arXiv:2608.25034 2026 Architecture 2 ideas extracted · analyzed Aug 29, 2026
The paper establishes an exact equivalence between two-terminal DAG path computations, min-plus polynomials, and extended Newton polyhedra. This provides a transferable canonicalization mechanism: alternatives that are not on the lower convex hull can be removed without changing the computed function for any input. The same algebra suggests neural modules with explicit min-plus states, where serial composition is Minkowski addition and parallel composition is lower-hull reduction. These structures are especially promising for dynamic-programming tasks and length extrapolation, where opaque recurrent states often fail.
Build a sequence or graph module whose state is a canonical set of affine tropical pieces rather than an opaque hidden vector. Compose consecutive segments by Minkowski addition of their lifted supports, merge alternatives by union followed by lower-hull reduction, and evaluate the resulting piecewise-linear function with a minimum.
Represent each alternative in a min-plus router or dynamic-programming layer by an affine score \(c_i+\langle\alpha_i,x\rangle\). Remove every alternative whose lifted point \((\alpha_i,c_i)\) is not on the lower convex hull, because it can never be the unique minimum for any input and its deletion preserves the exact output function.