Quantifying Function Simplicity by Arc Length: A Geometric Foundation for Explainable AI

Human understanding of functions is inherently tied to geometric simplicity; linear or smoothly varying functions are intuitively easier to interpret, whereas highly oscillatory or irregular functions appear complex and opaque. Modern AI models, however, often correspond to extremely nonlinear functions whose inner mechanisms are difficult to explain, giving rise to the so-called "black-box problem." This study proposes a geometric approach to explainability, hypothesizing that functions with shorter arc lengths are inherently more interpretable to humans. By generating 50,000 nonlinear functions and analyzing their arc-length distributions, we establish a robust threshold for distinguishing "simple" from "complex" functions. The results demonstrate a 14× median difference between the two groups, validating arc length as a quantitative indicator of human-understandable simplicity. This framework provides a theoretical foundation for developing explainable AI models that approximate black-box functions with geometrically simple surrogates. This study contributes to bridging mathematical geometry and human cognition, providing a new direction for designing inherently interpretable AI systems.