Encoding syntactic objects and Merge operations in function spaces

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.

💡 Research Summary

The paper presents a constructive mathematical proof that, given an embedding of lexical items as functions (for example, wavelets) in a suitable function space, one can extend this embedding to all syntactic objects while preserving the algebraic structure that characterizes the core operation of Minimalist syntax, namely free symmetric Merge. The authors first assume that each lexical item is represented by a localized waveform—such as a wavelet or a sinusoid—so that the set of lexical items becomes a subset of a Hilbert (or more generally, Banach) space of functions.

To capture the non‑associative, commutative magma structure of syntactic objects, they introduce a non‑standard addition operation on this function space. This operation is defined via the second Rényi entropy, which serves as a diversity measure on the distribution of the underlying waveforms. Concretely, for two functions f and g the “sum” f ⊕ g is the function h that minimizes the Rényi‑2 entropy of the probability distribution induced by the normalized energy of the linear combination αf + βg. Because the minimization depends only on the unordered pair {f,g}, the operation is commutative; however, the minimization does not satisfy associativity, yielding a genuine magma. The resulting algebraic structure is a non‑associative semiring (called a thermodynamic semiring) where the usual multiplication is the ordinary linear combination (associative) and the addition is the entropy‑based ⊕.

With this semiring in place, the authors model Merge as the application of a binary gate that takes two input waveforms, computes the ⊕‑sum, and outputs a new waveform. The gate is described as a composition of two elementary operations: (i) a minimization step that implements the entropy‑based addition, and (ii) an entropy computation that can be realized by a small neural circuit estimating Rényi entropy from spike‑timing statistics. The overall system is endowed with a coproduct Δ: V → V ⊗ V, making the space of waveforms into a coalgebra, and the dynamics of repeated Merge on a workspace are captured by a Hopf‑algebraic Markov chain. This formalism guarantees that the action of Merge on syntactic workspaces is faithfully mirrored by the action on the corresponding circuits.

A concrete instantiation is given using cross‑frequency phase synchronization of sinusoidal waves. Two sinusoids with frequencies ω₁ and ω₂ in a rational ratio p:q are phase‑aligned; their interaction produces a third sinusoid whose frequency is the sum ω₁ + ω₂ and whose phase encodes the “successor” operation. This demonstrates that Merge can be expressed in terms of a successor function of the underlying semiring, echoing Chomsky’s observation of a similarity between Merge and arithmetic succession.

The paper also discusses neuro‑computational plausibility. The minimization gate can be implemented by inhibitory‑excitatory neural microcircuits that converge to an energy minimum, while Rényi‑2 entropy estimation can be performed by populations that compute pairwise spike‑time coincidences. The authors relate their construction to existing models such as the Compositional Neural Architecture (CNA) and the ROSE model, emphasizing that both rely on cross‑frequency coupling and phase‑phase synchronization, which fit naturally into the presented formalism.

In summary, the authors achieve four main results: (1) they formalize an embedding of lexical items as waveforms; (2) they define a non‑associative, commutative addition on the function space via second Rényi entropy, yielding a thermodynamic semiring; (3) they show that free symmetric Merge can be realized as a binary gate operating within this semiring, with its dynamics captured by a Hopf‑algebraic Markov chain; and (4) they provide a concrete sinusoidal phase‑synchronization example that links Merge to a successor function. The work bridges formal linguistics, algebraic topology, and computational neuroscience, offering a theoretically sound pathway for future empirical investigations into how the brain might encode hierarchical syntactic structure using oscillatory dynamics.