3 research outputs found
Attribute implications with unknown information based on weak Heyting algebras
Simplification logic, a logic for attribute implications, was originally defined for Boolean sets. It was extended to distributive fuzzy sets by using a complete dual Heyting algebra. In this paper, we weaken this restriction in the sense that we prove that it is possible to define a simplification logic on fuzzy sets in which the membership value structure is not necessarily distributive. For this purpose, we replace the structure of the complete dual Heyting algebra by the so-called weak complete dual Heyting algebra. We demonstrate the soundness and completeness of this simplification logic, and provide a characterisation of the operations defining weak complete dual Heyting algebras.Funding for open access charge: Universidad de MĂĄlaga/CBU
Parameterized Simplification Logic: Reasoning With Implications in an Automated Way
In this sequel to our previous article (Cordero et al., 2020) on general inference systems for reasoning with ifâthen dependencies, we study transformations of ifâthen rules to semantically equivalent collections of ifâthen rules suitable to solve several problems related to reasoning with data dependencies.Wework in a framework of general lattice-based ifâthen rules whose semantics is parameterized by systems of isotone Galois connections. This framework allows us to obtain theoretical insight as well as algorithms on a general level and observe their special cases by choosing types of parameterizations. This way, we study methods
for automated reasoning with different types of ifâthen rules in a single framework that covers existing as well as novel types of rules. Our approach supports a large family of ifâthen rules, including fuzzy ifâthen rules with various types of semantics. Themain results in this article include new observations on the syntactic inference of ifâthen rules, complete collections of rules, reduced normal forms of collections of rules, and automated reasoning methods. We demonstrate the generality of the framework and the results by examples of their particular cases focusing on fuzzy ifâthen rules.This work was supported in part by the Spanish Ministry of Science, Innovation, and Universities (MCIU), in part by the State Agency of Research (AEI), in part by the Junta de AndalucĂa (JA), in part by the Universidad de MĂĄlaga (UMA), and in part by the European Regional Development Fund (FEDER) under Grant TIN2017-89023-P (MCIU/AEI/FEDER), Grant UMA2018-FEDERJA-001, Grant UMA18-FEDERJA-158, and Grant UMACEIATECH- 24 (JA/UMA/FEDER)
Simplification logic for the management of unknown information
This paper aims to contribute to the extension of classical Formal Concept Analysis (FCA), allowing the management of unknown information. In a preliminary paper, we define a new kind of attribute implications to represent the knowledge from the information currently available. The whole FCA framework has to be appropriately extended to manage unknown information. This paper introduces a new logic for reasoning with this kind of implications, which belongs to the family of logics with an underlying Simplification paradigm. Specifically, we introduce a new algebra, named weak dual Heyting Algebra, that allows us to extend the Simplification logic for these new implications. To provide a solid framework, we also prove its soundness and completeness and show the advantages of the Simplification paradigm. Finally, to allow further use of this extension of FCA in applications, an algorithm for automated reasoning, which is directly built from logic, is defined.Funding for open access charge: Universidad de MĂĄlaga / CBUA
This article is Supported by Grants TIN2017-89023-P, PRE2018-085199 and PID2021-127870OB-I00 of the Ministry of Science
and Innovation of Spain and UMA2018-FEDERJA-001 of the Junta de Andalucia and European Social Fund