Source-linked AI summary
Mappings on Soft Classes
Athar Kharal, B. Ahmad
TL;DR
The paper addresses the formal treatment of mappings between collections of soft sets and the behavior of soft images and inverse images. It introduces these notions, establishes their properties with examples and counterexamples, and applies them to medical diagnosis in expert systems. The paper concludes that these concepts support approximate reasoning about information systems and may advance Soft Set Theory.
Problem
Classical methods for incomplete knowledge require parameter specification, motivating a formalism and mappings that accommodate approximate descriptions.
Method
The paper defines mappings on soft classes, studies soft images and inverse images, and supports the analysis with examples and counterexamples before applying it to medical diagnosis.
Results
The paper establishes several properties of soft images and inverse images and applies these notions to medical diagnosis in medical expert systems.
Takeaways & Limitations
Soft sets provide a mathematical formulation of approximate reasoning about information systems, with the developed notions applied to medical diagnosis.
Takeaways & Limitations
Some inequalities in Theorem 14 cannot generally be reversed, as demonstrated by counterexamples.
Abstract
from arXiv · showhide
In this paper, we define the notion of a mapping on soft classes and study several properties of images and inverse images of soft sets supported by examples and counterexamples. Finally, these notions have been applied to the problem of medical diagnosis in medical expert systems.
1. Introduction
Classical approaches to incomplete knowledge require pre-specified parameters, whereas Soft Set Theory uses approximate descriptions without requiring a membership function or similar parameter. The paper situates soft sets among related theories and develops mappings on soft classes, soft images, and inverse images for medical diagnosis.
- Motivation: Classical probability, fuzzy, interval, and rough-set methods require pre-specified parameters when addressing incomplete knowledge.Examples include probability density functions, membership functions, and equivalence relations.
- Motivation: Soft Set Theory avoids this parameter-specification requirement by starting from approximate descriptions of objects.It permits parametrization using words, sentences, numbers, functions, or mappings.
- Relation to Existing Theories: Soft sets are collections of information granules closely resembling rough sets and can express arbitrary rough or fuzzy sets.The paper describes Soft Set Theory as more general than rough-set or fuzzy-set theory in this respect.
- Applications: Soft-set applications have extended to analysis, probability, measurement, optimization, game theory, operations research, and multicriteria decision making.The cited applications include rough-set-based knowledge reduction in decision making.
- Paper Scope: This paper introduces mappings on soft classes, studies soft images and inverse images with examples and counterexamples, and applies them to medical diagnosis.Soft classes are collections of soft sets.
2. Preliminaries
The preliminaries define soft sets as parameterized families of subsets and establish operations and inclusion relations used later. They also refine soft-set intersection to exclude the degenerate case of disjoint parameter sets.
- Soft Sets: A soft set over X is a pair (F, A), where F maps parameters in A to subsets of X.For each parameter, F(ε) is the set of ε-approximate elements.
- Soft Relations: Soft-subset inclusion requires both parameter inclusion, A ⊆ B, and pointwise set inclusion, F(ε) ⊆ G(ε).The reverse relation is called a soft superset.
- Operations: Soft union uses parameter set A ∪ B and combines the associated approximations by case: F, G, or F(ε) ∪ G(ε).The cases correspond respectively to parameters exclusive to A, exclusive to B, or shared by both.
- Operations: Earlier intersection definitions were revised because F(ε) and G(ε) need not be identical for shared parameters.The revised operation uses F(ε) ∩ G(ε) on A ∩ B.
- Operations: The paper requires A ∩ B ≠ ∅ when defining intersection, avoiding the degenerate disjoint-parameter case.Definition 6 applies the intersection operation only under this condition.
3. Mappings on Soft Classes
The paper defines mappings between soft classes through mappings on universes and attributes, then establishes properties of soft images and inverse images. Examples and counterexamples show how these operations behave, including cases where inclusion reversals fail.
- Definitions: A soft class is the collection of all soft sets over a universe X whose attributes come from E.
- Definitions: A mapping between soft classes uses u:X→Y and p:E→E′, with the image of (F,A) assigned attributes B=p(A) and membership values determined through u.
- Definitions: The inverse image of a soft set (G,C) uses D=p^-1(C) and assigns each parameter α the set u^-1(G(p(α))).
- Properties of soft images: Theorem 14 gives preservation properties for soft images, including monotonicity: (F,A)⊆(G,B) implies f(F,A)⊆f(G,B).
- Counterexamples: The examples demonstrate that the reversals of inequalities (2) and (4) in Theorem 14 do not hold in general.
- Properties of soft inverse images: Theorem 16 gives corresponding properties for soft inverse images, including monotonicity: (F,A)⊆(G,B) implies f^-1(F,A)⊆f^-1(G,B).
4. An Application in Medical Expert Systems
The medical expert-system application encodes patient symptoms and their importance as a soft set, then maps them to possible causes and treatment preferences using stored medical knowledge. The resulting soft set groups causes by mapped importance levels.
- Patient representation: A patient’s complaints are represented as a soft set whose symptoms are assigned high, medium, or low importance.The narration includes burning stomach, headache, sleeplessness, semi-conscious sleep, joint and backbone pain, depression, and anxiety.
- Knowledge encoding: Medical knowledge is encoded as lookup tables defining mappings from symptoms to causes and from importance levels to treatment preferences.The mappings assign symptoms to conditions such as acidity, blood pressure, wrong posture, fatigue, and low energy level.
- Soft-class mapping: The application uses mappings u : X → Y and p : E → E′ to transform a soft set of symptoms and importance into a soft set of causes and medical preferences.The symptom soft class is mapped to causes, while the importance parameters are mapped to treatment-potency categories.
- Mapped diagnosis output: The resulting representation assigns infrequent high potency to acidity and blood pressure, and frequent low potency to fatigue.The mapped output is presented as the medical system’s transformed soft set.
- Application significance: The paper concludes that soft images and inverse images support an application to medical diagnosis in medical expert systems.The authors present the application as an illustration of the mapping notions developed in the paper.