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Encoding syntactic objects and Merge operations in function spaces

Matilde Marcolli, Robert C. Berwick

TL;DR

This work tackles the challenge of realizing the core syntactic operation Merge in a neurocomputationally plausible way by embedding lexical items, encoded as functions (e.g., wavelets), into a common function space and extending this embedding to cover arbitrary syntactic objects. The authors build a nonassociative commutative magma representation using a thermodynamic semiring based on the second Rényi entropy, ensuring that Merge’s algebraic structure is preserved in the target space and that the embedding is faithful (reconstructible from the image). They prove faithfulness under a finite leaf-size bound and use topological transversality to argue generic injectivity, while addressing repetitions via a Fulton–MacPherson-type configuration space. The paper then connects this algebraic framework to neural realizations via phase synchronization, showing how Merge can be implemented as entropy-optimized cross-frequency synchronization of sinusoidal waves, and discusses the relation to existing neurolinguistic models (Martin and Murphy). The results provide a constructive, theory-guided blueprint for neurocomputational realizations of the core syntax, with a clear path toward integrating algebraic structure, Hopf-algebra dynamics on workspaces, and synchronization-based neural substrates.

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.

Encoding syntactic objects and Merge operations in function spaces

TL;DR

This work tackles the challenge of realizing the core syntactic operation Merge in a neurocomputationally plausible way by embedding lexical items, encoded as functions (e.g., wavelets), into a common function space and extending this embedding to cover arbitrary syntactic objects. The authors build a nonassociative commutative magma representation using a thermodynamic semiring based on the second Rényi entropy, ensuring that Merge’s algebraic structure is preserved in the target space and that the embedding is faithful (reconstructible from the image). They prove faithfulness under a finite leaf-size bound and use topological transversality to argue generic injectivity, while addressing repetitions via a Fulton–MacPherson-type configuration space. The paper then connects this algebraic framework to neural realizations via phase synchronization, showing how Merge can be implemented as entropy-optimized cross-frequency synchronization of sinusoidal waves, and discusses the relation to existing neurolinguistic models (Martin and Murphy). The results provide a constructive, theory-guided blueprint for neurocomputational realizations of the core syntax, with a clear path toward integrating algebraic structure, Hopf-algebra dynamics on workspaces, and synchronization-based neural substrates.

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.

Paper Structure

This paper contains 33 sections, 18 theorems, 146 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Syntactic Objects and Weighted Combinations
  3. Summary of Merge and its mathematical model
  4. Binary trees in spaces of functions
  5. Polynomial functions from trees
  6. Reconstructing the tree
  7. Combination operations
  8. Thermodynamic Semirings and the Rényi Entropy
  9. Probabilities and entropy
  10. Thermodynamic semirings of functions
  11. The second Rényi entropy
  12. Embedding syntactic objects
  13. Uniform bound and high temperature regime
  14. Single wall-crossing
  15. Transversality and embeddings of syntactic objects
  16. ...and 18 more sections

Key Result

Proposition 2.3

Given an injective map $\varphi: {\mathcal{S}}{\mathcal{O}}_0 \to {\mathcal{F}}$ whose image consists of a collection of linearly independent vectors in ${\mathcal{F}}$, and a syntactic object $T\in {\mathcal{S}}{\mathcal{O}}={\mathfrak T}_{{\mathcal{S}}{\mathcal{O}}_0}$ that is in ${\rm Dom}(h)$, l where the $a^T_\ell(\Lambda)$ are polynomials in the variables $\lambda_v$. The polynomials $a^T_\e

Figures (4)

  • Figure 1: The behavior of the function $\lambda_{\min}(x,y)$, in the variable $u=\beta (y-x)/2$ in the range $|u|<1$.
  • Figure 2: The behavior of the successor function $\Upsilon(x,\beta)$ for $S={\rm Ry}_2$, shown in the case of $\beta=1$.
  • Figure 3: The successor function $\Upsilon(x,\beta)$ for $S={\rm Ry}_2$ and $\beta=1$, in the interval $|u|<1$ for $u=\beta x/2$.
  • Figure 4: Phase-amplitude coupling in sinusoidal waves.

Theorems & Definitions (36)

  • Remark 2.1
  • Proposition 2.3
  • Remark 2.4
  • Corollary 2.5
  • Proposition 2.6
  • Corollary 2.7
  • Lemma 2.8
  • Lemma 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 26 more