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.

翻译：我们提出一个数学论证，表明在给定词汇项作为函数（例如小波）在某个函数空间中的表示时，可以在同一函数空间中构造任意句法对象的忠实表示。该空间可通过使用二阶Rényi熵构建一个交换非结合半环结构。所得句法对象表示与岩浆结构兼容。由此产生的函数集是操作数上的代数，其中操作数中的运算模拟将输入波形转换为编码句法结构的组合输出的电路。合并操作在工作空间上的作用通过余积和Hopf代数马尔可夫链，忠实地实现为对这些电路的作用。本文获得的结果提供了一个建设性论证，表明句法核心计算结构的神经计算实现具有理论可能性。我们还展示了这一一般构造的特殊情形，其中此类合并操作的实现被实施为正弦波上的跨频率相位同步。这也表明合并操作可以用半环的后继函数来表达，从而澄清了其与算术后继函数相似性的著名观察。