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arXiv 1003.4394cs.CLcs.LOmath.CT
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意义组成分布模型的数学基础

Mathematical Foundations for a Compositional Distributional Model of Meaning

Oxford University Computing Laboratory(牛津大学计算实验室);University of Cambridge Computer Laboratory(剑桥大学计算机实验室)

Bob Coecke, Mehrnoosh Sadrzadeh, Stephen Clark

摘要:

我们提出了一个数学框架,用于统一向量空间模型中的意义分布理论,以及语法类型的合成理论,其中我们依赖于Lambek引入的预群代数。这个数学框架使我们能够从句子成分的含义中计算出一个好类型句子的含义。具体地说,预群的类型约简被“提升”为范畴中的态射,这是一个将成分的意义转化为(良好类型的)整体的意义的过程。重要的是,整个句子的意义存在于一个空间中,独立于句子的语法结构。因此,内积可以用来比较任意句子的意义,就像它是用来比较分布模型中单词的意义一样。我们采用的数学结构允许一个纯粹的图解演算,它揭示了信息如何在句子中的单词之间流动,以构成整个句子的意义。我们的“分类模型”的一个变化,其中涉及约束标量的向量空间的布尔半环的结果在蒙塔格风格的布尔值语义。

英文摘要:

We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the meaning of a well-typed sentence from the meanings of its constituents. Concretely, the type reductions of Pregroups are `lifted' to morphisms in a category, a procedure that transforms meanings of constituents into a meaning of the (well-typed) whole. Importantly, meanings of whole sentences live in a single space, independent of the grammatical structure of the sentence. Hence the inner-product can be used to compare meanings of arbitrary sentences, as it is for comparing the meanings of words in the distributional model. The mathematical structure we employ admits a purely diagrammatic calculus which exposes how the information flows between the words in a sentence in order to make up the meaning of the whole sentence. A variation of our `categorical model' which involves constraining the scalars of the vector spaces to the semiring of Booleans results in a Montague-style Boolean-valued semantics.

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