🤖 AI Summary
This study addresses the uneven distribution of instance-space complexity in higher-order atomic concept learning by introducing a locality-of-complexity perspective grounded in the geometric structures of hypercubes and hyperplanes. It reveals that logical complexity concentrates along the full diagonal, while complexity collapses on other hyperplanes due to constraint-induced simplifications. Through high-dimensional geometric modeling, analysis of logical equivalence classes, and construction of constrained hypothesis spaces—combined with canonical simple concepts, minimal orderings, and representative reduction mechanisms—the work systematically classifies the behavior of high-dimensional hyperplanes. The analysis fully resolves binary and ternary cases, characterizing properties of orthogonal families, partial diagonals, and full diagonals, and establishes an upper bound on the number of equivalence classes for non-full-diagonal hyperplanes that is independent of term depth.
📝 Abstract
We revisit higher-arity atomic concept learning through the geometry of hypercubes and hyperplanes of ground instances. Our starting point is the observation that the ambient r-dimensional hypercube of ground atoms is not structurally uniform. Its logical complexity is organized by hyperplanes: every hyperplane other than the full diagonal collapses into finitely many elementary-equivalence classes, with a bound independent of the term depth, while the full diagonal is exceptional and its class count grows without bound. This asymmetry is not merely geometric. It reflects the reduction-theoretic structure of the concepts themselves. Building on a higher-dimensional framework developed in the author's earlier work, we reinterpret these results through canonical simple concepts, minimal orderings, and representative reductions. This yields a taxonomy of hyperplane behavior in higher dimensions and shows that complexity is localized rather than spread uniformly through the instance space. The paper includes a fully worked binary case, an explicit treatment of the ternary hypercube, and an unpacked account of the reduction machinery that drives the collapse. The three-dimensional case already exhibits the essential phenomenon of orthogonal families, partial diagonals, and the exceptional full diagonal. This geometric-logical perspective clarifies where complexity is concentrated in atomic concept learning and suggests a modern interpretation in terms of constrained hypothesis spaces and structured classification.