类比比例
Analogical proportions
AI总结：
本文从第一性原理提出泛代数中的类比比例抽象框架，可统一跨域比较数学对象，并证明其可嵌入一阶逻辑且与结构保持映射兼容，为类比推理与学习理论奠定基础。
摘要：
类比制作处于人类和人工智能及创造力的核心，其应用涵盖证明数学定理和构建数学理论、常识推理、学习、语言习得以及故事讲述等多样任务。本文从第一性原理出发，在泛代数的一般设置中引入了一种形式为“`a`之于`b`犹如`c`之于`d`”的类比比例的抽象代数框架。这使我们能够以统一的方式比较可能跨不同领域的数学对象，这对于人工智能系统至关重要。事实证明，我们的类比比例概念具有吸引人的数学性质。由于我们仅使用泛代数的基本概念从第一性原理构建模型，并且由于我们的模型质疑了文献中预设的类比比例的一些基本性质，为了说服读者相信我们模型的合理性，我们展示了它可以自然地通过模型论类型嵌入一阶逻辑，并从该视角证明类比比例与结构保持映射兼容。这为其适用性提供了概念性证据。在更广泛的意义上，本文是迈向类比推理和学习系统理论的第一步，这些系统可能应用于常识推理、计算学习和创造力等基础人工智能问题。
英文摘要：
Analogy-making is at the core of human and artificial intelligence and creativity with applications to such diverse tasks as proving mathematical theorems and building mathematical theories, common sense reasoning, learning, language acquisition, and story telling. This paper introduces from first principles an abstract algebraic framework of analogical proportions of the form `$a$ is to $b$ what $c$ is to $d$' in the general setting of universal algebra. This enables us to compare mathematical objects possibly across different domains in a uniform way which is crucial for AI-systems. It turns out that our notion of analogical proportions has appealing mathematical properties. As we construct our model from first principles using only elementary concepts of universal algebra, and since our model questions some basic properties of analogical proportions presupposed in the literature, to convince the reader of the plausibility of our model we show that it can be naturally embedded into first-order logic via model-theoretic types and prove from that perspective that analogical proportions are compatible with structure-preserving mappings. This provides conceptual evidence for its applicability. In a broader sense, this paper is a first step towards a theory of analogical reasoning and learning systems with potential applications to fundamental AI-problems like common sense reasoning and computational learning and creativity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
