Hypercube Geometry Reveals Complexity Collapse in Concept Learning

This paper revisits the learning of higher-arity atomic concepts by examining the geometric properties of hypercubes and hyperplanes formed by ground instances. The core observation is that the r-dimensional hypercube of ground atoms exhibits structural non-uniformity in its logical complexity. Specifically, all hyperplanes, except for the full diagonal, collapse into a finite number of elementary-equivalence classes, with this bound being independent of term depth. In contrast, the full diagonal is unique, with its class count growing without limit. This geometric asymmetry is not merely superficial but reflects the underlying reduction-theoretic structure of the concepts themselves. Building on prior work, the authors reframe these findings using canonical simple concepts, minimal orderings, and representative reductions. This approach provides a taxonomy of hyperplane behavior in higher dimensions, demonstrating that complexity is concentrated in specific areas of the instance space rather than being uniformly distributed. The paper includes detailed examples for binary and ternary hypercubes, illustrating how orthogonal families, partial diagonals, and the exceptional full diagonal contribute to this complexity localization.