{"slug": "on-the-representational-geometry-of-dynamic-programs", "title": "On the Representational Geometry of Dynamic Programs", "summary": "A new arXiv paper (2608.25034v1) proves that finite min-plus dynamic programs are isomorphic to tropical polynomials and extended Newton polyhedra, and shows two structural negatives for length generalization: dimension-reduction operations are not injective or closed, and series/parallel composition cannot construct all DAG topologies from smaller sub-DAGs.", "body_md": "arXiv:2608.25034v1 Announce Type: new\nAbstract: Standard neural architectures often fail to generalize to longer inputs for dynamic programming (DP) targets. We investigate what makes this hard geometrically. Every finite min-plus DP is a shortest path on a DAG, which is equivalently a tropical polynomial whose extended Newton polyhedron encodes the decision boundary of which path wins. We prove these three descriptions (graph, polynomial, polyhedron) form isomorphic semirings at two levels --- formal polynomials and their computed functions --- connected by operations that characterize all structural redundancies. We then address the length-generalization question geometrically: does the decision boundary at length $T$ decide the boundary at $T+1$? We present two structural negatives. The semiring's two native ways to reduce dimension (setting a variable to each identity) are neither injective nor always closed within the DP. Series and parallel composition fail to construct all DAG topologies from smaller sub-DAGs, and even all terminal-only operations do not capture all DP compositions.", "url": "https://wpnews.pro/news/on-the-representational-geometry-of-dynamic-programs", "canonical_source": "https://arxiv.org/abs/2608.25034", "published_at": "2026-08-27 04:00:00+00:00", "updated_at": "2026-08-27 04:19:22.713794+00:00", "lang": "en", "topics": ["artificial-intelligence", "machine-learning", "ai-research"], "entities": ["arXiv"], "alternates": {"html": "https://wpnews.pro/news/on-the-representational-geometry-of-dynamic-programs", "markdown": "https://wpnews.pro/news/on-the-representational-geometry-of-dynamic-programs.md", "text": "https://wpnews.pro/news/on-the-representational-geometry-of-dynamic-programs.txt", "jsonld": "https://wpnews.pro/news/on-the-representational-geometry-of-dynamic-programs.jsonld"}}