Internalizing Tools as Morphisms in Graded Transformers
Tony Shaska
TL;DR
The paper addresses the challenge of integrating symbolic computation into neural transformers by internalizing tools as typed morphisms on a graded representation space $V=\bigoplus_{g\in G}V_g$. It introduces a graded transformer framework with a differentiable routing policy that selects morphic activations $\phi_{h\leftarrow g}:V_g\to V_h$ to reduce next-token loss, yielding sparse, interpretable computations and unifying symbolic reasoning with self-supervised learning. Theoretical foundations are developed in category-theoretic and information-geometric terms, including an internal model category $\mathcal{M}$, functorial internalization of external tools, adjunctions for typed round-trips, and a graded utility that connects to KL gain and Fisher geometry. Analytic constructions, minimal toy examples, and a concrete PyTorch-like implementation sketch illustrate the approach, quantify complexity reductions from sparsity, and establish conditions for identifiability and stable, monotone loss下降. Overall, the graded Toolformer framework provides a principled, end-to-end differentiable path to modular AI, where symbolic operations are embedded as internal morphisms within the model’s geometry, enabling scalable, verifiable, and composable integration of external-like tooling.
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) $φ_{h\leftarrow g}:V_g\to V_h$ 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.