Fuzziness, Indeterminacy and Soft Sets: Frontiers and Perspectives

• URL: http://arxiv.org/abs/2211.15408v1
• Date: Thu, 10 Nov 2022 07:09:07 GMT
• Title: Fuzziness, Indeterminacy and Soft Sets: Frontiers and Perspectives
• Authors: Michael Gr. Voskoglou
• Abstract summary: The paper comes across the main steps that laid from Zadeh's fuzziness ana Atanassov's intuitionistic fuzzy sets to Smarandache's indeterminacy and to Molodstov's soft sets. It is described how the concept of topological space can be extended to fuzzy structures and how to generalize the fundamental mathematical concepts of limit, continuity compactness and Hausdorff space within such kind of structures.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The present paper comes across the main steps that laid from Zadeh's fuzziness ana Atanassov's intuitionistic fuzzy sets to Smarandache's indeterminacy and to Molodstov's soft sets. Two hybrid methods for assessment and decision making respectively under fuzzy conditions are also presented through suitable examples that use soft sets and real intervals as tools. The decision making method improves an earlier method of Maji et al. Further, it is described how the concept of topological space, the most general category of mathematical spaces, can be extended to fuzzy structures and how to generalize the fundamental mathematical concepts of limit, continuity compactness and Hausdorff space within such kind of structures. In particular, fuzzy and soft topological spaces are defined and examples are given to illustrate these generalizations.

Related papers

• A Dynamic Survey of Soft Set Theory and Its Extensions [2.2667044928324747]
  Soft set theory provides a direct framework for parameterized decision modeling.<n>The theory has expanded into numerous variants including hypersoft sets, superhypersoft sets, TreeSoft sets, bipolar soft sets, and dynamic soft sets.
  arXiv  Detail & Related papers  (2026-02-24T11:58:45Z)
• Towards A Unified PAC-Bayesian Framework for Norm-based Generalization Bounds [63.47271262149291]
  We propose a unified framework for PAC-Bayesian norm-based generalization.<n>The key to our approach is a sensitivity matrix that quantifies the network outputs with respect to structured weight perturbations.<n>We derive a family of generalization bounds that recover several existing PAC-Bayesian results as special cases.
  arXiv  Detail & Related papers  (2026-01-13T00:42:22Z)
• Native Logical and Hierarchical Representations with Subspace Embeddings [25.274936769664098]
  We introduce a novel paradigm: embedding concepts as linear subspaces.<n>It naturally supports set-theoretic operations like intersection (conjunction) and linear sum (disjunction)<n>Our method achieves state-of-the-art results in reconstruction and link prediction on WordNet.
  arXiv  Detail & Related papers  (2025-08-21T18:29:17Z)
• A Set-to-Set Distance Measure in Hyperbolic Space [50.134086375286074]
  We propose a hyperbolic set-to-set distance measure for computing dissimilarity between sets in hyperbolic space.<n>By considering the topological differences, HS2SD provides a more nuanced understanding of the relationships between two hyperbolic sets.
  arXiv  Detail & Related papers  (2025-06-23T11:31:40Z)
• FUSE: Measure-Theoretic Compact Fuzzy Set Representation for Taxonomy Expansion [36.714348668366]
  We propose a sound and efficient formulation of set representation learning based on its volume approximation as a fuzzy set.<n>The resulting embedding framework, Fuzzy Set Embedding (FUSE), satisfies all set operations and compactly approximates the underlying fuzzy set.
  arXiv  Detail & Related papers  (2025-06-10T03:28:32Z)
• A Canonicalization Perspective on Invariant and Equivariant Learning [54.44572887716977]
  We introduce a canonicalization perspective that provides an essential and complete view of the design of frames. We show that there exists an inherent connection between frames and canonical forms. We design novel frames for eigenvectors that are strictly superior to existing methods.
  arXiv  Detail & Related papers  (2024-05-28T17:22:15Z)
• Foundational theories of hesitant fuzzy sets and families of hesitant fuzzy sets [24.890815871172745]
  Hesitant fuzzy sets find extensive application in specific scenarios involving uncertainty and hesitation. As a type of sets, hesitant fuzzy sets necessitate a clear and explicit definition of the inclusion relationship.
  arXiv  Detail & Related papers  (2023-11-07T14:03:28Z)
• Algebraic, Topological, and Mereological Foundations of Existential Granules [0.0]
  Existential granules are those that determine themselves initially, and interact with their environment subsequently. It is shown that they fit into multiple theoretical frameworks (axiomatic, adaptive, and others) of granular computing. The characterization is intended for algorithm development, application to classification problems and possible mathematical foundations of generalizations.
  arXiv  Detail & Related papers  (2023-08-30T17:22:11Z)
• Isotropic Gaussian Processes on Finite Spaces of Graphs [71.26737403006778]
  We propose a principled way to define Gaussian process priors on various sets of unweighted graphs. We go further to consider sets of equivalence classes of unweighted graphs and define the appropriate versions of priors thereon. Inspired by applications in chemistry, we illustrate the proposed techniques on a real molecular property prediction task in the small data regime.
  arXiv  Detail & Related papers  (2022-11-03T10:18:17Z)
• Edge modes as dynamical frames: charges from post-selection in generally covariant theories [0.0]
  We develop a framework based on the covariant phase space formalism that identifies gravitational edge modes as dynamical reference frames. We study the symmetries consistent with such an embedding. We explain how the boundary conditions and presymplectic structure can be encoded into boundary actions.
  arXiv  Detail & Related papers  (2022-05-02T13:51:45Z)
• Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents [102.42623636238399]
  We identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. We derive algorithms that exploit these geometric structures to solve these problems efficiently.
  arXiv  Detail & Related papers  (2022-03-20T16:23:17Z)
• A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces [0.0]
  We define a symmetric monocave category fuzzy concepts and fuzzy conceptual reasoning within G"ardenfors' spaces. We propose log-concave functions, showing that fuzzy concepts are the most general choice satisfying a criterion.
  arXiv  Detail & Related papers  (2021-10-12T13:22:27Z)
• Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization [98.0595480384208]
  We propose a generalization extraient spaces which converges to a stationary point. The algorithm applies not only to general $p$-normed spaces, but also to general $p$-dimensional vector spaces.
  arXiv  Detail & Related papers  (2020-10-31T21:35:42Z)
• Orthologics for Cones [5.994412766684843]
  In this paper we study logics for such geometric structures. We describe an extension of minimal orthologic with a partial modularity rule that holds for closed convex cones. This logic combines a feasible data structure (exploiting convexity/conicity) with sufficient expressivity, including full orthonegation.
  arXiv  Detail & Related papers  (2020-08-07T13:28:27Z)
• Geodesics in fibered latent spaces: A geometric approach to learning correspondences between conditions [62.997667081978825]
  This work introduces a geometric framework and a novel network architecture for creating correspondences between samples of different conditions. Under this formalism, the latent space is a fiber bundle stratified into a base space encoding conditions, and a fiber space encoding the variations within conditions.
  arXiv  Detail & Related papers  (2020-05-16T03:14:52Z)
• Algebraic Approach to Directed Rough Sets [0.0]
  In relational approach to general rough sets, ideas of directed relations are supplemented with additional conditions. The relations are also specialized to representations of general parthood that are upper-directed, reflexive and antisymmetric.
  arXiv  Detail & Related papers  (2020-04-25T15:39:14Z)
• A General Framework for Consistent Structured Prediction with Implicit Loss Embeddings [113.15416137912399]
  We propose and analyze a novel theoretical and algorithmic framework for structured prediction. We study a large class of loss functions that implicitly defines a suitable geometry on the problem. When dealing with output spaces with infinite cardinality, a suitable implicit formulation of the estimator is shown to be crucial.
  arXiv  Detail & Related papers  (2020-02-13T10:30:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.