Developing Constrained Interval Operators for Fuzzy Logic with Interval Values
A well-known problem in the interval analysis literature is the overestimation and loss of information. In this article, we define new interval operators, called <i>constrained interval operators</i>, that preserve information and mitigate overestimation. These operators are investigated...
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| Language: | English |
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MDPI AG
2023-12-01
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| Online Access: | https://www.mdpi.com/2075-1680/12/12/1115 |
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| author | Jocivania Pinheiro Regivan H. N. Santiago Benjamin Bedregal Flaulles Bergamaschi |
| author_facet | Jocivania Pinheiro Regivan H. N. Santiago Benjamin Bedregal Flaulles Bergamaschi |
| author_sort | Jocivania Pinheiro |
| collection | DOAJ |
| description | A well-known problem in the interval analysis literature is the overestimation and loss of information. In this article, we define new interval operators, called <i>constrained interval operators</i>, that preserve information and mitigate overestimation. These operators are investigated in terms of correction, algebraic properties, and orders. It is shown that a large part of the properties studied is preserved by this operator, while others remain preserved with the condition of continuity, as is the case of the exchange principle. In addition, a comparative study is carried out between this operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>g</mi><mo>¨</mo></mover></semantics></math></inline-formula> and the best interval representation: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>g</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula>. Although <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>g</mi><mo>¨</mo></mover><mo>⊆</mo><mover accent="true"><mi>g</mi><mo stretchy="false">^</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>g</mi><mo>¨</mo></mover></semantics></math></inline-formula> do not preserve the Moore correction, we do not have a loss of relevant information since everything that is lost is irrelevant, mitigating the overestimation. |
| first_indexed | 2024-03-08T20:59:32Z |
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| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-08T20:59:32Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-442bf830d50e481da93031c49df2e6c42023-12-22T13:53:17ZengMDPI AGAxioms2075-16802023-12-011212111510.3390/axioms12121115Developing Constrained Interval Operators for Fuzzy Logic with Interval ValuesJocivania Pinheiro0Regivan H. N. Santiago1Benjamin Bedregal2Flaulles Bergamaschi3Departamento de Ciências Naturais, Matemática e Estatística, Universidade Federal Rural do Semi-Árido, Mossoró 59625-900, BrazilDepartamento de Informática e Matemática Aplicada, Universidade Federal do Rio Grande do Norte, Natal 59072-970, BrazilDepartamento de Informática e Matemática Aplicada, Universidade Federal do Rio Grande do Norte, Natal 59072-970, BrazilDepartamento de Ciências Exatas e Tecnológicas, Universidade Estadual do Sudoeste da Bahia (UESB), Vitória da Conquista 45083-900, BrazilA well-known problem in the interval analysis literature is the overestimation and loss of information. In this article, we define new interval operators, called <i>constrained interval operators</i>, that preserve information and mitigate overestimation. These operators are investigated in terms of correction, algebraic properties, and orders. It is shown that a large part of the properties studied is preserved by this operator, while others remain preserved with the condition of continuity, as is the case of the exchange principle. In addition, a comparative study is carried out between this operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>g</mi><mo>¨</mo></mover></semantics></math></inline-formula> and the best interval representation: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>g</mi><mo stretchy="false">^</mo></mover></semantics></math></inline-formula>. Although <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>g</mi><mo>¨</mo></mover><mo>⊆</mo><mover accent="true"><mi>g</mi><mo stretchy="false">^</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>g</mi><mo>¨</mo></mover></semantics></math></inline-formula> do not preserve the Moore correction, we do not have a loss of relevant information since everything that is lost is irrelevant, mitigating the overestimation.https://www.mdpi.com/2075-1680/12/12/1115interval-valued fuzzy logicinterval mathematicsfuzzy logicfuzzy implications |
| spellingShingle | Jocivania Pinheiro Regivan H. N. Santiago Benjamin Bedregal Flaulles Bergamaschi Developing Constrained Interval Operators for Fuzzy Logic with Interval Values Axioms interval-valued fuzzy logic interval mathematics fuzzy logic fuzzy implications |
| title | Developing Constrained Interval Operators for Fuzzy Logic with Interval Values |
| title_full | Developing Constrained Interval Operators for Fuzzy Logic with Interval Values |
| title_fullStr | Developing Constrained Interval Operators for Fuzzy Logic with Interval Values |
| title_full_unstemmed | Developing Constrained Interval Operators for Fuzzy Logic with Interval Values |
| title_short | Developing Constrained Interval Operators for Fuzzy Logic with Interval Values |
| title_sort | developing constrained interval operators for fuzzy logic with interval values |
| topic | interval-valued fuzzy logic interval mathematics fuzzy logic fuzzy implications |
| url | https://www.mdpi.com/2075-1680/12/12/1115 |
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