Geometric Semantic Genetic Programming for Evolving Real-Valued Functions with Order Awareness

In many applications, the relative order of the outputs is more crucial than their absolute numerical values. Geometric semantic genetic programming (GSGP) typically operates on function outputs directly but lacks an inherent behaviour to represent the order structure of a function, making it less suitable for problems where decisions are driven by comparison of outputs. In this study, we explore a novel perspective of semantics in the context of GSGP rooted in the order structure of real-valued functions referred to as order semantics. We show that existing geometric semantic operators for real-valued functions retain their geometric properties under order semantics when an alternative notion of semantic distance is considered instead of Euclidean distance. Consequently, the fitness landscape seen by these operators is unimodal with respect to this choice of distance in order semantic space. We validate our method through experiments comparing standard GSGP and a newly proposed GSGP based on order semantics on randomly generated functions. Our results demonstrate that the proposed GSGP improves ranking accuracy at the cost of numerical precision.