We give a single compositional setting in which gradient-based learning and Hamiltonian-style mechanics appear as functorial semantics. The syntax is an operad Arr whose objects are input-output interfaces (pairs of manifolds) and whose morphisms are *smooth adaptive arrangements*, which consist of a responsive parameter space, a lens given by smooth output and input maps, and a real-valued potential. The main technical result of the paper is what we call *lens internalization*, a lax symmetric monoidal functor Lens(C) $\to$ C associated to any symmetric monoidal closed category C. Using it, we provide two functors $Φ_\text{phase}$, $Φ_\text{conf}$: Arr $\to$ PC into the 2-category of polynomial coalgebras -- input-output discrete dynamical systems -- which we take as the semantics category. $Φ_\text{phase}$ stores both position and momentum, whereas $Φ_\text{conf}$ stores only position. When applied to a parameterized function, $Φ_\text{conf}$ recovers the gradient descent training algorithm, with backpropagation as the lens' backward pass. When applied to harmonic particles wired together -- in series, or according to any finite directed graph -- one diagram yields two different regimes, both of which are governed by the graph Laplacian: $Φ_\text{phase}$ gives the discrete wave equation, which is conservative and second-order, and $Φ_\text{conf}$ gives the discrete heat equation, which is dissipative and first-order. They are two semantics of one adaptive arrangement, e.g. with the same potential in each case. And because Arr is an operad, such diagrams nest -- larger systems wired from smaller ones -- and each semantics assembles a system's dynamics functorially from its parts. These dynamics are moreover executable: a parameterized neural network and a graph of particles both compile, by the same construction, to explicit state machines one can run.
Assurance arguments provide a clear and structured way to explain why stakeholders should trust that a system satisfies certain properties, yet widely used notations, e.g.Goal Structuring Notation (GSN), typically lack an operational semantics for deriving assurance confidence. Existing approaches address structure and soundness but largely reason over truth values, not over confidence in the justification of claims. Subjective Logic (SL) offers a calculus of belief, disbelief, and uncertainty with operators for combining opinions, enabling confidence propagation under incomplete, conflicting, or subjective evidence. However, existing SL-based approaches do not provide a uniform, compositional semantics that covers all argument elements and relations to enable overall confidence assessment. We propose a confidence semantics that represents argument elements as SL opinions and maps relations between elements to SL operators modelling how confidence flows, effectively turning the argument into an analyzable confidence network. The approach provides explicit warrants, principled handling of context, preserved provenance, and compatibility with GSN, along with practical guidance using an exemplary assurance confidence assessment.
Transformer based pre-trained large language models have become ubiquitous. There is increasing evidence to suggest that even with large scale pre-training, these models do not capture complete compositional context and certainly not, the full human analogous context. Besides, by the very nature of the architecture, these models hallucinate, are difficult to maintain, are not easily interpretable and require enormous compute resources for training and inference. Here, we describe Gyan, an explainable language model based on a novel non-transformer architecture, without any of these limitations. Gyan achieves SOTA performance on 3 widely cited data sets and superior performance on two proprietary data sets. The novel architecture decouples the language model from knowledge acquisition and representation. The model draws on rhetorical structure theory, semantic role theory and knowledge-based computational linguistics. Gyan's meaning representation structure captures the complete compositional context and attempts to mimic humans by expanding the context to a 'world model'. AI model adoption critically depends on trust and transparency especially in mission critical use cases. Collectively, our results demonstrate that it is possible to create models which are trustable and reliable for mission critical tasks. We believe our work has tremendous potential for guiding the development of transparent and trusted architectures for language models.
Distributional and neural approaches to natural language semantics have been built almost exclusively on conventional linear algebra: vectors, matrices, tensors, and the operations that accompany them. These methods have achieved remarkable empirical success, yet they face persistent structural limitations in compositional semantics, type sensitivity, and interpretability. I argue in this paper that geometric algebra (GA) -- specifically, Clifford algebras -- provides a mathematically superior foundation for semantic representation, and that a Functional Geometric Algebra (FGA) framework extends GA toward a typed, compositional semantics capable of supporting inference, transformation, and interpretability while retaining full compatibility with distributional learning and modern neural architectures. I develop the formal foundations, identify three core capabilities that GA provides and linear algebra does not, present a detailed worked example illustrating operator-level semantic contrasts, and show how GA-based operations already implicit in current transformer architectures can be made explicit and extended. The central claim is not merely increased dimensionality but increased structural organization: GA expands an $n$-dimensional embedding space into a $2^n$ multivector algebra where base semantic concepts and their higher-order interactions are represented within a single, principled algebraic framework.