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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.

Keywords:

Computability theory Ershov Hierarchy Fuzzy set n-Computably enumerable set

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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.