Approximating approximate reasoning: Fuzzy sets and the Ershov hierarchy

TL;DR

This paper introduces the Fuzzy Ershov Hierarchy, blending classical computability hierarchies with fuzzy set theory, and explores the complexity and representation of fuzzy sets within this new framework.

Contribution

It extends the Ershov Hierarchy to fuzzy sets, defining fuzzy $n$-c.e. sets, and analyzes their properties and limitations compared to classical hierarchies.

Findings

Fuzzy $n$-c.e. sets can be represented as Boolean combinations of fuzzy c.e. sets.

The hierarchy does not collapse, maintaining distinct levels.

Not all $ ext{Delta}^0_2$ fuzzy sets are captured by the hierarchy.

Abstract

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies $\Delta^0_2$ sets with respect to the number of mistakes that are needed to approximate them. Biacino and Gerla extended the Kleene Hierarchy to the realm of fuzzy sets, whose membership functions range in a complete lattice $L$ (e.g., the real interval $[0; 1]_\mathbb{R}$). In this paper, we combine the Ershov Hierarchy and fuzzy set theory, by introducing and investigating the Fuzzy Ershov Hierarchy. In particular, we focus on the fuzzy $n$-c.e. sets which form the finite levels of this hierarchy. Intuitively, a fuzzy set is $n$-c.e. if its membership function can be approximated by changing monotonicity at most $n-1$ times. We prove that the Fuzzy Ershov Hierarchy does not collapse; that, in analogy with the classical case, each fuzzy $n$-c.e. set can be represented as a Boolean combination of fuzzy c.e. sets; but that, contrary to the classical case, the Fuzzy Ershov Hierarchy does not exhaust the class of all $\Delta^0_2$ fuzzy sets.