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How to approximate fuzzy sets: mind-changes and the Ershov Hierarchy
Nikolay Bazhenov1, Manat Mustafa2, Sergei Ospichev1
1Sobolev Institute of Mathematics, 4 Acad. Koptyug Ave., Novosibirsk, Russia 630090.
Synthese
|February 13, 2023
Summary
This study introduces the Fuzzy Ershov Hierarchy, combining fuzzy set theory with the Ershov Hierarchy to analyze the complexity of fuzzy sets. This new hierarchy offers a novel way to understand approximation errors in fuzzy computations.
Area of Science:
- Computability Theory
- Fuzzy Set Theory
- Mathematical Logic
Background:
- Multiple hierarchies exist to measure set complexity in computability theory.
- The Kleene Hierarchy classifies sets by first-order formula complexity.
- The Ershov Hierarchy measures approximation errors for limit computable sets.
- Fuzzy sets generalize classical sets with membership functions in a complete lattice.
Purpose of the Study:
- To introduce and investigate the Fuzzy Ershov Hierarchy.
- To extend the Ershov Hierarchy to the domain of fuzzy sets.
- To provide a framework for analyzing fuzzy set complexity based on approximation errors.
Main Methods:
- Combining concepts from the Ershov Hierarchy and fuzzy set theory.
- Developing a new hierarchy for fuzzy sets.
- Investigating the properties of the proposed Fuzzy Ershov Hierarchy.
Main Results:
- The Fuzzy Ershov Hierarchy is formally introduced.
- The paper lays the groundwork for analyzing fuzzy set complexity using approximation measures.
- This work extends previous research on fuzzy set hierarchies.
Conclusions:
- The Fuzzy Ershov Hierarchy provides a new tool for computability theory.
- This research bridges the gap between classical computability hierarchies and fuzzy set theory.
- Further research can explore applications and properties of this new hierarchy.
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