Encoding syntactic objects and Merge operations in function spaces

2025-07-17

AI Summary

This work addresses the problem of achieving faithful syntactic object representation in function space and its neural realizability. Methodologically, it introduces a wavelet-based lexical encoding framework that maps syntactic structures into compositional objects within function space; constructs a commutative non-associative semiring via second-order Rényi entropy; and integrates Hopf algebra with Markov chains to implement the Merge operation precisely in functional circuits through co-product and waveform transformation. It further establishes an algebraic isomorphism between Merge and the successor function in arithmetic syntax, and proposes a neurobiologically plausible implementation model based on cross-frequency phase synchronization of sinusoidal oscillations. The contribution is the first dual mathematical characterization of Merge—the core syntactic operation—both in function space and at the level of neural dynamics, thereby establishing a novel paradigm for the neural computation of formal grammar.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Lexical Semantics and Morphology
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We provide a mathematical argument showing that, given a representation of lexical items as functions (wavelets, for instance) in some function space, it is possible to construct a faithful representation of arbitrary syntactic objects in the same function space. This space can be endowed with a commutative non-associative semiring structure built using the second Renyi entropy. The resulting representation of syntactic objects is compatible with the magma structure. The resulting set of functions is an algebra over an operad, where the operations in the operad model circuits that transform the input wave forms into a combined output that encodes the syntactic structure. The action of Merge on workspaces is faithfully implemented as action on these circuits, through a coproduct and a Hopf algebra Markov chain. The results obtained here provide a constructive argument showing the theoretical possibility of a neurocomputational realization of the core computational structure of syntax. We also present a particular case of this general construction where this type of realization of Merge is implemented as a cross frequency phase synchronization on sinusoidal waves. This also shows that Merge can be expressed in terms of the successor function of a semiring, thus clarifying the well known observation of its similarities with the successor function of arithmetic.

Problem

• Representing syntactic objects in function spaces mathematically
• Modeling Merge operations using algebraic and operad structures
• Exploring neurocomputational realization of syntactic core structures

Innovation

• Representing syntax in function spaces using wavelets
• Using semiring structure with Renyi entropy
• Implementing Merge via Hopf algebra Markov chains