A Comprehensive Study on the Different Approaches of the Symmetric Difference in Nilpotent Fuzzy Systems - Luca Sára Pusztaházi; György Eigner; Orsolya Csiszár (Mathematics)
This paper comprehensively examines symmetric difference operators within logical systems generated by nilpotent t-norms and t-conorms, specifically addressing their behavior and applicability in bounded and Łukasiewicz fuzzy logic systems. We identify two distinct symmetric difference operators and analyze their fundamental properties, revealing their inherent non-associativity. Recognizing the limitations posed by non-associative behavior in practical multi-step logical operations, we introduce a novel aggregated symmetric difference operator constructed through the arithmetic mean of the previously defined operators. The primary theoretical contribution of our research is establishing the associativity of this new aggregated operator, significantly enhancing its effectiveness for consistent multi-stage computations. Moreover, this operator retains critical properties including symmetry, neutrality, antitonicity, and invariance under negation, thus making it particularly valuable for various computational and applied domains such as image processing, pattern recognition, fuzzy neural networks, cryptographic schemes, and medical data analysis. The demonstrated theoretical robustness and practical versatility of our associative operator provide a clear improvement over existing methodologies, laying a solid foundation for future research in fuzzy logic and interdisciplinary applications. Our broader aim is to derive and study symmetric difference operators in both bounded and Łukasiewicz systems, as this represents a new direction of research.