Formal semantics clarify coordinated transformations in modern AI systems
From the Transformer to a Formal Semantics of Coordinated Transformations
Artificial IntelligenceComputation and LanguageLogic in Computer Science
Summary
Transformers are a key AI system used for processing information, but how their parts work together at a deep mathematical level is not fully clear. The authors develop a formal way to describe how different parts of these systems transform data both numerically and conceptually. They explain that these transformations coordinate so properties are preserved or combined in meaningful ways. Their approach helps understand not only Transformers but also more complex or different designs built on the same principles.
What this means in practice
- •For machine learning engineers: Design and analyze attention mechanisms and residual connections based on a unified semantic framework that applies beyond standard Transformers.
- •For ai system architects: Decompose complex AI models into semantically coordinated modules to build varied architectures sharing a common formal foundation.
A theory result. No direct application yet.
Authors
Marco Giunti, Fabrizia Giulia Garavaglia
Abstract
We develop a formal semantics for a general class of coordinated systems of parameterized transformations, taking the Transformer as a paradigmatic starting point but abstracting from its particular mathematical instantiation. Representational vectors receive both extensional reference and ordered intensional interpretations, while parameter structures acquire relative, second-level interpretations that determine how argument properties are preserved, replaced, or combined. This yields the notion of a transforming concept: a parameterized transformation whose numerical action is coordinated with a corresponding property-level map. We then distinguish the common algebraic-semantic core of these systems from their architecture-specific functional organization, allowing different mathematical instantiations and mechanical realizations of the same abstract system type. The framework is applied to attention heads and residual connections, providing semantic explanations that depend only on the shared algebraic-semantic core and their local functional organization and therefore generalize beyond the standard Transformer. These analyses also reveal the modularity of the framework: complex architectures can be decomposed into simpler semantically coordinated subsystems whose organization may recur across different abstract system types.