Hesitation-Gated Fuzzy–Neutrosophic Prototype Learning under

Asymmetric Label Noise

Necati Olgun1,* Ahmed Hatip1

1 Department of Mathematics, Gaziantep University, Gaziantep, Turkey

Received: September 30, 2025 Revised: November 27, 2025 Accepted: January 30, 2026 ⋆ Corresponding author

ABSTRACT

Label corruption is difficult for prototype classifiers because a mislabeled observation does two things at once: it

perturbs the prototype vyi associated with the supplied label and obscures whether the observation is genuinely

ambiguous or simply inconsistent with its assigned class. This paper develops an adaptive fuzzy–neutrosophic

prototype learning algorithm that separates these effects. For each training observation, the membership vector

ui = (ui1, . . . ,uiK) ∈ ΔK−1 induced by the current prototypes is converted into the evidence state zi = (Ti, Ii,Fi) ∈

[0,1]3: truth is the membership assigned to the observed class, falsity is the strongest competing membership,

and indeterminacy is the normalized membership entropy. These quantities drive three coupled mechanisms:

a contradiction margin ci = [Fi −Ti]+ that attenuates unreliable labels, an entropy-dependent fuzzy exponent

mi ∈ [mmin,mmax] that adapts membership weighting near class overlap, and a conservative soft-label correction

activated only when Fi > Ti and the hesitation Ii is sufficiently small. The resulting Adaptive Fuzzy–Neutrosophic

Prototype Learning (AFNPL) algorithm remains a lightweight prototype method with linear cost in the number of

observations, classes and features per iteration. A reproducible three-class study evaluates 0–40% cyclic asymmetric

label corruption under low, medium and high class overlap. At medium overlap and 40% corruption, AFNPL

obtains 91.54% test accuracy and 91.55% macro-F1, compared with 72.52%/72.37% for noisy class means and

80.23%/80.17% for trimmed class means. Its internal contradiction score also detects corrupted labels with mean

AUC between 0.959 and 0.971 across the contaminated settings. The contribution is therefore not only a robust

prototype update, but a fuzzy learning mechanism in which neutrosophic truth, indeterminacy and falsity have

explicit algorithmic roles in label reliability and adaptive fuzzy weighting.

Keywords: Fuzzy prototype learning Single-valued neutrosophic information Label noise Adaptive fuzzy exponent

Uncertainty Robust classification

1. INTRODUCTION

Fuzzy learning was introduced to avoid the brittle assumption

that an observation must belong entirely to one class

or cluster, replacing a hard indicator 1{yi = k} by graded

memberships uik ∈ [0,1]. In fuzzy c-means (FCM), membership

degrees distribute an observation over several prototypes

rather than forcing a hard assignment [1]. Possibilistic clustering

later relaxed the unit-sum membership constraint to

reduce sensitivity to atypical points [2], while type-2 fuzzy

sets provided an additional layer for representing uncertainty

in the membership function itself [3]. These ideas remain

relevant whenever the data geometry is uncertain, boundaries

overlap, or supervision cannot be treated as completely reliable;

in such cases, a membership vector with maxk uik < 1

conveys information that a hard label suppresses.