Hypercubes, Hyperplanes, and Constraint-Induced Complexity Collapse in Atomic Concept Learning

Computer Science > Artificial Intelligence [Submitted on 3 Aug 2026] Hypercubes, Hyperplanes, and Constraint-Induced Complexity Collapse in Atomic Concept Learning Irene Tsapara 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. Subjects: Artificial Intelligence (cs.AI); Computation and Language (cs.CL); Logic in Computer Science (cs.LO) Cite as: arXiv:2608.02930 [cs.AI]   (or arXiv:2608.02930v1 [cs.AI] for this version)   https://doi.org/10.48550/arXiv.2608.02930 Focus to learn more Submission history From: Irene Tsapara [view email] [v1] Mon, 3 Aug 2026 22:31:30 UTC (850 KB) Access Paper: HTML (experimental) view license Current browse context: cs.AI < prev   |   next > new | recent | 2026-08 Change to browse by: cs cs.CL cs.LO References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)