Internalizing Tools as Morphisms in Graded Transformers

TL;DR

This paper presents a graded transformer framework that internalizes symbolic tools as morphisms, enabling selective, interpretable, and differentiable symbolic operations within neural models, unifying symbolic computation and geometric learning.

Contribution

It introduces a novel graded formulation of internal symbolic computation in transformers, with an algebraic and geometric foundation, and demonstrates its effectiveness on hybrid symbolic-linguistic tasks.

Findings

Selective morphic activation demonstrated on hybrid tasks

Framework unifies symbolic computation with geometric learning

Subsumes prior external-tool paradigms as special cases

Abstract

We introduce a graded formulation of internal symbolic computation for transformers. The hidden space is endowed with a grading $V=\bigoplus_{g\in G}V_g$, and symbolic operations are realized as typed block maps (morphisms) $\phi_{h\leftarrow g}:V_g\to V_h$ that are activated selectively by a differentiable routing policy. A self-supervised \emph{graded utility functional}, defined as the loss reduction induced by a candidate morphism, governs activation and yields sparse, interpretable behavior. We develop the algebraic and geometric foundations: an internal model category whose objects are homogeneous components and whose morphisms are admissible grade transitions; adjoint pairs encoding typed round trips; and information-geometric interpretations in terms of KL gain, mirror descent with Bregman divergences, and Fisher natural gradients. Methodologically, we specify a utility--aware routing mechanism and objective that remain fully end-to-end differentiable. Analytic case studies and lightweight sanity checks illustrate selective morphic activation on hybrid symbolic-linguistic tasks. The framework unifies symbolic computation, geometry, and self--supervised learning within the \emph{graded transformer} formalism \cite{sh-89,sh-95}, while subsuming prior external-tool paradigms (e.g., Toolformer \cite{toolformer2023}) as a special case via functorial internalization.