Internalizing Tools as Morphisms in Graded Transformers

2025-11-21

AI Summary

Transformer models struggle to internalize symbolic computation, limiting their ability to perform rigorous reasoning over structured symbolic domains. Method: We propose a hierarchical Transformer framework that models symbolic operations as categorical morphisms in latent space, selectively activated via differentiable routing; for the first time, we internalize external tool invocation as a functor within the Transformer’s algebraic structure, and introduce KL-gain-driven natural gradient optimization with type-aware categorical modeling. Our approach integrates hierarchical vector decomposition, Bregman divergence-based mirror descent, and morphism-block mapping to enable sparse, interpretable, self-supervised symbol-geometry joint learning. Contribution/Results: Theoretical analysis and lightweight experiments demonstrate that the framework unifies symbolic reasoning, geometric structure modeling, and self-supervised learning. It significantly improves selectivity and interpretability of morphism activation on hybrid symbol-language tasks, subsuming and extending prior work such as Toolformer.

Technology Category

• Computer Vision: Visual Reasoning & Symbolic Representations
• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We introduce a graded formulation of internal symbolic computation for transformers. The hidden space is endowed with a grading $V=igoplus_{gin G}V_g$, and symbolic operations are realized as typed block maps (morphisms) $φ_{hleftarrow g}:V_g o V_h$ 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.

Problem

• Developing graded transformers with internal symbolic computation using typed morphisms
• Creating self-supervised routing mechanisms for sparse interpretable activation
• Unifying symbolic computation and geometry within differentiable transformer frameworks

Innovation

• Graded hidden space with typed morphisms for symbolic operations
• Self-supervised utility functional governs sparse activation
• End-to-end differentiable routing mechanism for tool internalization