum operator or any other S-norm, i.e.

                                        πœ‡π΅β€² (𝑦) = S πœ‡π΅π‘˜β€² (𝑦).                                                (19)
                                                         π‘˜=1,𝑁

From (18), (9) and (19) we get
                                           {             {                                      }}
                                                                           T4     T2
                       βˆ‘π‘˜=1,𝑁 𝑦 π‘˜ β‹… S           sup          πœπ‘¨π’‹ |𝑨′ (𝑣) βˆ— (𝑣 βˆ— πœ‡π΅π‘— (𝑦 π‘˜ ))
                                   𝑗=1,𝑁       π‘£βˆˆ[0;1]
                  𝑦=                       {             {                                      }} .         (20)
                                                                           T4     T2
                             βˆ‘π‘˜=1,𝑁 S           sup          πœπ‘¨π’‹ |𝑨′   (𝑣) βˆ— (𝑣 βˆ— πœ‡π΅π‘— (𝑦 π‘˜ ))
                                   𝑗=1,𝑁       π‘£βˆˆ[0;1]
Let us denote πœ‡π΅π‘— (𝑦 π‘˜ ) = π‘π‘—π‘˜ . From (14) follows π‘π‘˜π‘˜ = πœ‡π΅π‘˜ (𝑦 π‘˜ ) = 1. According to (12), the S-norm
can be written as follows:
      {        {                        }}                    {          {                      }}
                           T4      T2                                              T4    T2
   S    sup πœπ‘¨π’‹ |𝑨 (𝑣) βˆ— (𝑣 βˆ— π‘π‘—π‘˜ )
                     β€²                       = S Π𝑨𝒋 |𝑨 , S
                                                       β€²            sup πœπ‘¨π’‹ |𝑨 (𝑣) βˆ— (𝑣 βˆ— π‘π‘—π‘˜ )
                                                                              β€²                       .
 𝑗=1,𝑁 π‘£βˆˆ[0;1]                                  (        𝑗=1,𝑁 π‘£βˆˆ[0;1]                              )
                                                            π‘—β‰ π‘˜
                                                                                            (21)
The network architecture corresponding to expression (20) with substitution (21) is represented
in figure 3. If T4 = T2 = T, then
                    {           {                     }}                    {          }
                                         T    T                                  T
              S          sup πœπ‘¨π’‹ |𝑨′ (𝑣) βˆ— (𝑣 βˆ— π‘π‘—π‘˜ )    = S Π𝑨𝒋 |𝑨′ , S Π𝑨𝒋 |𝑨′ βˆ— π‘π‘—π‘˜    .       (22)
            𝑗=1,𝑁       π‘£βˆˆ[0;1]                             (         𝑗=1,𝑁              )
                                                                     π‘—β‰ π‘˜

In this case the network architecture of the system takes the form represented in figure 4. If

                                    π‘π‘—π‘˜ β‰ˆ 0 for 𝑗, π‘˜ = 1, 𝑁 ,     𝑗 β‰  π‘˜,                          (23)

then expressions (21) and (22) will take the form of (17), and the network architectures given
in figures 3 and 4 take the form of the architecture depicted in figure 2. Figure 5 provides an
example of fuzzy sets π΅π‘˜ , π‘˜ = 1, 𝑁 that meet condition (23). Therefore, the center average and
center of gravity (defined by expression (13)) defuzzification methods lead to same results for
the same input data.


7. Conclusion
Inference based on fuzzy truth value enables us to spread Mamdani’s approach onto systems
with multiple fuzzy inputs regardless of the t-norms used, thereby eliminating exponential
computational complexity.
   Moreover, the most important advantage of using the concept of fuzzy truth value is the
fact that the relation between the premise and fact is represented as a fuzzy set, in contrast to
methods [Rut10, Als06], which reduce this relation to a scalar value.
   Representing all the relationships between the premises and facts within the same space of
truthfulness reduces the computational complexity of the inference result from exponential to
polynomial.
   Expressions of output values for fuzzy systems utilizing measure of possibility generaliza-
tion (11) with the use of center average and center of gravity defuzzification methods were
introduced in the article.
   Formulas (17), (20), (21), (22) were used to build network structures. Using learning algo-
rithms for their parameters they can be transformed into neuro-fuzzy systems.


Acknowledgments
This work is partially supported by RFBR (grant β„–20-07-00030).
                                                                       1
             𝐢𝑃(𝐴11 , 𝐴′1 )
                                                                𝑏12                   𝑦1
             𝐢𝑃(𝐴21 , 𝐴′2 )                                                𝐹12
                                   TΜƒ           Π𝐴1 |𝐴′                           S
                                                                                      𝑦2
                                                                𝑏1𝑛                            Ξ£
             𝐢𝑃(𝐴𝑛1 , 𝐴′𝑛 )                                                𝐹1𝑛
                                                  1

             𝐢𝑃(𝐴12 , 𝐴′1 )                                     𝑏21
                                                                           𝐹21        𝑦𝑁
  𝐴′1        𝐢𝑃(𝐴22 , 𝐴′2 )
                                                Π𝐴2 |𝐴′
                                   T̃                                      1
  𝐴′2
                                                                                  S                 Γ·   𝑦
             𝐢𝑃(𝐴𝑛2 , 𝐴′𝑛 )                                     𝑏2𝑛
                                                  1                        𝐹2𝑛

                                                                                      1
  𝐴′𝑛

             𝐢𝑃(𝐴1𝑁 , 𝐴′1 )                                     𝑏𝑁 1
                                                                           𝐹𝑁 1
                                                                                      1
             𝐢𝑃(𝐴2𝑁 , 𝐴′2 )                                                                    Ξ£
                                                Π𝐴𝑁 |𝐴′
                                                                𝑏𝑁 2              S
                                   TΜƒ                                      𝐹𝑁 2


             𝐢𝑃(𝐴𝑛𝑁 , 𝐴′𝑛 )                                                           1
                                                  1                    1
                     1              2                 3                    4      5        6        7

                                                            T              T
                              where πΉπ‘—π‘˜ = sup {πœπ‘¨π’‹ |𝑨′ (𝑣) βˆ—4 (𝑣 βˆ—2 π‘π‘—π‘˜ )}
                                           π‘£βˆˆ[0;1]

Figure 3: Network structure of inference process based on (21)




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Figure 4: Network structure of inference process based on (22)



        1       πœ‡π΅1 (𝑦)             πœ‡π΅2 (𝑦)             πœ‡π΅3 (𝑦)                πœ‡π΅4 (𝑦)




        0            𝑦1                 𝑦2                    𝑦3                   𝑦4             𝑦

Figure 5: An example of set of terms meeting condition (23)




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