DEPARTMENT OFINFORMATICS ANDAPPLIEDMATHEMATICS
PROGRAM OFGRADUATESTUDIES INSYSTEMS ANDCOMPUTING
DOCTORATE IN COMPUTER SCIENCE
On Algebras for Interval-Valued Fuzzy Logic
Antˆonia Jocivania Pinheiro
Natal-RN August, 2019
On Algebras for Interval-Valued Fuzzy Logic
This thesis was submitted to the Postgraduate Program in Systems and Computing of Federal University of Rio Grande do Norte.
Area of Concentration: Fundations for Com-puting.
Advisor:
Prof. Dr. Regivan Hugo Nunes Santiago
Natal-RN August, 2019
Pinheiro, Antônia Jocivania.
On algebras for interval-valued fuzzy logic / Antonia Jocivania Pinheiro. - 2019.
135f.: il.
Tese (Doutorado)-Universidade Federal do Rio Grande do Norte, Centro de Ciências Exatas e da Terra, Pós-Graduação em Sistemas e Computação, Natal, 2019.
Orientador: Dr. Regivan Hugo Nunes Santiago.
1. Interval Mathematics - Tese. 2. Fuzzy Logic - Tese. 3. BCI Algebras - Tese. 4. SBCI Algebras - Tese. 5. Fuzzy Implications - Tese. I. Santiago, Regivan Hugo Nunes. II. Título.
RN/UF/BCZM CDU 004.032.26
This work aims to introduce other approaches to the interval-valued fuzzy logic. These new approaches were inspired by Lodwick and Chalco’s works on constraint intervals. These constraint intervals were used in this thesis to extend the fuzzy operators into two modes, named Single-Level Constrained Interval Operatorsand Constrained Interval Operators and studied their properties. A new algebra, called SBCI algebra, which arises from the intervalization of BCI-algebras, is also introduced. These algebras aims to be the algebraic model for interval-valued fuzzy logics, which take into account the notion of correctness.
A new class of fuzzy implications, called (T, N)-implications has also been studied. The author investigated the behavior of the BCI/SBCI algebras and(T, N)-implications.
Keywords: Interval-valued fuzzy logic, Interval Mathematics, Fuzzy Logic, BCI algebras, SBCI algebras, Fuzzy Implications.
Este trabalho visa introduzir outras abordagens para a l´ogica fuzzy com valores intervalares. Essas novas abordagens foram inspiradas nos trabalhos de Lodwick e Chalco sobre intervalos restritos. Esses intervalos restritos foram usados para estender os operadores fuzzy, nos quais eles foram chamados Operadores Intervalares Restritos de N´ıvel ´Unico (C-operador) e suas propriedades foram estudadas. Al´em disso, esses operadores foram estendidos a operadores cor-retos chamados Operadores Intervalares Restritos. Uma nova ´algebra, chamada SBCI ´algebra, que surge da intervalizac¸˜ao de BCI ´algebras, tamb´em ´e introduzida. Essas ´algebras tˆem como objetivo ser o modelo alg´ebrico para l´ogicas fuzzy com valores intervalares que levam em conta a noc¸˜ao de correc¸˜ao.
Tamb´em foi estudada uma nova classe de implicac¸˜oes fuzzy, chamada(T, N)-implicac¸˜oes. O autor investigou o comportamento das BCI/SBCI ´algebras e das(T, N)-implicac¸˜oes.
Palavras-chave: L´ogica Fuzzy com Valores Intervalares, Matem´atica Intervalar, L´ogica Fuzzy, BCI ´algebras, SBCI ´algebras, Implicac¸˜oes Fuzzy.
1 Introduction p. 8
I Preliminaries
12
2 Interval Arithmetics and Logics p. 13 2.1 Arithmetics . . . p. 13 2.1.1 Standard Interval Arithmetic . . . p. 13 2.1.1.1 Intervalization of Structures . . . p. 14 2.1.2 Constrained Interval Arithmetic and Single-Level Constrained
Inter-val Arithmetic . . . p. 15 2.2 Logics . . . p. 17 2.2.1 Fuzzy Connectives . . . p. 18 2.2.2 Interval Fuzzy Connectives . . . p. 22 2.3 Final Remarks . . . p. 23 3 BCI Algebras p. 24 3.1 Pseudo-BCI Algebras . . . p. 26 3.2 Final Remarks . . . p. 28
II Contributions
29
4 (T , N )-Implications p. 304.1 Definition and Basic Properties . . . p. 30 4.2 Functional equations and(T, N)-implications . . . p. 42 4.3 Applying(T, N)-implications to generate fuzzy subsethood measures . . . . p. 47 4.4 Generalization of(T, N)-implications . . . p. 52 4.4.1 Characterizations of(N′, T, N)-Implications . . . p. 64 4.4.2 Aggregating(N′, T, N)-Implications . . . p. 66
5 Intervalization of BCI algebras and Semi-BCI algebras p. 68 5.1 Intervalization of BCI algebras . . . p. 68 5.2 Semi-BCI algebras . . . p. 74 5.3 Comparing Semi-BCI and Pseudo-BCI Algebras . . . p. 78 5.4 Final Remarks . . . p. 79
6 Single-Level Constraint Interval Fuzzy Connectives p. 80 6.1 Single-Level Constrained Interval Operators . . . p. 82 6.2 Composition of C-Operators . . . p. 88
6.2.1 Properties of Some Interval Fuzzy Connectives with Respect to Eager
Composition . . . p. 89 6.2.2 Properties of Some Interval Fuzzy Connectives with Respect to
Single-Level Composition . . . p. 95 6.3 Final Remarks . . . p. 100
7 Constrained Interval Fuzzy Connectives p. 101 7.1 Some Properties of the Constrained Interval Fuzzy Connectives with Respect
to Order≤KM . . . p. 111 7.1.1 (T, N)-Implications Generated from Constrained Interval Fuzzy
Con-nectives . . . p. 115 7.2 Some Properties of the Constrained Interval Fuzzy Connectives with Respect
to Order⊴ . . . p. 117 7.3 On the Extension of BCI and Semi-BCI Algebras via Constrained Interval
Fuzzy Operators . . . p. 120 7.3.1 On Extension of BCI and Semi-BCI Algebras with Respect to Eager
Composition . . . p. 121 7.3.2 On Extension of BCI and Semi-BCI Algebras with Respect to
Con-strained Punctual Composition . . . p. 122 7.4 Final Remarks . . . p. 123
8 Concluding remarks p. 125
1
Introduction
Interval-valued fuzzy logic was developed in order to deal with the uncertainties concerning not only on fuzzy rules, in which fuzzy logic is very successful, but also on inputs-outputs. In cases in which the fuzzification of inputs derives from choices made by experts, fuzzy logic is not always suitable, as the expert may be unsure when defining the membership function. This was one of the motivations for the research on interval-valued fuzzy logic. Another case lies in the situation in which the inputs represent imprecise numerical data. Interval-valued fuzzy logic is a particular case of interval type-2 fuzzy logic (ZADEH, 1975;LIANG; MENDEL, 2000;
BUSTINCE et al., 2015;CASTILLO et al., 2016).
Interval-valued fuzzy logic was applied in a wide variety of domains, for example: Melin and Castillo (CERVANTES; CASTILLO; MELIN, 2011) used it in the context of plant control; Figueroa et al. (FIGUEROA et al., 2005) for non-autonomous robots in the context of a robot football game; Lynch et al. (LYNCH; HAGRAS; CALLAGHAN, 2005) built an interval control sys-tem for large marine diesel engines; Chourasia et al. (CHOURASIA; TIWARI; GANGOPADHYAY, 2014) developed a new method for assessing fetal health status based on interval type-2 fuzzy logic through fetal phonocardiography; Nguyen et al. (NGUYEN et al., 2015) used the wavelet feature in interval type-2 fuzzy logic system (IT2FLS) to reduce the computation burden and time of IT2FLS; Leow et al. (LEOW et al., 2019) developed a hybrid of Generalized Adaptive Resonance Theory (GART) and interval type-2 fuzzy logic system algorithm; among many others.
The field of interval analysis has a long term development. The idea of bounding round-ing errors by computround-ing with intervals was first given by Warmus (WARMUS, 1956), Sunaga (SUNAGA, 1958) and Moore (MOORE, 1959). However, it can be said that interval mathemat-ics and analysis began with the appearance of R. E. Moore’s book, Interval Analysis in 1966 (MOORE, 1966). It deals with numerical data in the form of compact intervals in order to encode computational errors or inaccuracies. The interval analysis has been applied in several areas (JAULIN et al., 2001;KEARFOTT; KREINOVICH, 2013), like: Electrical power systems (BARBOZA; DIMURO; REISER, 2004), mechanical engineering (MUHANNA; ZHANG; MULLEN, 2007), chem-ical engineering (STADTHERR et al., 2007), artificial intelligence (HU et al., 2008), multi-agent systems(DIMURO; COSTA, 2004) and geophysics (AGUIAR; DIMURO; COSTA, 2004).
Warmus (WARMUS, 1956), Teruo Sunaga (SUNAGA, 1958) and Ramon Moore (MOORE, 1959, 1962) independently developed the interval arithmetic. The Moore’s arithmetic is ac-cepted as the standard approach and is called here standard interval arithmetic (SIA). There are two most important criteria for an interval arithmetic, namely: Correctness (accuracy) and op-timality(HICKEY; JU; EMDEN, 2001;MOORE, 1979). The first criterion establishes that the result of an interval computation must always contain the value of the respective real function (see (MOORE, 1979, Theorem 3.1)). Although correctness is a desirable property, not every interval
method is correct. Santiago et al. (SANTIAGO; BEDREGAL; ACI ´OLY, 2006) investigates the no-tion of correctness for interval funcno-tions and its impact on some interval topological viewpoints. They call correctness as interval representations, since interval entities (algorithms and inter-vals) are seen as linguistic entities which represent real entities (functions and numbers). An interval function, F , is said to represent a real function, f , whenever it satisfies the following property: x ∈ [a, b] ⇒ f(x) ∈ F([a, b]) and F([a, a]) = f(a). The second criterion estab-lishes that the resulting interval of a computation should not be greater than necessary, which is captured by the notion of canonical interval representation.
The process of giving the correct and optimal interval version F for a function f is called intervalization. There are many proposals of intervalization of algebraic structures further than that of real numbers proposed by Moore and Sunaga, see for example the case for Łukasiewicz algebras and MV-algebras in (BEDREGAL; SANTIAGO, 2013) and (CABRER; MUNDICI, 2014), respectively. In most cases, interval algebras fail to satisfy some properties that are satisfied by the algebras from which they came.
In order to solve the algebraic incompatibility between the real arithmetic and Moore arith-metic, Lodwick (LODWICK, 1999) defined a new interpretation for intervals called constraint intervals. With this new approach, Lodwick proposed an alternative to Moore arithmetic on intervals in order to have X− X = [0, 0], X ÷ X = [1, 1] if 0 ∉ X and the distributive property. This new arithmetic, called constrained interval arithmetic (CIA), is an extension of Moore’s interval arithmetic, in the sense that they coincide in the case that there is no variable depen-dence and are distinct when there are dependencies. In this case, the CIA arithmetic presents a smaller width interval, thus improving the overestimation of Moore’s arithmetic.
This thesis presents a new approach to interval-valued fuzzy logic, in which interval oper-ators preserve some of the main algebraic properties, the overestimation problem is mitigated and there is no loss of information. These operators were defined using the constrained interval introduced by Lodwick (LODWICK, 1999). In what follows, it is described how this study was done and also the contributions that were studied or compared with this new approach. The reader can find a more detailed discussion in Part II.
The first contribution of this work lies on the investigation of a new class of fuzzy implica-tions, called(T, N)-implications (BEDREGAL, 2007), in which it is obtained from the compo-sition of a fuzzy negation and a t-norm. It is not difficult to find in the literature implications that are obtained through other operators, among which we can mention the(S, N)-, R- and QL-implications that have been widely investigated. Many applications have already been de-veloped using fuzzy implications, such as (MAS et al., 2007;BACZY ´NSKI, 2013;BACZY ´NSKI et al., 2013), and they can still be applied in areas of study such as approximate reasoning, control the-ory, decision making thethe-ory, expert systems, diffuse mathematical morphology (BLOCH, 2009;
YAGER, 2004;BACZY ´NSKI, 2013;BANDLER; KOHOUT, 1980;BUSTINCE et al., 2013), among oth-ers. In this document, the main properties of(T, N)-implications (PINHEIRO et al., 2017, 2018a) with respect to different fuzzy negations were studied. In addition, an application to fuzzy subsethood measure was presented, in which a new subsethood measure was defined, namely P B-subsethood measure (PINHEIRO et al., 2018b), and it was verified that it is possible to gen-erate this measure from a family of(T, N)-implications. Finally, it was defined (N′, T, N )-implications(PINHEIRO et al., 2018), generalizes(T, N)-implications. The author also presents a characterization of the (N′, T, N)-implications and verify that it is possible to aggregate a family of(N′, T, N)-implications and this aggregation is still an (N′, T, N)-implication.
The second contribution lies on the application of intervalization on BCI algebras (IS ´EKI, 1966; HUANG, 2006). BCI algebras are the algebraic counterpart of a common fragment of several important logics, like fuzzy logic, which is modeled by BL algebras (a kind of BCI algebra). Thus, the intervalization of BCI algebras are important for the construction of an algebraic model for some interval fuzzy logics. Here, a class of BCI algebras was interval-ized and an investigation of the resulting structures was provided. Like for MV-algebras and Łukasiewicz algebras, the resulting interval structure does not belong to the same category of its starting algebra, it is a new mathematical structure. This new structure is a generalization of the BCI algebras and came to be called semi-BCI algebra (SANTIAGO et al., 2019). The semi-BCI algebras were studied in detail and, in addition, the relationship between the semi-semi-BCI and pseudo-BCI algebras has been investigated. It is verified that the only intersection between the two is the class of BCI algebra.
The third contribution introduces a new approach to interval-valued fuzzy logic, in which the notion of constraint interval, proposed by Lodwick in (LODWICK, 1999), is applied. In 2014, Chalco-Cano et al. in (CHALCO-CANO; LODWICK; BEDE, 2014) proposed a variant of constraint interval operators that uses a single parameter (level), instead of using two parameters proposed by Lodwick. Following Lodwick and Chalco, the author extends the fuzzy operators to the so-called Single-Level Constrained Interval Operators (C-operators) and studied their main properties. Also, it has been shown that the fuzzy connectives are extended to their respective C-operators, however, not all are correct. The composition of C-operators provides two methods of evaluation for interval compositions, called: Eager evaluation and single-level evaluation. Important properties such as the exchange principle, contraposition law (also, left and right contraposition law), among others, were investigated by using both methods.
The main and final contribution of this thesis presents another approach to interval-valued fuzzy logic, in which the new operators, called Constrained Interval Operators, are very close to Moore’s correctness, in fact, they satisfy a new correction that will be suggested, in which it is as efficient as Moore’s correction, being, however, less demanding, which was called here Constraint Interval Correctness. Two methods of evaluating the compositions of these opera-tors, namely: Eager and constrained punctual compositions, have been defined. In addition to the comparative study between single-level constrained interval operators and constrained inter-val operators, the main properties of this operator have been verified in relation to two orders, namely: Kulisch-Miranker and Moore order. It has been found that this new approach guar-antees the extension of many algebraic properties and that its fuzzy operators are extended to their respective constrained interval operators, when considering the order of Kulisch-Miranker. Also, a special class of fuzzy implications was extended, called (T, N)-implications. It has been found that the(T, N)-implications generated from constrained interval operators coincide with the best representation of the original(T, N)-implication. Finally, both algebras, BCI and semi-BCI, are extended to their respective constrained interval algebras with respect to punctual composition. Here correctness is maintained from the perspective of Constrained Interval.
This thesis is organized as follows: Part I recalls some definitions and concepts used throughout the text in order to provide a self-contained document. It is divided into two chap-ters: Chapter 2 presents interval arithmetics and logics; and Chapter 3 introduces the BCI alge-bras. Part II presents the contributions of this work, it is divided into five chapters: Chapter 4 provides a new class of fuzzy implications called(T, N)-implication. Chapter 5 proposes a new algebra called Semi-BCI algebra, which generalizes BCI algebras. Chapter 6 proposes a new way of making interval-valued fuzzy logics, in which the operators were called Single-Level
Constrained Interval Operatorsand Chapter 7 proposes another way of making interval-valued fuzzy logics, in which the operators were called Constrained Interval Operators, which gener-alize the operators of the previous chapter. The last chapter includes some conclusions, future works, and the bibliography.
Part I
Preliminaries
2
Interval Arithmetics and Logics
2.1
Arithmetics
The limited capacity of machines to store just a finite set of finitely represented objects con-strains the automatic calculation (computation) of structures in which a machine representation of some objects exceeds such capacity. In the case of real numbers, although programs often provide highly accurate results, it can happen that rounding errors built up during each step in the computation produce results which are not even meaningful. One of the proposals to over-come this problem is due, almost simultaneously, to M. Warmus (WARMUS, 1956), T. Sunaga (SUNAGA, 1958) and R. Moore (MOORE, 1959, 1962), with the development of the so-called interval arithmetic, as the following section shows:
2.1.1
Standard Interval Arithmetic
Interval arithmetic is a set of operations on the set of all closed intervals I(R) = {X ∣ X = [x, x]; x, x ∈ R and x ≤ x}. The operations are defined in the following way:
1. X+ Y = [x + y, x + y], 2. X− Y = [x − y, x − y] ,
3. X⋅ Y = [min{x ⋅ y, x ⋅ y, x ⋅ y, x ⋅ y}, max{x ⋅ y, x ⋅ y, x ⋅ y, x ⋅ y}], 4. X/Y = [x, x] ⋅ ([1/y, 1/y]), provided that 0 ∉ [y, y].
Observe that for each operation ∗ ∈ {+, −, ⋅, /}, X ⊛ Y = {x ∗ y ∈ R ∶ x ∈ X and y ∈ Y } is always an interval.
Moore’s development of interval arithmetic is accepted as the standard approach to interval arithmetic, which will be called standard interval arithmetic (SIA), and it is the approach to interval arithmetic in common use. Here are some properties associated with SIA (see (MOORE, 1979)), for X, Y and Z in I(R):
(1) X+ (Y + Z) = (X + Y ) + Z – the associative law for addition (2) X⋅ (Y ⋅ Z) = (X ⋅ Y ) ⋅ Z – the associative law for multiplication (3) X+ Y = Y + X – the commutative law for addition
(4) X⋅ Y = Y ⋅ X – the commutative law for multiplication (5) [0, 0] + X = X + [0, 0] = X – additive identity
(6) [1, 1] ⋅ X = X ⋅ [1, 1] = X – multiplicative identity
(7) X⋅ (Y + Z) ⊆ X ⋅ Y + X ⋅ Z – the subdistributive property
Moore’s interval arithmetic presents a problem of overestimation associated with multiple occurrences of the same variable in an expression. Also, it is clear that X− X is never 0, unless X is a real number (a zero width interval) and X÷ X is never 1, unless X is a real number (a zero width interval).
In this theory, there are two important criteria called correctness (accuracy) and optimality (HICKEY; JU; EMDEN, 2001;MOORE, 1979), which were formalized by Santiago et al. in ( SAN-TIAGO; BEDREGAL; ACI ´OLY, 2006), and was called interval representation and best interval representation, respectively (see, e.g., (BEDREGAL; SANTIAGO, 2013; BEDREGAL; TAKAHASHI, 2005, 2006)). Both are defined in the following subsection.
2.1.1.1 Intervalization of Structures
Assuming that the set X ⊛ Y = {x ∗ y ∈ R ∶ x ∈ X and y ∈ Y } always corresponds to an interval, where ∗ ∈ {+, −, ⋅, /}, this reveals two important properties of this arithmetic (a) Correctnessand (b) Optimality.
“Correctness. The criterion for correctness of a definition of interval arithmetic is that the “Fundamental Theorem of Interval Arithmetic” holds 1: when an
ex-pression is evaluated using intervals, it yields an interval containing all results of pointwise evaluations based on point values that are elements of the argument in-tervals.
[. . . ]
Optimality. By optimality, it is meant that the computed floating-point interval is not wider than necessary.”
Hickey et.al(HICKEY; JU; EMDEN, 2001, p.1040)
The application of interval methods follows the following paradigm: Enclosure in intervals the values which are not exact by whatever reason (e.g. the value comes from an imprecise measurement) and applying correct and optimal operations on such intervals in order to obtain the best interval which contains the desired output.
The property of correctness was investigated in 2006 by Santiago et al (SANTIAGO; BEDRE-GAL; ACI ´OLY, 2006; BEDREGAL; SANTIAGO, 2013). Instead of correctness, they used the term interval representation, since an interval computation could be understood not just as a machine representation of real numbers, but also as a mathematical representation of real numbers (this idea is confirmed by the Representation Theorems of Euclidean continuous functions in ( SANTI-AGO; BEDREGAL; ACI ´OLY, 2006;BEDREGAL; SANTIAGO, 2013)). Also, the notion of optimality
1Moore (
MOORE, 1979, Theorem 3.1, p. 21): If F is an inclusion monotonic interval extension of f , then
→
f (X1, ..., Xn) ⊆F (X1, ..., Xn), where
→
was named as best interval representation, or best representation for short. And, in what fol-lows, this notion is shown for binary operations: A binary interval operation ⊛ represents a binary real operation,∗, whenever:
(x, y) ∈ X × Y ⇒ x ∗ y ∈ X ⊛ Y.
This can be easily extended to n-ary operations. The author showed that this notion is more general than what is stated by the Fundamental Theorem of Interval Arithmetic, given that there are representations which are not inclusion monotonic (see (SANTIAGO; BEDREGAL; ACI ´OLY, 2006, p. 238)). The formal definition follows:
Definition 2.1. An interval operation F ∶ U([0, 1])n→ U([0, 1]) is Moore-correct or interval
representation with respect to a function f ∶ [0, 1]n → [0, 1] whenever, for all (A
1, . . . , An) ∈
U([0, 1])n and a
i ∈ Ai, f(a1, . . . , an) ∈ F(A1, . . . , An). In addition, F is best interval
rep-resentation with respect to function f , denoted by ˆf , if F(A1, . . . , An) is the least interval
containing the set{f(a1, . . . , an) ∣ ai∈ Ai} for all Ai∈ U([0, 1]) and i ∈ {1, . . . , n}, i.e.,
ˆ
f(A1, . . . , An) = [inf{f(a1, . . . , an) ∣ ai ∈ Ai}, sup{f(a1, . . . , an) ∣ ai∈ Ai}] . (2.1)
The process of giving the correct and optimal interval version F for a function f is called intervalization. There are many proposals of intervalization of algebraic structures further than that of real numbers proposed by Moore, Warmus and Sunaga. In the literature, the reader can find proposals even for the field of Logic. For example: The Łukasiewicz implicative algebra ⟨[0, 1], →LK, 1⟩, where x →LK y = min(1, 1 − x + y) interprets some many-valued logics and
was intervalized by Bedregal et al in (BEDREGAL; SANTIAGO, 2013). Its MV algebra counterpart was intervalized by Cabrer et al in (CABRER; MUNDICI, 2014), also, in order to overcome the same problems already stated for I(R). In both cases, the interval algebras did not satisfy the same properties that are satisfied by the algebras from which they came.
In order to solve the algebraic incompatibility between the real arithmetic and Moore arith-metic, Lodwick (LODWICK, 1999) defined a new interpretation for intervals called constraint intervals, as seen below below.
2.1.2
Constrained Interval Arithmetic and Single-Level Constrained
In-terval Arithmetic
In 1999, Lodwick (LODWICK, 1999) proposed an alternative to Moore arithmetic on inter-vals in order to have X− X = [0, 0] and X ÷ X = [1, 1] if 0 ∉ X. For this purpose, he defined a new way of interpreting intervals, called constrained intervals.
Definition 2.2. Given an interval X = [x, x] a constrained interval associated to X is the functionfX ∶ [0, 1] → [0, 1], s.t for 0 ≤ λx ≤ 1,
fX(λx) = (1 − λx)x + λxx
= x + λxωx,
The resulting arithmetic, called constrained interval arithmetic (CIA), is defined as follows: fX(λx) ∗ fY(λy), (2.2)
for λx, λy ∈ [0, 1] and ∗ ∈ {+, −, ×, ÷}, where the resulting interval, X ⊛ Y , can be extracted
from (2.2) by computing the minimum and maximum,
[ min0≤λ x,λy≤1{f
X(λx) ∗ fY(λy)}, max 0≤λx,λy≤1{f
X(λx) ∗ fY(λy)}] .
This arithmetic retains the desirable properties, i.e., X− X = [0, 0], X ÷ X = [1, 1] and the distributive property X × (Y + Z) = (X × Y ) + (X × Z). Note that this new arithmetic is an extension of Moore’s interval arithmetic, in the sense that they coincide in the case that there is no dependence and are distinct when there are dependencies present. In this case, the CIA arithmetic presents a smaller width interval, thus improving the overestimation of Moore’s arithmetic. See the following example.
Example 2.1. Consider the expression X+ Y − X for the intervals X = [1, 2] and Y = [−1, 1]. Using Moore’s interval arithmetic we obtainX+ Y − X = ([1, 2] + [−1, 1]) − [1, 2] = [0, 3] − [1, 2] = [−2, 2]. While by CIA arithmetic, fX(λx) = 1+λxandfY(λy) = −1+2λy, so(fX(λx)+
fY(λy)) − fX(λx) = fY(λy) and hence X + Y − X = Y = [−1, 1].
Instead of using independent lambdas for each variable, Chalco-Cano et al. in ( CHALCO-CANO; LODWICK; BEDE, 2014) proposed another arithmetic, called single level constrained in-terval arithmetic (SLCIA). This arithmetic is the constrained inin-terval arithmetic in which only one parameter, i.e. λy = λx. That is, if we consider two intervals X and Y , we take a level
λ∈ [0, 1] for both intervals instead for of a λxfor X and λy for Y , i.e. we take
fX(λ) = (1 − λ)x + λx and fY(λ) = (1 − λ)y + λy.
After operating at all levels, the minimum and maximum are calculated, i.e.:
X⊛ Y = [min
0≤λ≤1{fX(λ) ∗ fY(λ)}, max0≤λ≤1{fX(λ) ∗ fY(λ)}].
The SLCIA arithmetic is a restriction of the CIA, in which it maintains the desirable properties, that is, X− X = [0, 0], X ÷ X = [1, 1] and the distributive law. However, in this approach we also have X− Y = [0, 0] when X = Y , unlike CIA.
Chalco-Cano extended the single-level interval arithmetic for expressions with interval operands. The evaluation of an expression is performed according to the following rule:
E(A1, . . . , An) = [min
0≤λ≤1{E(fA1(λ), . . . , fAn(λ))}, max
0≤λ≤1{E(fA1(λ), . . . , fAn(λ))}].
Considering this way of evaluating the expressions, they showed the following algebraic properties: For all interval X, Y, Z and α, β∈ R,
(1) X⊕ (−Y ) = X ⊖ Y ; (2) X⊖ (−Y ) = X ⊕ Y ;
(4) ⊕ is commutative: X ⊕ Y = Y ⊕ X;
(5) [0, 0] is the only neutral element for ⊕: X ⊕ [0, 0] = X; (6) α⊙ (X ⊕ Y ) = (α ⊙ X) ⊕ (α ⊙ Y ); (7) (α + β) ⊙ X = (α ⊙ X) ⊕ (β ⊙ X); (8) X⊖ X = [0, 0]; (9) (X ⊕ Y ) ⊖ Y = X; (10) (X ⊖ Y ) ⊖ X = (−1) ⊙ Y ; (11) [0, 0] ⊖ (X ⊖ Y ) = Y ⊖ X; (12) X⊖ Y = (−1) ⊙ (Y ⊖ X); (13) X⊖ Y = ((−1) ⊙ Y ) ⊖ ((−1) ⊙ X); (14) X⊖ Y = Y ⊖ X iff X ⊖ Y is symmetric; (15) X⊗ (Y ⊕ Z) = X ⊗ Y ⊕ X ⊗ Z; (16) (Y ⊕ Z) ⊗ X = Y ⊗ X ⊕ Z ⊗ X; (17) X⊗ (Y ⊖ Z) = X ⊗ Y ⊖ X ⊗ Z; (18) (Y ⊖ Z) ⊗ X = Y ⊗ X ⊖ Z ⊗ X.
Intervals not only provide a way to express a number approximations, they can also be used as a logical value. The next section shows how intervals play this role.
2.2
Logics
In conventional or classical logic, a statement is either false or true and can not be partially false and partially true. However, in the real world, it is very common to meet complicated problems that are not always bivalent, nor are they always made of absolutely true or false facts. To model such problems, multivalent logics have emerged, such as Fuzzy Logic, in which it reflects the way people think, trying to shape their sense of words, decision making, or common sense.
Fuzzy Logic was introduced by Lofti Zadeh in 1965 in the article titled: Fuzzy Sets (ZADEH, 1965). Fuzzy logic is a formalism suitable for modeling the human capacity for approximate reasoning and support for decision making in environments where there is imperfect informa-tion and gradual set belonging, allowing a sufficient variety of physical and mental tasks to be performed without any measure or computation. It has been applied in several areas, such as control systems (GUANRONG; TAT, 2001), decision making (CHANG; WANG, 2009), expert systems (SILER; BUCKLEY, 2005), pattern recognition (CHOI; RHEE, 2009), etc.
2.2.1
Fuzzy Connectives
Among the most important operators in fuzzy logic, the author highlights t-norms, t-conorms, fuzzy negations and fuzzy implications. These operators are generalizations of the classical con-junctions, discon-junctions, negations and implications to fuzzy logic, respectively . The definitions are given as follows:
Definition 2.3. (SCHWEIZER; SKLAR, 1958, 1960, 1961) A functionT ∶ [0, 1]2→ [0, 1] is said to
be atriangular norm (t-norm, for short) if it satisfies the following conditions, for all x, y, z ∈ [0, 1]:
(T1) Symmetry: T(x, y) = T(y, x);
(T2) Associativity: T(x, T(y, z)) = T(T(x, y), z);
(T3) Monotonicity: Ifx1≤ x2andy1 ≤ y2 thenT(x1, y1) ≤ T(x2, y2);
(T4) 1-identity: T(x, 1) = x. (boundary condition) In fuzzy logic, the conjunction is often represented by a t-norm. The standard fuzzy con-junction TM ∶ [0, 1]2 → [0, 1], given by TM(x, y) = min{x, y}, called minimum t-norm, is the
only idempotent t-norm (see (KLIR; YUAN, 1995) - Theorem 3.9). Another example of t-norms is the product, denoted by TP.
Proposition 2.1. (BEDREGAL, 2007) LetT be a t-norm. Then T(0, y) = 0 for each y ∈ [0, 1]. Definition 2.4. A t-norm T is called positive if, for all x, y ∈ [0, 1], it satisfies the condition: T(x, y) = 0 if and only if x = 0 or y = 0.
Definition 2.5. (SCHWEIZER; SKLAR, 1961) Atriangular conorm (t-conorm for short) is a binary operationS on the unit interval[0, 1], i.e., a function S ∶ [0, 1]2 → [0, 1], which, for all x, y, z ∈
[0, 1], satisfying (T1), (T2), (T3) and
(S4) S(x, 0) = x. (boundary condition) The standard fuzzy disjunction SM ∶ [0, 1]2 → [0, 1] given by SM(x, y) = max{x, y},
called maximum t-conorm, is the only idempotent t-conorm (see (KLIR; YUAN, 1995) - Theorem 3.14).
From an axiomatical point of view, t-norms and t-conorms differ only with respect to their boundary conditions.
In the following, the notion of fuzzy negation is recalled.
Definition 2.6. (FODOR; ROUBENS, 1994) A functionN ∶ [0, 1] → [0, 1] is a fuzzy negation if (N1) N is antitonic, i.e. N(x) ≤ N(y) whenever y ≤ x;
(N2) N(0) = 1 and N(1) = 0. It isstrict whenever
(N3) N is continuous and
(N4) N(x) < N(y) whenever y < x. It isstrong if
(N5) N(N(x)) = x, for each x ∈ [0, 1]. A fuzzy negationN is crisp if
(N6) N(x) ∈ {0, 1}, for all x ∈ [0, 1];
A fuzzy negationN is frontier if it satisfies the property: (N7) N(x) ∈ {0, 1} if and only if x = 0 or x = 1;
A fuzzy negationN is non-vanishing if (N8) N(x) > 0 whenever x < 1.
Example 2.2. The least fuzzy negation, N⊥, and the greatest fuzzy negation, N⊺, are defined, respectively, by N⊥(x) = { 1, if x= 0 0, if x> 0 and N⊺(x) = { 0, if x= 1 1, if x< 1 .
Definition 2.7. Given a t-norm T and a fuzzy negation N , it can be said that the pair(T, N) satisfies thelaw of contradiction whenever
T(x, N(x)) = 0, x ∈ [0, 1]. (LC) Definition 2.8. Let T be a t-norm, S be a t-conorm and N be a fuzzy negation. Then S is said to beN -dual to T if for all x, y ∈ [0, 1] we have N(S(x, y)) = T(N(x), N(y)). It will be denoted byST. Analogously, T is said to be N -dual to S if for all x, y ∈ [0, 1] we have
N(T(x, y)) = S(N(x), N(y)). It will be denoted by TS.
Fuzzy implication generalizes the usual material implications. In what follows they are presented with some of their properties.
Definition 2.9. (FODOR; ROUBENS, 1994) A functionI ∶ [0, 1]2 → [0, 1] is a fuzzy implication
if the following properties are satisfied, for allx, y, z ∈ [0, 1]: (I1) If x≤ y then I(y, z) ≤ I(x, z);
(I2) If y≤ z then I(x, y) ≤ I(x, z); (I3) I(0, y) = 1;
(I5) I(1, 0) = 0.
The set of all fuzzy implications will be denoted byFI.
Definition 2.10. Let I∈ FI. The function NI∶ [0, 1] → [0, 1] defined by
NI(x) = I(x, 0), x ∈ [0, 1] (2.3)
is called thenatural negation of I or the negation induced by I.
In the following, some of the most important properties of some fuzzy implications are presented, which will be useful in this work (see (SMETS; MAGREZ, 1987;TRILLAS; VALVERDE, 1993;FODOR; ROUBENS, 1994)).
Definition 2.11. A fuzzy implication I is said to satisfy:
(i) theidentity property, if, for all x∈ [0, 1]
I(x, x) = 1; (IP)
(ii) theleft neutrality property, if, for all y ∈ [0, 1]
I(1, y) = y; (NP)
(iii) theexchange principle, if, for all x, y, z∈ [0, 1]
I(x, I(y, z)) = I(y, I(x, z)). (EP) (iv) theleft-ordering property, if, for all x, y ∈ [0, 1]
I(x, y) = 1 whenever x ≤ y; (LOP) (v) theright-ordering property, if, for all x, y ∈ [0, 1]
I(x, y) ≠ 1 whenever x > y. (ROP) (vi) theorder property iff I satisfy (LOP) and (ROP), i.e., for all x, y∈ [0, 1]
I(x, y) = 1 iff x ≤ y; (OP)
(vii) thelaw of left self-distributivity2, if, for allx, y, z ∈ [0, 1]
I(x, I(y, z)) = I(I(x, y), I(x, z)). (LSD) Definition 2.12. Let I∈ FI and let N be a fuzzy negation. I is said to satisfy the:
(i) contraposition law with respect to N , if
I(x, y) = I(N(y), N(x)), for all x, y ∈ [0, 1]; (CP) (ii) left contraposition law with respect to N , if
I(N(x), y) = I(N(y), x), for all x, y ∈ [0, 1]; (L-CP)
2This law was studied in (
CRUZ; BEDREGAL; SANTIAGO, 2018) under the name Boolean-Like. The name given here is more appropriate because it is closer to self left-distributive law in (FRINK, 1955).
(iii) right contraposition law with respect to N , if
I(x, N(y)) = I(y, N(x)), for all x, y ∈ [0, 1]. (R-CP) If I satisfies the (left, right) contraposition with respect to a specific N , the following nota-tion will be used: L−CP(N), R−CP(N) and CP(N), respectively.
Proposition 2.2. (BACZY ´NSKI; JAYARAM, 2008) LetI ∶ [0, 1]2 → [0, 1] be an any function and
NIbe a strong negation.
(i) IfI satisfies CP(NI), then I satisfies (NP).
(ii) IfI satisfies(EP), then I satisfies (I3), (I4), (I5), (NP) and (CP) only with respect toNI.
Proposition 2.3. (BACZY ´NSKI; JAYARAM, 2008) If a functionI ∶ [0, 1]2 → [0, 1] satisfies (EP)
andNI is a fuzzy negation, thenI satisfies R−CP(NI).
Proposition 2.4. (BACZY ´NSKI; JAYARAM, 2008) If a functionI ∶ [0, 1]2→ [0, 1] satisfies (R−CP)
with respect to continuous fuzzy negationN , then I satisfies(I1) if and only if it satisfies (I2). Definition 2.13. Let I ∈ FI and T be any t-norm and I be a fuzzy implication for T. The pair (I, T) satisfies the T-conditionality property for T if, for each x, y ∈ [0, 1],
T(x, I(x, y)) ≤ y . (TC) Definition 2.14. Let I be a fuzzy implication and T be a t-norm. It may be said that I satisfies theLaw of importation (LI) with respect to a t-norm T if
I(T(x, y), z) = I(x, I(y, z)), (2.4) for allx, y, z ∈ [0, 1].
It is well known that the fuzzy implications are generalizations of the implications of classi-cal logic to fuzzy logic, just as a t-norm and a t-conorm are generalizations of classiclassi-cal conjunc-tion and disjuncconjunc-tion, respectively. There are some ways to generate fuzzy implicaconjunc-tions from logical connectives. The main fuzzy implications are generalizations of the following tautolo-gies of classical logic:
p→ q = ¬p ∨ q, p → q = ¬p ∨ (p ∧ q) and p → q = (¬p ∧ ¬q) ∨ q,
namely(S, N), QL and D-implications, respectively (see (FODOR, 1991; MAS; MONSERRAT; TORRENS, 2006; BACZY ´NSKI; JAYARAM, 2008; BACZY ´NSKI, 2004)). In addition, there is an implication that arises from the isomorphism that exists between the classical logic of two values and the classical set theory by the following identity
A′∪ B = (A/B)′= ⋃{C ⊆ X ∣ A ∩ C ⊆ B},
where A and B are subsets of some universal set X and A′is the complement of set A. Fuzzy implications obtained as generalization of identity above form the family of residual implica-tions, usually called in the literature of R-implications.
Definition 2.15. Let I be a fuzzy implication. It can be said that I is called:
(i) (S, N)-implications, if I(x, y) = S(N(x), y) for a given t-conorm S and a fuzzy negation N . If N is strong, then I is simply called S-implications.
(ii) R-implications, if I(x, y) = sup{z ∈ [0, 1] ∣ T(x, z) ≤ y} for a given left-continuous t-normT .
(iii) QL-implications, if I(x, y) = S(N(x), T(x, y)) for a given t-conorm S, a t-norm T and the greatest fuzzy negationN⊺.
(i) D-implications, if I(x, y) = S(T(N(x), N(y)), y) for a given t-conorm S, a t-norm T and a strong negationN .
Fuzzy connectives can be extended to take into account imprecision. The next section shows how some of the above fuzzy connectives can be extended to operate with interval values.
2.2.2
Interval Fuzzy Connectives
Let U([0, 1]) be the set of closed intervals on [0, 1], i.e U([0, 1]) = {[x, x] ∣ 0 ≤ x ≤ x ≤ 1}. Let⟨U([0, 1]), ≤⟩ be a bounded poset with [0, 0] as bottom and [1, 1] as top elements.
Definition 2.16. A function N∶ U([0, 1]) → U([0, 1]) is an interval fuzzy negation on ⟨U([0, 1]), ≤ ⟩ if it is decreasing and satisfies N([0, 0]) = [1, 1] and N([1, 1]) = [0, 0]. If N(N(A)) = A, ∀A ∈ U([0, 1]), then N is called strong interval fuzzy negation.
Definition 2.17. A t-norm on U([0, 1]), called interval triangular norm (it-norm) on ⟨U([0, 1]), ≤ ⟩, is a commutative, associative, increasing mapping T ∶ U([0, 1])2 → U([0, 1]) which satisfies
T(A, [1, 1]) = A, for all A ∈ U([0, 1]).
Proposition 2.5. Let T∶ U([0, 1])2 → U([0, 1]) be an it-norm on the bounded poset ⟨U([0, 1]), ≤
⟩. Then, T(A1,[0, 0]) = [0, 0] for all A1∈ U([0, 1]).
Proof. Indeed, for all A1 ∈ U([0, 1]), we have that [0, 0] ≤ A1≤ [1, 1]. So, since T is increasing,
T([0, 0], A1) ≤ T([0, 0], [1, 1]). From the commutativity and boundary condition of T, we
obtain T(A1,[0, 0]) ≤ [0, 0], therefore, T(A1,[0, 0]) = [0, 0].
Definition 2.18. A function I ∶ U([0, 1])2 → U([0, 1]) is an interval fuzzy implication on
⟨U([0, 1]), ≤⟩ if, for all A1, A2, A3 ∈ U([0, 1]), I satisfies the following properties:
(I1) If A1≤ A2 then I(A2, A3) ≤ I(A1, A3) ;
(I2) If A2≤ A3 then I(A1, A2) ≤ I(A1, A3);
(I3) I([0, 0], A2) = [1, 1];
(I4) I(A1,[1, 1]) = [1, 1];
In (BEDREGAL; TAKAHASHI, 2006), Bedregal and Takahashi presented a characterization for the best interval representation of fuzzy implications. See the following theorem:
Theorem 2.1. (BEDREGAL; TAKAHASHI, 2006, Theorem 6.2) LetI be a fuzzy implication. Then, the best interval representation ofI, denoted by ˆI, is given by:
ˆ
I(A1, A2) = [I(a1, a2), I(a1, a2)] ,
for allA1, A2 ∈ U([0, 1]).
Following the notion of interval representation (formalized by Santiago et al. in (SANTIAGO; BEDREGAL; ACI ´OLY, 2006)), Bedregal et al. (BEDREGAL; SANTIAGO, 2013) showed that the intervalization of Łukasiewicz implication does not preserve (OP), however weakening the right side of (OP), gave rise to the pair of properties, in which the implication of Łukasiewicz satisfies:
(*) the interval r-weak order property, for all A1, A2∈ U([0, 1]), if A1≤ A2 then
I(A1, A2) = [1, 1]; (IR-WOP)
(**) the interval l-order property, for all A1, A2 ∈ U([0, 1]), if I(A1, A2) = [1, 1] then
A1≤ A2. (IL-OP)
Logics are usually interpreted by algebras; e.g. Classical Propositional Logics is interpreted by the usual boolean algebra{0, 1}, Łukasiewicz Logics is interpreted by MV-algebras. Most of the algebraic models for logics are BCI algebras (IS ´EKI, 1966) with additional axioms. The next chapter introduces those algebras and an extension for that. Chapter 5 provides another generalization called Semi-BCI algebras, which rises from the process of intervalization of BCI algebras which captures the properties (IR-WOP) and (IL-OP).
2.3
Final Remarks
The definitions of some interval arithmetic in the literature and their properties were pre-sented. Besides, the definitions and main properties of fuzzy connectives and interval fuzzy connectives were also exposed. Still in this interval context, two of the main concepts of inter-val theory were presented, namely: Correctness and optimality, where the first states that the result of an interval computation must always contain the value of the respective real function and the second establishes that the interval result should be as small as possible meeting the correctness criterion.
In the next chapter we present the BCI-algebras, which will be later intervalized (see Chap-ter 5) using the concepts of correctness and optimality.
3
BCI Algebras
Artificial intelligence has the important task of making computers simulate humans to deal with certainty and uncertainty in information. Certain information processing is based on the classical logic of two values, however, it is natural and necessary to try to establish some rational logical system as the logical basis for the uncertain information processing. This type of logic is an extension of two-valued logic. In order to construct natural and efficient inference systems to deal with uncertainty, several types of non-classical logic systems have been developed, such as BCI-logic.
BCI algebras were introduced by Is´eki (IS ´EKI, 1966) in the 60s and since then have been extensively investigated. The term, BCI-algebra, originates from the combinatories B, C, I in combinatory logic. There are several (equivalent) definitions of a BCI algebra, differing in type and notation. Some of them contain a∗ symbol for the binary operation and the symbol 0 (or ) for the null element. Here the BCI algebras will be used as algebras ⟨A, →, ⊺⟩ and one of the convincing arguments for this notation is that it makes obvious the connection with logic (the original approach is done in signature⟨A, ∗, 0⟩, see e.g. (IS ´EKI, 1966) and (IMAI; IS ´EKI, 1965)). Definition 3.1. A BCI algebra is a structureC = ⟨A, →, ⊺⟩, where → is a binary operation on A and⊺ is an element of A, verifying, the axioms: for all x, y, z ∈ X,
(C-1) (y → z) → ((z → x) → (y → x)) = ⊺, (C-2) x → ((x → y) → y) = ⊺,
(C-3) x → x = ⊺,
(C-4) if x → y = ⊺ and y → x = ⊺ then x = y.
On any such BCI algebra it is possible to define a partial order “⪯”: (C-5) x ⪯ y iff x → y = ⊺.
Wheneverx→ ⊺ = ⊺, i.e. x ⪯ ⊺, the BCI algebra C = ⟨A, →, ⊺⟩ will be called a BCK algebra. This relation⪯ is called induced order of A. It is not mandatory for ⊺ be the greatest element of(A, ⪯) (however, it is maximal), contrary to the case of BCK-algebras.
BCI algebras are the stronghold of several algebras that model important logics, including the fuzzy logics that are modeled by BL algebras. Thus, the intervalization of BCI algebras are important for the construction of an algebraic model for some interval fuzzy logics.
Example 3.1.
(1) The Łukasiewicz implicative algebra([0, 1], →LK, 1), where x →LK y= min(1, 1 − x +
y), is a BCI algebra.
(2) Given an abelian group(G, ⋅, e) with e as the unit element, (G, →, e) is a BCI algebra, wherex→ y = y ⋅ x−1.
(3) Given a setA, consider the parts of A, denoted by P(A). The structure (P(A), ⇒, ∅) is a BCI algebra, with⇒ such that X ⇒ Y = Y ∩ XC, whereXC is the complement of
X.
Let us now recall some useful properties of BCI algebras (for more details see (HUANG, 2006)):
(A-1) ⊺ ⪯ x implies x = ⊺;
(A-2) x⪯ y implies y → z ⪯ x → z; (First place antitonicity) (A-3) x⪯ y implies z → x ⪯ z → y; (Second place isotonicity) (A-4) x⪯ y and y ⪯ z implies x ⪯ z; (Transitivity) (A-5) x→ (y → z) = y → (x → z); (Exchange – EP) (A-6) x⪯ y → z implies y ⪯ x → z;
(A-7) x→ y ⪯ (z → x) → (z → y);
(A-8) ⊺ → x = x; (Left Neutrality)
(A-9) ((y → x) → x) → x = y → x; (A-10) x→ y ⪯ (y → x) → ⊺; (A-11) (x → y) → ⊺ = (x → ⊺) → (y → ⊺); (A-12) (x → y) → ⊺ = ((y → x) → ⊺) → ⊺; (A-13) y→ ((x → ⊺) → ⊺) = ((y → x) → ⊺) → ⊺; (A-14) x→ ⊺ = ((x → y) → y) → ⊺; (A-15) x⪯ y implies x → ⊺ = y → ⊺; (A-16) y→ ⊺ ⪯ x implies x = y → ⊺.
Note that properties (A-7), (A-5) and (C-3) model the combinators B, C and I of BCI Logic (HINDLEY; SELDIN, 1986).
Proposition 3.1. Let⟨A, →, ⊺⟩ be a BCI algebra. ⟨A, →, ⊺⟩ is a BCK algebra if and only if for eachx∈ A there exists y ∈ A such that y ⪯ x and y ⪯ ⊺.
Proof. (⇒) Straightforward because in BCK algebras ⊺ is the greatest element, i.e. x ⪯ ⊺ for each x∈ A.
(⇐) Suppose, by contradiction, that ⟨A, →, ⊺⟩ is not a BCK algebra. Then, there exists a∈ A such that a /⪯ ⊺. By hypothesis there exists b ∈ A such that b ⪯ a and b ⪯ ⊺. So, by (A-8) and the definition of⪯, (b → ⊺) → ((⊺ → a) → (b → a)) = ⊺ → (a → ⊺) = a → ⊺ ≠ ⊺. Therefore, (C-1) fails.
BCK and BCI algebras have been extensively investigated by many researchers (see (JUN; SHIM, 2005;LIU; XU; MENG, 2007;ZHAN; LIU, 2005;LIU et al., 2000;LIU; ZHANG, 1994)). There are three important classes of BCI algebras: commutative BCI-algebras (MENG; XIN, 1992a), implicative BCI-algebras (MENG; XIN, 1992b) and positive implicative BCI-algebras (MENG; XIN, 1993). In addition to important concepts such as ideals/filters (these are dual concepts and depend on the definition used, see e.g. (MENG, 1993;LIU; ZHANG, 1994;WEI; JUN, 1995)). For a more detailed view of BCI-algebra, see e.g. (HUANG, 2006).
There are several generalizations of the BCI algebras as shown by, for example, Iorgulescu in (IORGULESCU, 2016a), in which he found thirty-one new generalizations distinct from BCI or BCK algebras and showed the hierarchies existing among these algebras (see also (IORGULESCU, 2016b)). In this work the author presents a generalization (see the Pseudo-BCI algebra in the following section) in which it has the signature different from the one previously mentioned, since it brings two binary operators instead of one, and propose another generalization aim-ing to capture the intervalization of point algebras that model logics (see Chapter 5). Because Pseudo-BCI has the same signature as the new generalization of BCI algebra shown in Chapter 5, the author decided to show the relationship between them.
3.1
Pseudo-BCI Algebras
In (GEORGESCU; IORGULESCU, 2001a), G. Georgescu and A. Iorgulescu introduced the no-tion of pseudo-BCK algebras as an extension of BCK-algebras. The motivano-tion was the fol-lowing: since bounded commutative BCK algebra corresponds (is categorically equivalent) to MV algebra (MUNDICI, 1986), they wanted to verify which structure corresponds to the pseudo-MV algebra, which pseudo-MV algebra is a non-commutative extension of MV al-gebras (GEORGESCU; IORGULESCU, 1999, 2001b). Years later, W. A. Dudek and Y. B. June (DUDEK; JUN, 2008) proposed a generalization of the BCI algebras, called pseudo-BCI alge-bras, as an extension of BCI-algebras.
Definition 3.2. A pseudo-BCI algebra, or PBCI algebra for short, is a structure⟨A, ≤, →, ↝, ⊺⟩ such that “≤” is a binary relation on the set A, “→” and “↝” are binary operations on A, ⊺ ∈ A and for allx, y, z∈ A:
(PB-1) x→ y ≤ (y → z) ↝ (x → z), (PB-2) x↝ y ≤ (y ↝ z) → (x ↝ z), (PB-3) x≤ (x → y) ↝ y,
(PB-5) x≤ x,
(PB-6) if x≤ y and y ≤ x, then x = y, (PB-7) x≤ y ⇔ x → y = ⊺ ⇔ x ↝ y = ⊺.
Whenever x → ⊺ = ⊺, i.e. x ⪯ ⊺ for all x, the PBCI algebra ⟨A, ≤, →, ↝, ⊺⟩ will be called a PBCK algebra.
Note that every PBCI algebra satisfying x→ y = x ↝ y for all x, y ∈ X is a BCI algebra. Example 3.2.
(1) The structure A = ⟨R2,⪯, ↠, →, (0, 0)⟩, where (x
1, y1) ↠ (x2, y2) = (x2 − x1,(y2−
y1)e−x1) and (x1, y1) → (x2, y2) = (x2− x1, y2− y1ex2−x1), is a PBCI algebra (proper,
i.e. it is not a PBCK-algebra).
(2) The structureA = ⟨(−∞, 0] ⪯, ↠, →, (0, 0)⟩, where x↠ y = { 0, if x2y ⪯ y πarctan(ln( y x)), if y < x and x→ y = { 0, if x⪯ y ye−tan(πx2y), if y< x is a PBCK algebra.
Proposition 3.2. LetA = ⟨A, ≤, →, ↝, ⊺⟩ be a PBCI algebra, then the following properties holds for allx, y, z ∈ A: (P-1) ⊺ ⪯ x implies x = ⊺; (P-2) x⪯ y implies y → z ⪯ x → z and y ↝ z ⪯ x ↝ z; (P-3) x⪯ y and y ⪯ z implies x ⪯ z; (P-4) x→ (y ↝ z) = y → (x ↝ z) (P-5) x⪯ y → z iff y ⪯ x ↝ z; (P-6) x→ y ⪯ (z → x) → (z → y), x ↝ y ⪯ (z ↝ x) ↝ (z ↝ y); (P-7) x⪯ y implies z → x ⪯ z → y and z ↝ x ⪯ z ↝ y; (P-8) ⊺ → x = ⊺ ↝ x = x; (P-9) ((x → y) ↝ y) → y = x → y and ((x ↝ y) → y) ↝ y = x ↝ y; (P-10) x→ y ⪯ (y → x) ↝ ⊺ and x ↝ y ⪯ (y ↝ x) → ⊺; (P-11) (x → y) → ⊺ = (x → ⊺) ↝ (y ↝ ⊺), (x ↝ y) ↝ ⊺ = (x ↝ ⊺) → (y → ⊺);
(P-12) x→ ⊺ = x ↝ ⊺.
Since its definition, PBCI algebras have been investigated by many researchers, in which they have defined new concepts and applications. Among these researchers worth mentioning is Xiaohong Zhang and Grzegorz Dymek, for their many published works on the subject. For more results see for example (ZHANG, 2010; LEE; PARK, 2009; HALA ˇS; K ¨UHR, 2009; DYMEK, 2012, 2013; ZHANG; PARK; WU, 2018; ZHANG; LU; MAO, 2010; ZHANG; MA; SMARANDACHE, 2017).
3.2
Final Remarks
The definition of BCI algebras and its main properties have been presented; which algebra is well known in the literature. In addition, it was exposed one of several generalizations of BCI algebras, namely: Pseudo-BCI algebras.
The previous section closes the part preliminary of the present thesis. In what follows, the reader finds my published and submitted contributions. They can be divided into two categories: (1) Fuzzy Connectives and (2) An extension of BCI algebras, which was proposed to capture the process of intervalization of those algebras (Semi-BCI algebras).
Part II
Contributions
4
(
T , N )-Implications
In this chapter, it is introduced a class of fuzzy implication called(T, N)-implication, ob-tained from the composition of a fuzzy negation and a t-norm. It has been shown under what constraints,(T, N)-implications preserve some main properties of fuzzy implications, such as property of order, the principle of exchange, the law of contraposition, among others. In ad-dition, we apply this new implication class to generate a new fuzzy subsethood measure. And finally the(T, N)-implications were extended to (N′, T, N)-implications.
4.1
Definition and Basic Properties
As well as the (S, N), QL and D-implications, defined in Chapter 2, are generalizations of implications of classical logic for fuzzy logic, the(T, N)-implications also satisfy the same family of implications, in which it generalizes the tautology:
p→ q = ¬(p ∧ ¬q).
This session is focused on verifying if the classical law of double negation,¬(¬p) = p, is satis-fied thus the(T, N)-implications coincide with (S, N)-implications. See below the definition of(T, N)-implication, initially defined by Bedregal in (BEDREGAL, 2007).
Proposition 4.1. (BEDREGAL, 2007) Let T be a t-norm and N be a fuzzy negation. Then the functionIN
T ∶ [0, 1]2 → [0, 1] defined by
ITN(x, y) = N(T(x, N(y))) (4.1) is a fuzzy implication.
Definition 4.1. Let T be a t-norm and N be a fuzzy negation. The function IN
T defined by
equa-tion (4.1) is called the(T, N)-implication.
Proposition 4.2. (BEDREGAL, 2007) LetN be a strong fuzzy negation and T be a t-norm. Then,
T(x, y) = N(ITN(x, N(y))).
If I is a(T, N)-implication and N is a strong fuzzy negation, then by (KLEMENT; MESIAR; PAP, 2000)(p.234), we get that I is an S-implication, i.e.
where, ST is as in Definition 2.8. The reciprocal is also true, since by duality we have:
ST(N(x), y) = N(T(N(N(x)), N(y))) = N(T(x, N(y))).
Remark 4.1. If N is not a strong negation and I(x, y) = S(N(x), y), then we say that I is a (S,N)-implication. It will be denoted byI(S,N).
Proposition 4.3. Let IN
T be a (T, N)-implication. If N is a strong fuzzy negation, then ITN
satisfies(NP), (EP), R−CP(N) and CP(N).
Proof. (NP) Since T is a t-norm, then by the symmetry and 1-identity properties, for any y ∈ [0, 1], we have
ITN(1, y)(4.1)= N(T(1, N(y)))T 1= N(N(y))/T 4 N 5= y. (EP) Since N a strong fuzzy negation and T a t-norm, we have
IN T (x, ITN(y, z)) (4.1) = N(T(x, N(N(T(y, N(z))))))N 5 = N(T(x, T(y, N(z)))) T 1 = N(T(x, T(N(z), y)))T 2 = N(T(T(x, N(z)), y)) T 1 = N(T(y, T(x, N(z))))N 5 = N(T(y, N(N(T(x, N(z)))))) (4.1)= IN T (y, ITN(x, z)).
(R-CP) Because N is strong and from the symmetry of T, we have
ITN(x, N(y)) (4.1)= N(T(x, N(N(y))))N 5= N(T(x, y))T 1= N(T(y, x))
N 5
= N(T(y, N(N(x))))(4.1)= IN
T (y, N(x)).
(CP) Again, because N is strong and from the symmetry of T, we have
ITN(N(y), N(x)) (4.1)= N(T(N(y), N(N(x))))N 5= N(T(N(y), x))
T 1
= N(T(x, N(y)))(4.1)= IN T (x, y).
Proposition 4.4. Given a(T, N)-implication ITN, the following properties are satisfied: (i) NIN
T = N; (ii) L−CP(N);
(iii) IfN is strict, then R−CP(N−1).
Proof. (i) Since N is a fuzzy negation, then N(0) = 1, therefore from the 1-identity property of the t-norm T , we have, for all x∈ [0, 1],
NIN T(x) (2.3)= IN T (x, 0) (4.1)= N(T(x, N(0)))N 2 = N(T(x, 1))T 4 = N(x).
(ii) From the symmetry of T , we have IN T (N(x), y) (4.1) = N(T(N(x), N(y)))T 1 = N(T(N(y), N(x)))(4.1)= IN T (N(y), x), for all x, y∈ [0, 1].
(iii) Again, from the symmetry property, we have
ITN(x, N−1(y)) (4.1)= N(T(x, N(N−1(y)))) = N(T(x, y))T 1= N(T(y, x)) = N(T(y, N(N−1(x))))(4.1)= IN
T (y, N−1(x)),
for all x, y∈ [0, 1].
Under some conditions, there are methods available for obtaining t-conorms and t-norms of a(T, N)-implication and a fuzzy negation. The following propositions present these methods: Proposition 4.5. Given a(T, N)-implication IN
T , define the functionSIN
T ∶ [0, 1] 2 → [0, 1] by SIN T(x, y) = I N T (N(x), y)
for allx, y∈ [0, 1]. Then: (i) SIN
T(1, x) = SI N
T(x, 1) = 1, ∀x ∈ [0, 1]; (ii) SIN
T is increasing in both the arguments, i.e., ∀x, y, z ∈ [0, 1] with y ≤ z we have SIN T(x, y) ≤ SI N T(x, z) and SI N T(y, x) ≤ SI N T(z, x); (iii) SIN T is commutative; (iv) IfN is strong, then SIN
T satisfies(S4), i.e., SI N
T(x, 0) = x; (v) ItN is strong, then SIN
T satisfies(S2), i.e., SITN(x, SITN(y, z)) = SITN(SITN(x, y), z). Proof. (i) As N is a fuzzy negation and T is a t-norm, then from Proposition 2.1, we have
SIN T(1, x) = I N T (N(1), x) N 2 = IN T (0, x) (4.1) = N(T(0, N(x)))P rop.2.1 = N(0)N 2 = 1 and SIN T(x, 1) = I N T (N(x), 1) (4.1)= N(T(N(x), N(1))N 2 = N(T(N(x), 0)) T 1 = N(T(0, N(x)))P rop.2.1 = N(0)N 2 = 1. (ii) For any x, y∈ [0, 1] we get
SIN
T(x, y) = I
N
T (N(x), y)
and SIN T(x, z) = I N T (N(x), z) (4.1)= N(T(N(x), N(z))),
hence, by the monotonicity of the t-norm T , we get that
y≤ zN 1⇒ N(z) ≤ N(y)⇒ T(N(x), N(z)) ≤ T(N(x), N(y)),T 3 applying(N1), we have N(T(N(x), N(y))) ≤ N(T(N(x), N(z))) i.e., SIN T(x, y) ≤ SI N T(x, z). The other part follows similar.
(iii) By Proposition 4.4(ii), we have SIN T(x, y) = I N T (N(x), y) L−CP = IN T (N(y), x) = SIN T(y, x). (iv) Being N a strong fuzzy negation, we have by Proposition 4.3 that
SIN T(x, 0) (iii)= S IN T(0, x) = I N T (N(0), x) N 2 = IN T (1, x) N P = x. (v) By Proposition 4.4(ii), we have
SIN T(x, SITN(y, z)) = I N T (N(x), I N T (N(y), z)) L−CP = IN T (N(x), I N T (N(z), y)).
However, being N strong, Proposition 4.3(ii) ensures that
ITN(N(x), ITN(N(z), y)) = ITN(N(z), ITN(N(x), y)), thus: SIN T(x, SITN(y, z)) = I N T (N(z), ITN(N(x), y)) = SIN T(z, SITN(x, y)) (iii)= S IN T(SITN(x, y), z).
From this proposition, it is possible to conclude that, if N is strong, then SIN
T is a t-conorm. Thus, IN
T is an S-implication.
Proposition 4.6. Given a(T, N)-implication IN1
T , define the functionT N2 ITN1 ∶ [0, 1] 2→ [0, 1] by TN2 ITN1(x, y) = N2(I N1 T (x, N2(y))), x, y ∈ [0, 1],
whereN2 is a fuzzy negation. Then,TN2
ITN1 is a t-norm iffN2 is strict andN1= N −1 2 .
Proof. Let us assume, firstly, that TN2
ITN1 is a t-norm. To prove that N2 is strict, we must show
that N2 is decreasing and continues.
(i) N2is decreasing.
In fact, suppose there are x0, y0 ∈ [0, 1] com y0 < x0, such that N2(x0) = N2(y0). Thus,
for(I2), IN1 T (1, N2(x0)) = ITN1(1, N2(y0)) N 1 ⇒ TN2 ITN1(1, x0) = T N2 ITN1(1, y0) T 1/T 4 ⇒ x0 = y0.
Contradiction. Therefore, as N2is a fuzzy negation, we have by(N1) that N2is
decreas-ing.
(ii) N2is continuous.
In fact, first there is a need to check that N2 is injective. Given x, y ∈ [0, 1] with x ≠ y
we can suppose that x< y. Then, as N2is decreasing, we have that N2(y) < N2(x), and
therefore, N2 is injective. N2 is also surjective, since given y ∈ [0, 1] any, as TN2 ITN1 is a t-norm, we have y T 4= TN2 ITN1(y, 1) = N2(I N1 T (y, N2(1))) N 2 = N2(ITN1(y, 0)) (4.1) = N2(N1(T(y, 1))) T 4 = N2(N1(y)),
therefore, there is N1(y) ∈ [0, 1] such that
N2(N1(y)) = y, (4.2)
so, N2 is surjective. Thus, N2 is bijective. It then follows that N2 is continuous
be-cause otherwise there would be y ∈ [0, 1] such that y ≠ N2(x), for all x ∈ [0, 1], which
contradicts the bijectivity of N2.
Finally, to ensure that N1 = N2−1, by Equation (4.2), just suffice to show that N1(N2(y)) = y.
Suppose N1(N2(y)) ≠ y. Then N1(N2(y)) < y or N1(N2(y)) > y. If N1(N2(y)) < y, then by
(i), we have
N2(y) < N2(N1(N2(y)))
Eq.(4.2)
= N2(y).
Contradiction. Analogously, we come to a contradiction when N1(N2(y)) > y. Let us assume
now that N2is strict and N1= N2−1. Then, for all x, y∈ [0, 1], we have
TN2 ITN1(x, y) = N2(I N1 T (x, N2(y))) (4.1) = N2(N1(T(x, N1(N2(y))))) N1=N2−1 = T(x, y). Therefore, TN2 ITN1 is a t-norm.
Lemma 4.1. If I∈ FI satisfies (NP), (EP) and NI is a strong fuzzy negation, then
TI(x, y) = NI(I(x, NI(y)))
Proof. By Propositions 2.2 and 2.3, we have that I satisfies CP(NI) and R−CP(NI). It will
verified now that TI satisfies the conditions of the definition of t-norm:
(T1) From law of right contraposition, we get that
TI(x, y) = NI(I(x, NI(y))) R−CP
= NI(I(y, NI(x))) = TI(y, x).
(T2) By virtue of NIis strong and I satisfies(EP) and (R−CP), we have
TI(x, TI(y, z)) N 5
= NI(I(x, I(y, NI(z)))) R−CP
= NI(I(x, I(z, NI(y)))) (EP )
= NI(I(z, I(x, NI(y)))) N 5
= TI(z, TI(x, y)) T 1
= TI(TI(x, y), z).
(T3) It must be shown that, given x, y, z∈ [0, 1] with y ≤ z, then TI(x, y) ≤ TI(x, z), i.e.,
NI(I(x, NI(y))) ≤ NI(I(x, NI(z))).
Indeed,
y≤ z N 2⇒ NI(z) ≤ NI(y) I2
⇒ I(x, NI(z)) ≤ I(x, NI(y)) N 2
⇒ NI(I(x, NI(y))) ≤ NI(I(x, NI(z))).
(T4) As I satisfies(NP), we have for x ∈ [0, 1], TI(x, 1) T 1 = TI(1, x) = NI(I(1, NI(x))) N P = NI(NI(x)) N 5 = x. Portanto, TI uma t-norm.
Theorem 4.1. For a function I ∶ [0, 1]2 → [0, 1], the following statements are equivalent:
(i) I = IN
T is a(T, N)-implication, with N strong fuzzy negation;
(ii) I satisfies(I1), (EP) and (NI) is a strong fuzzy negation.
Moreover, the representation ofI= IN
T is unique in this case.
Proof. (i) ⇒ (ii) Because of Proposition 4.1 we get that I = IN
T ∈ FI, so (I1) is satisfies. Now,
by Proposition 4.3, I satisfies(EP) and of Proposition 4.4(i) it can be concluded NI= N, and
therefore NIis strong.
(ii) ⇒ (i) As NIis strong, then by Propositions 2.2, 2.3 and 2.4 it can be concluded I ∈ FI.
As I∈ FI satisfies (NP), (EP) and NIis a strong fuzzy negation, we have by the Lemma 4.1
that TI is a t-norm. Now, it will be demonstrated that I = ITNII. In fact, for all x, y∈ [0, 1],
INI
TI (x, y)
(4.1)
= NI(TI(x, NI(y))) = NI(NI(I(x, NI(NI(y))))) N 5
Finally, to prove unity assume that there are two strong fuzzy negations N1, N2 and two
t-norms T1, T2 such that
I(x, y) = IN1
T1 (x, y) = I
N2
T2 (x, y) for all x, y∈ [0, 1]. In particular, for y = 0 we obtain, for all x ∈ [0, 1],
NI(x) (2.3)= I(x, 0) = IN1 T1 (x, 0) = I N2 T2 (x, 0), so, NI(x) (2.3)= INi Ti(x, 0) (4.1)= N i(Ti(x, Ni(0))) N 2 = Ni(Ti(x, 1)) T 4 = Ni(x), i = 1, 2,
thus NI = N1 = N2. Now, since NI is strong, we obtain from Proposition 4.2 that
Ti(x, y) = Ni(ITNii(x, Ni(y))), i = 1, 2, then, for all x, y∈ [0, 1],
T1(x, y) = N1(ITN11(x, N1(y)))
hip
= N1(ITN22(x, N1(y)))
N1=N2
= N2(ITN22(x, N2(y))) = T2(x, y).
Therefore, the representation I= IN
T is unique.
Proposition 4.7. Let Nα(x) = {
1, if x≤ α
0, if x> α , for some α ∈ (0, 1), and T be a t-norm. Then INα T is the implication INα T (x, y) = ⎧⎪⎪ ⎨⎪⎪ ⎩ 1, if y> α or x ≤ α 0, otherwise . Proof. Indeed, INα T (x, y) = Nα(T(x, Nα(y))) = { 1, if y> α or x = 0 Nα(x), if y ≤ α e x > 0 = { 1, if y> α or x ≤ α 0, otherwise . Remark 4.2.
1. The t-normT is irrelevant for INα
T .
2. There is no (S,N)-implication such that INα
T = I(S,N), since if it exists and N is a fuzzy
negation, thenN(1) = 0 and S(N(1), y) = y ∉ {0, 1}, therefore S(N(x), y) ≠ INα
T .
3. There is no t-normT such that INα
T = IT, since by definition, forx= 1:
IT(1, y) = sup{z ∈ [0, 1] ∣ T(1, z) ≤ y}
= sup{z ∈ [0, 1] ∣ z ≤ y} = y ≠ INα
T (1, y), ∀y ∈ (0, 1).
Theorem 4.2. Let T be a t-norm and N be a fuzzy negation. Then N is strong iff IN T is an
S-implication withN as underlying negation.
Proof. Assume that N is a strong fuzzy negation, then by (KLEMENT; MESIAR; PAP, 2000) (p.234), we get that IN
T is an S-implication. Conversely, let ITN be an S-implication. Then
there exist a fuzzy negation N′ and a t-conorm S such that ITN(x, y) = S(N′(x), y), for all x, y∈ [0, 1]. Therefore,
x= S(N′(1), x) = ITN(1, x) = N(T(1, N(x))) = N(N(x)), ∀x ∈ [0, 1], i.e., N(N(x)) = x for all x ∈ [0, 1]. Moreover,
N′(x) (S4)= S(N′(x), 0) = ITN(x, 0) = N(T(x, N(0)))
(N2)= N(T(x, 1))(T 4)= N(x) ∀x ∈ [0, 1],
so, N′= N.
From Theorem 4.2, it is implied that every result which holds for S-implications, where N is a strong fuzzy negation, also holds for IN
T . Therefore, for the purposes of this work, there is
no interest in this kind of outcomes (results with a strong N ). The study keeps on focusing on proving results for non-strong fuzzy negations.
Proposition 4.8. Let IN
T be a(T, N)-implication and let N be a non-strong fuzzy negation:
(i) IN
T does not satisfy(NP);
(ii) IfN is strict, then IN
T does not satisfy(EP);
(iii) IfN is strict, then IN
T does not satisfyR−CP(N);
(iv) IfN is strict, then IN
T does not satisfyCP(N).
Proof. (i) In fact, since N is not a strong fuzzy negation, then there exists x ∈ [0, 1] such that N(N(x)) ≠ x, so
ITN(1, x) = N(T(1, N(x))) = N(N(x)) ≠ x.
(ii) Since N is not strong, then there exists x∈ [0, 1] such that N(N(x)) ≠ x, so
ITN(1, ITN(x, 0)) = N(T(1, N(N(T(x, N(0))))))N 2= N(T(1, N(N(T(x, 1))))) T 4 = N(T(1, N(N(x))))T 4 = N(N(N(x))) and ITN(x, ITN(1, 0)) = N(T(x, N(N(T(1, N(0))))))N 2= N(T(x, N(N(T(1, 1))))) T 4 = N(T(x, N(N(1))))N 2 = N(T(x, 1))T 4 = N(x). Therefore, by N being strict, IN
(iii) Again, by hypothesis, there exists x∈ [0, 1] such that N(N(x)) ≠ x, then ITN(x, N(1)) = N(T(x, 1))T 4= N(x)
and
ITN(1, N(x)) = N(T(1, N(N(x))))T 4= N(N(N(x))). So, since N is strict, ITN(x, N(1)) ≠ ITN(1, N(x)).
(iv) Again, there exists x∈ [0, 1] such that N(N(x)) ≠ x, then ITN(x, 0) = N(T(x, 1))T 4= N(x) and
ITN(N(0), N(x)) = ITN(1, N(x)) = N(T(1, N(N(x))))T 4= N(N(N(x))). Therefore, once N is strict, IN
T (x, 0) ≠ ITN(N(0), N(x)).
Remark 4.3. As any QL-implication (R-implication, D-implication) (see Definition 2.15) sat-isfies (NP), we have by Proposition 4.8(i) that if a(T, N)-implication is a QL-implication (R-implication, D-implication), thenN is strong and therefore it is an S-implication.
Remark 4.4. By (DIMURO et al., 2017), a fuzzy negationN ∶ [0, 1] → [0, 1] is crisp if and only if there existsα∈ [0, 1[ such that N = Nαor there existsα∈ ]0, 1] such that N = Nα, where
Nα(x) =⎧⎪⎪⎨⎪⎪ ⎩ 0, ifx> α 1, ifx≤ α and Nα(x) =⎧⎪⎪ ⎨⎪⎪ ⎩ 0, ifx≥ α 1, ifx< α. Theorem 4.3. Let IN
T be a(T, N)-implication and let N be a crisp fuzzy negation. Then:
(i) IN T satisfies(EP); (ii) IN T satisfiesR− CP(N); (iii) IN T satisfiesCP(N); (iv) IN
T does not satisfy(NP);
(v) IN
T does not satisfy(ROP);
(vi) IN
T satisfies(LOP).
(i) Given x, y, z∈ [0, 1]: (1) If z ≤ α, then Nα(z) = 1, so
INα
T (x, I Nα
T (y, z)) = Nα(T(x, Nα(Nα(T(y, Nα(z)))))) = Nα(T(x, Nα(Nα(y))))
= ⎧⎪⎪⎨⎪⎪ ⎩ Nα(T(x, Nα(0))), if y > α Nα(T(x, Nα(1))), if y ≤ α =⎧⎪⎪⎨⎪⎪ ⎩ Nα(x), if y > α 1, if y≤ α = ⎧⎪⎪⎨⎪⎪ ⎩ 0, if x> α and y > α 1, otherwise and INα T (y, I Nα T (x, z)) = Nα(T(y, Nα(Nα(T(x, Nα(z)))))) = Nα(T(y, Nα(Nα(x)))) = ⎧⎪⎪⎨⎪⎪ ⎩ Nα(y), if x > α 1, if x≤ α = ⎧⎪⎪ ⎨⎪⎪ ⎩ 0, if x> α and y > α 1, otherwise. Therefore, for z ≤ α, INα T (x, I Nα T (y, z)) = I Nα T (y, I Nα T (x, z)). (2) If z > α, then Nα(z) = 0, so: INα T (x, I Nα T (y, z)) = Nα(T(x, Nα(Nα(T(y, Nα(z)))))) = Nα(T(x, Nα(Nα(0)))) = Nα(T(x, 0)) = Nα(0) = 1 and INα T (y, I Nα T (x, z)) = Nα(T(y, Nα(Nα(T(x, Nα(z)))))) = Nα(T(y, Nα(Nα(0)))) = Nα(T(y, 0)) = Nα(0) = 1. In any case, INα T (x, I Nα T (y, z)) = I Nα T (y, I Nα T (x, z)). (ii) Given x, y∈ [0, 1]: INα T (x, Nα(y)) = Nα(T(x, Nα(Nα(y)))) = ⎧⎪⎪ ⎨⎪⎪ ⎩ Nα(x), if y > α Nα(0), if y ≤ α = ⎧⎪⎪⎨⎪⎪ ⎩ 0, if x> α and y > α 1, otherwise and INα T (y, Nα(x)) = Nα(T(y, Nα(Nα(x)))) = ⎧⎪⎪ ⎨⎪⎪ ⎩ Nα(y), if x > α Nα(0), if x ≤ α = ⎧⎪⎪⎨⎪⎪ ⎩ 0, if y> α and x > α 1, otherwise. Therefore, INα T (x, Nα(y)) = I Nα T (y, Nα(x)).
(iii) Given x, y∈ [0, 1]: INα T (x, y) = Nα(T(x, Nα(y))) = ⎧⎪⎪ ⎨⎪⎪ ⎩ Nα(x), if y ≤ α Nα(0), if y > α = ⎧⎪⎪⎨⎪⎪ ⎩ 0, if x> α and y ≤ α 1, otherwise and INα T (Nα(y), Nα(x)) = Nα(T(Nα(y), Nα(Nα(x)))) =⎧⎪⎪⎨⎪⎪ ⎩ Nα(Nα(y)), if x > α Nα(0), if x≤ α = ⎧⎪⎪⎨⎪⎪ ⎩ 0, if x> α and y ≤ α 1, otherwise. Therefore, INα T (x, Nα(y)) = ITNα(y, Nα(x)). (iv) Indeed, INα T (1, y) = Nα(T(1, Nα(y))) =⎧⎪⎪⎨⎪⎪ ⎩ Nα(0), if y > α Nα(1), if y ≤ α = ⎧⎪⎪⎨⎪⎪ ⎩ 1, if y> α 0, if y≤ α. Therefore, INα T (1, y) ≠ y for all y ∈ (0, 1).
(v) Indeed, if y < x ≤ α, we have that INα
T (x, y) = Nα(T(x, Nα(y))) = Nα(T(x, 1)) =
Nα(x) = 1.
(vi) If x ≤ y, we have the following cases: (1) If x ≤ y ≤ α, then Nα(x) = Nα(y) = 1 and
Nα(T(x, Nα(y))) = Nα(x) = 1; (2) If x ≤ α < y, then Nα(x) = 1 and Nα(y) = 0,
so Nα(T(x, Nα(y))) = Nα(0) = 1; (3) If α < x ≤ y, then Nα(x) = Nα(y) = 0, so
Nα(T(x, Nα(y))) = Nα(0) = 1. In any case, ITNα(x, y) = 1.
The proof follows analogously for N = Nα, for some α∈ ]0, 1].
Remark 4.5. By item (v) of Theorem 4.3, it can be concluded IN
T does not satisfy (OP) whenN
is crisp.
Proposition 4.9. Let T be a positive t-norm and let N be a frontier fuzzy negation. Then IN
T (x, y) = 1, if and only if x = 0 or y = 1.
Proof. Assume ITN(x, y) = 1, then N(T(x, N(y))) = 1 and, therefore, T(x, N(y)) = 0, since N is a frontier fuzzy negation. Now, take a positive T , so we have x = 0 or N(y) = 0, and therefore x= 0 or y = 1. Conversely, if x = 0 or y = 1 then by (vi) of Theorem 4.3, it follows straightforward.
Proposition 4.10. Let N be a fuzzy negation and let T be the minimum t-norm. If N(N(x)) ≤ x, thenIN T (x, I N T (x, y)) = I N T (x, y).
