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.
翻译:分析是人类和人工智慧和创造力的核心,其应用包括证明数学理论和建立数学理论、常识推理、学习、语言获取和故事叙事等多种任务。本文从头等原则中引入了“a美元”形式模拟比例的抽象代数框架,即“a美元”形式在通用代数总体设置中“a美元”形式模拟比例的抽象代数框架,即美元等于美元等于美元等于美元等于美元等于美元,这使我们能够以对AI系统至关重要的统一方式比较可能跨越不同领域的数学对象。它证明我们模拟比例的概念具有吸引人的数学特性。我们从最初的原则中构建模型时,仅使用通用代数的基本概念,并且由于我们从文献中假定的一些模拟比例的基本特性问题,我们向读者展示了我们模型的可观性。我们表明,它可以通过模型理论类型自然地嵌入一阶逻辑,并从这个角度证明,类比比例与结构-保留绘图相容。它提供了概念性证据,说明其适用性。从广义上看,本文件是朝着基本理论学理论学理论学和基础性理论学和理论推理学系的第一步。