用函数空间建模语法结构，实现合并操作的神经计算可能
Encoding syntactic objects and Merge operations in function spaces
- 将词汇表示为函数，通过熵构造代数结构实现语法对象编码
- 合并操作由拓扑电路与霍普夫代数马尔可夫链实现，保持语义一致性
- 揭示了语法合并与算术后继函数的深层关联，适合形式语言研究者
我们提供了一个数学论证：若将词项表示为函数空间中的函数（如小波），则可在同一函数空间中构建任意语法对象的忠实表示。该空间可赋予基于二次瑞尼熵的交换但非结合半环结构，所得表示与叠合代数结构兼容。这些函数构成一个操集代数，其运算模拟将输入波形转化为编码语法结构的电路。合并操作在工作空间上的作用被忠实地实现为对这些电路的作用，通过余积与霍普夫代数马尔可夫链完成。结果表明，语法核心计算结构具有神经计算实现的理论可能性。我们还提出一个具体实例：通过正弦波跨频相位同步实现此类合并的实现。这亦表明合并可表达为半环的后继函数，澄清了其与算术后继函数相似性的长期观察。
原文摘要 · Abstract (English)
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.
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