Pub Date : 2024-12-18DOI: 10.1016/j.fss.2024.109248
Guilong Liu
A covering is a family of sets. This paper studies the generalization of rough set models via a generalization of the concept of neighborhoods when considering a general family of sets instead of a partition of a set. Given the notion of neighborhood, we propose four different rough set models based on a family of sets and study the reduction problem for such sets. The reduction relationship between rough sets and formal contexts is established. Three reduction algorithms are obtained and applied to identify all reducts. We extend these results to fuzzy sets. Using a family of fuzzy sets to replace a fuzzy covering or a fuzzy β covering, we define set-based fuzzy rough sets and consider three types of reduction. Finally, we give reduction algorithms that are required to identify all reducts.
{"title":"On some types of reduction for families of fuzzy sets","authors":"Guilong Liu","doi":"10.1016/j.fss.2024.109248","DOIUrl":"10.1016/j.fss.2024.109248","url":null,"abstract":"<div><div>A covering is a family of sets. This paper studies the generalization of rough set models via a generalization of the concept of neighborhoods when considering a general family of sets instead of a partition of a set. Given the notion of neighborhood, we propose four different rough set models based on a family of sets and study the reduction problem for such sets. The reduction relationship between rough sets and formal contexts is established. Three reduction algorithms are obtained and applied to identify all reducts. We extend these results to fuzzy sets. Using a family of fuzzy sets to replace a fuzzy covering or a fuzzy <em>β</em> covering, we define set-based fuzzy rough sets and consider three types of reduction. Finally, we give reduction algorithms that are required to identify all reducts.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109248"},"PeriodicalIF":3.2,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-18DOI: 10.1016/j.fss.2024.109249
Deli Zhang , Radko Mesiar , Endre Pap
As a generalization of Choquet integrals, the generalized Choquet type set-valued integral of functions w.r.t. set multifunctions and σ-additive measures has been performed in our previous paper [66]. The present paper is its continuation and it brings a novel set-valued type Choquet integral, named double set function multi-valued Choquet integral (DSMVCI), where the σ-additive measure is replaced by a fuzzy measure. Various kinds of its properties and convergence theorems are obtained, and set-valued type Jensen's and Markov's type inequalities are proved. Its application in multi-attribute decision-making with hesitant fuzzy information is given.
{"title":"Multi-valued Choquet integral based on a couple of set functions with an application in multi-attribute decision-making","authors":"Deli Zhang , Radko Mesiar , Endre Pap","doi":"10.1016/j.fss.2024.109249","DOIUrl":"10.1016/j.fss.2024.109249","url":null,"abstract":"<div><div>As a generalization of Choquet integrals, the generalized Choquet type set-valued integral of functions w.r.t. set multifunctions and <em>σ</em>-additive measures has been performed in our previous paper <span><span>[66]</span></span>. The present paper is its continuation and it brings a novel set-valued type Choquet integral, named double set function multi-valued Choquet integral (DSMVCI), where the <em>σ</em>-additive measure is replaced by a fuzzy measure. Various kinds of its properties and convergence theorems are obtained, and set-valued type Jensen's and Markov's type inequalities are proved. Its application in multi-attribute decision-making with hesitant fuzzy information is given.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109249"},"PeriodicalIF":3.2,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Under the assumption of probable input and fuzzy state (profust), profust (also named as generalized) failure probability function (G-FPF), which varies with random input distribution parameters (DP) in the interested region, can reflect the effect of DP on structure safety and decouples the generalized reliability-based design optimization. The direct double-loop analysis of G-FPF, which repeatedly estimates the G-FPF values at different DP realizations, is time-consuming. Thus, this paper proposes a single-loop importance sampling (IS) method to estimate G-FPF by combining a variance reduction technique with a sample information-sharing strategy. The proposed method has two innovations. The first is constructing an optimal unified IS density (ISD), which is independent of the DP and envelops the interested DP region. By sharing the sample of the unified ISD, the double-loop analysis for G-FPF can be avoided, and by fusing the IS variance reduction technique, the efficiency of estimating G-FPF can be improved further. The second is designing an adaptive strategy to update the Kriging model of performance function, so that the computational cost, which is measured by the number of performance function evaluations while ensuring the acceptable precision of G-FPF estimation, can be reduced in approaching and sampling the optimal unified ISD as well as predicting the performance function at the sample of the optimal unified ISD. Moreover, the proposed method has wide applicability, and it has no restriction on the nonlinearity of the performance function and the size of the interested DP region, which is sufficiently verified by the presented examples.
{"title":"An efficient algorithm for estimating profust failure probability function under the assumption of probable input and fuzzy state","authors":"Xiaomin Wu, Zhenzhou Lu, Yizhou Chen, Kaixuan Feng","doi":"10.1016/j.fss.2024.109250","DOIUrl":"10.1016/j.fss.2024.109250","url":null,"abstract":"<div><div>Under the assumption of <strong>pro</strong>bable input and <strong>fu</strong>zzy <strong>st</strong>ate (profust), profust (also named as generalized) failure probability function (G-FPF), which varies with random input distribution parameters (DP) in the interested region, can reflect the effect of DP on structure safety and decouples the generalized reliability-based design optimization. The direct double-loop analysis of G-FPF, which repeatedly estimates the G-FPF values at different DP realizations, is time-consuming. Thus, this paper proposes a single-loop importance sampling (IS) method to estimate G-FPF by combining a variance reduction technique with a sample information-sharing strategy. The proposed method has two innovations. The first is constructing an optimal unified IS density (ISD), which is independent of the DP and envelops the interested DP region. By sharing the sample of the unified ISD, the double-loop analysis for G-FPF can be avoided, and by fusing the IS variance reduction technique, the efficiency of estimating G-FPF can be improved further. The second is designing an adaptive strategy to update the Kriging model of performance function, so that the computational cost, which is measured by the number of performance function evaluations while ensuring the acceptable precision of G-FPF estimation, can be reduced in approaching and sampling the optimal unified ISD as well as predicting the performance function at the sample of the optimal unified ISD. Moreover, the proposed method has wide applicability, and it has no restriction on the nonlinearity of the performance function and the size of the interested DP region, which is sufficiently verified by the presented examples.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"504 ","pages":"Article 109250"},"PeriodicalIF":3.2,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-17DOI: 10.1016/j.fss.2024.109245
Xiaowei Wei , Yueli Yue
Choosing completely distributive lattices as lattice-valued environment, this paper introduces the strong L-upper set monad , studies Eilenberg-Moore algebras, Kleisli monoids and its applications of . Concretely, we give the monad by adjoints. Then we characterize fuzzy completely distributive lattices by its Eilenberg-Moore algebras and study Kleisli monoids of the monad. Moreover, we investigate the necessary and sufficient conditions for the establishment of . Finally, we construct two monads, study the relationships between these two monads and , moreover verify the rationality of the selection of lattice-valued environment by those relationships.
{"title":"Algebraic representations of (fuzzy) completely distributive lattices","authors":"Xiaowei Wei , Yueli Yue","doi":"10.1016/j.fss.2024.109245","DOIUrl":"10.1016/j.fss.2024.109245","url":null,"abstract":"<div><div>Choosing completely distributive lattices as lattice-valued environment, this paper introduces the strong <em>L</em>-upper set monad <span><math><mi>S</mi></math></span>, studies Eilenberg-Moore algebras, Kleisli monoids and its applications of <span><math><mi>S</mi></math></span>. Concretely, we give the monad <span><math><mi>S</mi></math></span> by adjoints. Then we characterize fuzzy completely distributive lattices by its Eilenberg-Moore algebras and study Kleisli monoids of the monad. Moreover, we investigate the necessary and sufficient conditions for the establishment of <span><math><mi>S</mi></math></span>. Finally, we construct two monads, study the relationships between these two monads and <span><math><mi>S</mi></math></span>, moreover verify the rationality of the selection of lattice-valued environment by those relationships.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109245"},"PeriodicalIF":3.2,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-17DOI: 10.1016/j.fss.2024.109244
Eszter K. Horváth , Leonard Kwuida , Branimir Šešelja , Andreja Tepavčević
Starting from a poset P and a set A, we introduce P-sets as a natural generalization of Ω-sets. A P-set on A is defined by adding to A a special map from to P, which generalizes the classical equality relation. We prove that P-sets on A are naturally obtained from centralized closure systems in a family of all weak equivalences on A. Moreover, for every P-set there is a canonical representation in which the used centralized closure system replaces the poset P.
Further, we present a classification of all P-sets by the family of cuts, where A and P are fixed. Different P-sets may have equal collections of cut sets. Necessary and sufficient conditions under which this happens are presented; i.e., all P-sets are classified according to the equality of collections of cut sets.
从正集 P 和集合 A 开始，我们引入 P 集作为 Ω 集的自然概括。A 上的 P 集是通过在 A 上添加一个从 A2 到 P 的特殊映射来定义的，它是对经典相等关系的概括。我们证明，A 上的 P 集是由 A 上所有弱等价关系族中的集中闭合系统自然获得的。此外，对于每个 P 集，都有一个规范表示，其中使用的集中闭合系统取代了 P 正集。不同的 P 集可能有相同的切集集合。我们提出了发生这种情况的必要条件和充分条件；也就是说，所有 P 集都是根据切集集合的相等性来分类的。
{"title":"P-sets","authors":"Eszter K. Horváth , Leonard Kwuida , Branimir Šešelja , Andreja Tepavčević","doi":"10.1016/j.fss.2024.109244","DOIUrl":"10.1016/j.fss.2024.109244","url":null,"abstract":"<div><div>Starting from a poset <em>P</em> and a set <em>A</em>, we introduce <em>P</em>-sets as a natural generalization of Ω-sets. A <em>P</em>-set on <em>A</em> is defined by adding to <em>A</em> a special map from <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> to <em>P</em>, which generalizes the classical equality relation. We prove that <em>P</em>-sets on <em>A</em> are naturally obtained from centralized closure systems in a family of all weak equivalences on <em>A</em>. Moreover, for every <em>P</em>-set there is a canonical representation in which the used centralized closure system replaces the poset <em>P</em>.</div><div>Further, we present a classification of all <em>P</em>-sets by the family of cuts, where <em>A</em> and <em>P</em> are fixed. Different <em>P</em>-sets may have equal collections of cut sets. Necessary and sufficient conditions under which this happens are presented; i.e., all <em>P</em>-sets are classified according to the equality of collections of cut sets.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109244"},"PeriodicalIF":3.2,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-17DOI: 10.1016/j.fss.2024.109246
Stefan Stanimirović , Linh Anh Nguyen , Miroslav Ćirić , Marko Stanković
We introduce two novel concepts: fuzzy weak (bi)simulations and their approximations, breadth-first fuzzy (bi)simulations, for fuzzy automata with membership values in an arbitrary complete residuated lattice. They offer improved approximations of the subsethood degree (equivalence degree) of fuzzy automata compared to previously established notions. Fuzzy weak (bi)simulations are defined as fuzzy relations, whereas breadth-first fuzzy (bi)simulations are characterized by decreasing sequences of fuzzy relations, where each component is a solution to a finite and linear system of fuzzy relation inequalities. Such systems consist of fuzzy sets that are nodes of the trees constructed in the classic breadth-first search manner. We provide an algorithm that computes the proposed simulations and bisimulations, along with a modified version that computes the subsethood/equality degree for given fuzzy automata. Both algorithms run in the exponential time complexity, reflecting the trade-off for achieving more precise approximations.
{"title":"Breadth-first fuzzy bisimulations for fuzzy automata","authors":"Stefan Stanimirović , Linh Anh Nguyen , Miroslav Ćirić , Marko Stanković","doi":"10.1016/j.fss.2024.109246","DOIUrl":"10.1016/j.fss.2024.109246","url":null,"abstract":"<div><div>We introduce two novel concepts: <em>fuzzy weak (bi)simulations</em> and their approximations, <em>breadth-first fuzzy (bi)simulations</em>, for fuzzy automata with membership values in an arbitrary complete residuated lattice. They offer improved approximations of the subsethood degree (equivalence degree) of fuzzy automata compared to previously established notions. Fuzzy weak (bi)simulations are defined as fuzzy relations, whereas breadth-first fuzzy (bi)simulations are characterized by decreasing sequences of fuzzy relations, where each component is a solution to a finite and linear system of fuzzy relation inequalities. Such systems consist of fuzzy sets that are nodes of the trees constructed in the classic breadth-first search manner. We provide an algorithm that computes the proposed simulations and bisimulations, along with a modified version that computes the subsethood/equality degree for given fuzzy automata. Both algorithms run in the exponential time complexity, reflecting the trade-off for achieving more precise approximations.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109246"},"PeriodicalIF":3.2,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-13DOI: 10.1016/j.fss.2024.109242
Xiaona Song , Zenglong Peng , Zhijia Zhao , Shuai Song
This paper mainly focuses on the interval observer design for nonlinear partial differential equation systems with non-homogeneous Dirichlet boundary conditions. Initially, under unknown but bounded external disturbances and uncertainties caused by unmeasurable premise variables, a Takagi–Sugeno fuzzy interval observer is established to estimate the upper and lower bounds of the system state. Then, by converting non-homogeneous Dirichlet boundary conditions into homogeneous ones, the solvable conditions to ensure the stability of the upper and lower observation errors are designed. Furthermore, to improve the observation accuracy of the designed interval observer, an -optimal fuzzy interval observer is designed to make the interval width more compact. Finally, the effectiveness and advantages of the designed interval observer are verified through numerical simulations.
{"title":"H∞-optimal interval observer design for nonlinear PDE systems","authors":"Xiaona Song , Zenglong Peng , Zhijia Zhao , Shuai Song","doi":"10.1016/j.fss.2024.109242","DOIUrl":"10.1016/j.fss.2024.109242","url":null,"abstract":"<div><div>This paper mainly focuses on the interval observer design for nonlinear partial differential equation systems with non-homogeneous Dirichlet boundary conditions. Initially, under unknown but bounded external disturbances and uncertainties caused by unmeasurable premise variables, a Takagi–Sugeno fuzzy interval observer is established to estimate the upper and lower bounds of the system state. Then, by converting non-homogeneous Dirichlet boundary conditions into homogeneous ones, the solvable conditions to ensure the stability of the upper and lower observation errors are designed. Furthermore, to improve the observation accuracy of the designed interval observer, an <span><math><msub><mrow><mi>H</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-optimal fuzzy interval observer is designed to make the interval width more compact. Finally, the effectiveness and advantages of the designed interval observer are verified through numerical simulations.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109242"},"PeriodicalIF":3.2,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180981","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-13DOI: 10.1016/j.fss.2024.109241
Jing He, Dexue Zhang
This paper presents a study of separation axioms and sobriety of bitopological spaces from the point of view of fuzzy topology via identifying bitopological spaces with topological spaces valued in the Boolean algebra of four elements. A system of separation axioms is proposed making use of Boolean-valued specialization order of bitopological spaces; The relationship between d-sobriety of bitopological spaces proposed by Jung and Moshier and sobriety of fuzzy topological spaces is studied; A Hofmann-Mislove theorem for bitopological spaces is established.
{"title":"A Boolean-valued space approach to separation axioms and sobriety of bitopological spaces","authors":"Jing He, Dexue Zhang","doi":"10.1016/j.fss.2024.109241","DOIUrl":"10.1016/j.fss.2024.109241","url":null,"abstract":"<div><div>This paper presents a study of separation axioms and sobriety of bitopological spaces from the point of view of fuzzy topology via identifying bitopological spaces with topological spaces valued in the Boolean algebra of four elements. A system of separation axioms is proposed making use of Boolean-valued specialization order of bitopological spaces; The relationship between d-sobriety of bitopological spaces proposed by Jung and Moshier and sobriety of fuzzy topological spaces is studied; A Hofmann-Mislove theorem for bitopological spaces is established.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109241"},"PeriodicalIF":3.2,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-13DOI: 10.1016/j.fss.2024.109243
Hongliang Lai, Qingzhu Luo
Let & be a continuous triangular norm on the unit interval and -Cat be the category of all real-enriched categories. Firstly, it is shown that a stable subconstruct A of -Cat is cartesian closed if and only if it is determined by a suitable subset of , where M is the set of all elements x in such that x&x is idempotent. Secondly, it is shown that all Yoneda complete real-enriched categories valued in the set M and Yoneda continuous -functors also form a cartesian closed category.
设 & 是单位区间 [0,1] 上的连续三角形规范，[0,1]-Cat 是所有实富集范畴。首先，本文证明了当且仅当[0,1]-Cat 的稳定子结构 A 是由[0,1]2 的合适子集 S⊆M2 确定时，它是卡特封闭的，其中 M 是[0,1]中所有元素 x 的集合，使得 x&x 是幂等的。其次，证明了所有在集合 M 和米氏连续 [0,1] 函数中取值的米氏完全实富集范畴也构成了一个笛卡尔封闭范畴。
{"title":"Cartesian closed and stable subconstructs of [0,1]-Cat","authors":"Hongliang Lai, Qingzhu Luo","doi":"10.1016/j.fss.2024.109243","DOIUrl":"10.1016/j.fss.2024.109243","url":null,"abstract":"<div><div>Let & be a continuous triangular norm on the unit interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> and <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>-<strong>Cat</strong> be the category of all real-enriched categories. Firstly, it is shown that a stable subconstruct <strong>A</strong> of <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>-<strong>Cat</strong> is cartesian closed if and only if it is determined by a suitable subset <span><math><mi>S</mi><mo>⊆</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <em>M</em> is the set of all elements <em>x</em> in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> such that <em>x</em>&<em>x</em> is idempotent. Secondly, it is shown that all Yoneda complete real-enriched categories valued in the set <em>M</em> and Yoneda continuous <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>-functors also form a cartesian closed category.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109243"},"PeriodicalIF":3.2,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-12DOI: 10.1016/j.fss.2024.109240
Marcelo E. Coniglio , Martín Figallo
Mathematical Fuzzy Logic studies fuzzyness from a foundational perspective based on many-valued logics. In the early twenties of the past century, almost simultaneously, systems of many-valued logic were introduced in the respective articles of Jan Łukasiewicz and Emil L. Post, making many-valued logics a respectable field of study. Emil L. Post gave a definition of many-valued logics that was a generalization of two-valued classical logic. He defined the most important operations and discussed some of their properties by means of truth-tables. Two decades later Paul C. Rosenbloom gave a definition of an algebraic structure that served as an interpretation of Post's system; these structures are called Post algebras. Post algebras were meant to capture the algebraic properties of Post's systems. On the other hand, in his 1969 PhD dissertation, Roberto Cignoli showed that n-valued () Post algebras are Łukasiewicz-Moisil algebras containing additional constants , ( and ).
In this paper, in first place, we give a representation of n-valued Post algebras by means of generalized twist structures. More precisely, we define n-valued twist structures as certain subsets of the product of 2n lattices satisfying a number of conditions; and present a suitable notion of morphism between these structures. Then, we prove that the category of n-valued Post algebras with its morphisms (homomorphisms in the sense of Universal Algebra) and the category of generalized twist structures with its morphisms are equivalent. On the other hand, we study some logics that arise from n-valued Post algebras. More precisely, we present different n-valued logics that naturally can be associated to n-valued Post algebras, namely, the non–falsity preserving logic, the truth–preserving logic, and the logic that preserves degrees of truth associated to n-valued Post algebras. We provide cut-free sequent calculus for the first two logics and a sequent calculus for the last one that presumably does not enjoy the cut-elimination property. Finally, it is shown that the logic that preserves degrees of truth w.r.t. n-valued Post algebras is a logic determined by logic matrices.
数理模糊逻辑学》从多值逻辑的基础角度研究模糊性。上世纪二十年代初，扬-卢卡谢维奇（Jan Łukasiewicz）和埃米尔-波斯特（Emil L. Post）几乎同时在各自的文章中提出了多值逻辑体系，使多值逻辑成为一个令人尊敬的研究领域。Emil L. Post 给出了多值逻辑的定义，这是对两值经典逻辑的概括。他定义了最重要的运算，并通过真值表讨论了它们的一些性质。二十年后，保罗-罗森布洛姆（Paul C. Rosenbloom）给出了一个代数结构的定义，作为对波斯特体系的解释；这些结构被称为波斯特代数。波斯特代数的目的是捕捉波斯特系统的代数特性。另一方面，罗伯托-西格诺里（Roberto Cignoli）在其 1969 年的博士论文中指出，n 值（n≥2）波斯特后代数是包含 n-2 个附加常数 ei、1≤i≤n-2（e0=0 和 en-1=1）的 Łukasiewicz-Moisil 后代数。更确切地说，我们把 n 值扭转结构定义为满足若干条件的 2n 格的乘积的某些子集；并提出了这些结构之间的态的适当概念。然后，我们证明了 n 值后代数范畴及其态式（泛代数意义上的同态）与广义扭曲结构范畴及其态式是等价的。另一方面，我们研究了由 n 值后代数产生的一些逻辑。更确切地说，我们提出了自然可以与 n 值后代数相关联的不同 n 值逻辑，即与 n 值后代数相关联的非虚假性保留逻辑、真值保留逻辑和真值度保留逻辑。我们为前两种逻辑提供了无剪切序列微积分，并为最后一种逻辑提供了一种序列微积分，它大概不享有剪切消除特性。最后，我们证明了在 n 值后验代数中保持真度的逻辑是由 n-1 逻辑矩阵决定的逻辑。
{"title":"On n-valued Post algebras and n-valued Post logics: Twist-style representation and proof theory","authors":"Marcelo E. Coniglio , Martín Figallo","doi":"10.1016/j.fss.2024.109240","DOIUrl":"10.1016/j.fss.2024.109240","url":null,"abstract":"<div><div>Mathematical Fuzzy Logic studies fuzzyness from a foundational perspective based on many-valued logics. In the early twenties of the past century, almost simultaneously, systems of many-valued logic were introduced in the respective articles of Jan Łukasiewicz and Emil L. Post, making many-valued logics a respectable field of study. Emil L. Post gave a definition of many-valued logics that was a generalization of two-valued classical logic. He defined the most important operations and discussed some of their properties by means of truth-tables. Two decades later Paul C. Rosenbloom gave a definition of an algebraic structure that served as an interpretation of Post's system; these structures are called Post algebras. Post algebras were meant to capture the algebraic properties of Post's systems. On the other hand, in his 1969 PhD dissertation, Roberto Cignoli showed that <em>n</em>-valued (<span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>) Post algebras are Łukasiewicz-Moisil algebras containing <span><math><mi>n</mi><mo>−</mo><mn>2</mn></math></span> additional constants <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>2</mn></math></span> (<span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span>).</div><div>In this paper, in first place, we give a representation of <em>n</em>-valued Post algebras by means of <em>generalized twist structures</em>. More precisely, we define <em>n</em>-valued twist structures as certain subsets of the product of 2<em>n</em> lattices satisfying a number of conditions; and present a suitable notion of morphism between these structures. Then, we prove that the category of <em>n</em>-valued Post algebras with its morphisms (homomorphisms in the sense of Universal Algebra) and the category of generalized twist structures with its morphisms are equivalent. On the other hand, we study some logics that arise from <em>n</em>-valued Post algebras. More precisely, we present different <em>n</em>-valued logics that naturally can be associated to <em>n</em>-valued Post algebras, namely, the non–falsity preserving logic, the truth–preserving logic, and the logic that preserves degrees of truth associated to <em>n</em>-valued Post algebras. We provide cut-free sequent calculus for the first two logics and a sequent calculus for the last one that presumably does not enjoy the cut-elimination property. Finally, it is shown that the logic that preserves degrees of truth w.r.t. <em>n</em>-valued Post algebras is a logic determined by <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> logic matrices.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"503 ","pages":"Article 109240"},"PeriodicalIF":3.2,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143180977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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