The author has recently introduced abstract algebraic frameworks of
analogical proportions and similarity within the general setting of universal
algebra. The purpose of this paper is to build a bridge from similarity to
analogical proportions by formulating the latter in terms of the former. The
benefit of this similarity-based approach is that the connection between
proportions and similarity is built into the framework and therefore evident
which is appealing since proportions and similarity are both at the center of
analogy; moreover, future results on similarity can directly be applied to
analogical proportions.

该论文通过在普适代数的一般框架内引入抽象代数的类比比例和相似性，旨在建立从相似性到类比比例之间的桥梁，通过将后者用前者来表述。这种基于相似性的方法的好处在于比例和相似性之间的联系已经融入到框架中，因此显而易见，这很吸引人，因为比例和相似性都处于类比的中心；此外，未来关于相似性的结果可以直接应用到类比比例。

基于相似性的类比比例

Similarity-based analogical proportions

Theorems from universal algebra such as that of Murski\u{i} from the 1970s
have a striking similarity to universal approximation results for neural nets
along the lines of Cybenko's from the 1980s. We consider here a discrete
analogue of the classical notion of a neural net which places these results in
a unified setting. We introduce a learning algorithm based on polymorphisms of
relational structures and show how to use it for a classical learning task.

通过广义代数中的定理（如 1970 年代的 Murski 定理），与 1980 年代 Cybenko 等人提出的神经网络的普遍逼近结果之间存在着惊人的相似性，我们在这里考虑了经典神经网络概念的离散模拟，将这些结果统一起来，并介绍了一种基于关系结构的多态函数的学习算法，并展示了如何将其应用于经典的学习任务。

离散神经网络与多态学习

Discrete neural nets and polymorphic learning

The rise of Artificial Intelligence (AI) recently empowered researchers to
investigate hard mathematical problems which eluded traditional approaches for
decades. Yet, the use of AI in Universal Algebra (UA) -- one of the fields
laying the foundations of modern mathematics -- is still completely unexplored.
This work proposes the first use of AI to investigate UA's conjectures with an
equivalent equational and topological characterization. While topological
representations would enable the analysis of such properties using graph neural
networks, the limited transparency and brittle explainability of these models
hinder their straightforward use to empirically validate existing conjectures
or to formulate new ones. To bridge these gaps, we propose a general algorithm
generating AI-ready datasets based on UA's conjectures, and introduce a novel
neural layer to build fully interpretable graph networks. The results of our
experiments demonstrate that interpretable graph networks: (i) enhance
interpretability without sacrificing task accuracy, (ii) strongly generalize
when predicting universal algebra's properties, (iii) generate simple
explanations that empirically validate existing conjectures, and (iv) identify
subgraphs suggesting the formulation of novel conjectures.

人工智能应用于通用代数领域的推测性预测，通过等式表达和拓扑表示来增强透明度和可解释性，验证现有猜想并提出新的猜想。

可解释图网络构建通用代数猜想

Interpretable Graph Networks Formulate Universal Algebra Conjectures

In this paper we determine the complexity of a broad class of problems that
extends the temporal constraint satisfaction problems. To be more precise we
study the problems Poset-SAT($\Phi$), where $\Phi$ is a given set of
quantifier-free $\leq$-formulas. An instance of Poset-SAT($\Phi$) consists of
finitely many variables $x_1,\ldots,x_n$ and formulas
$\phi_i(x_{i_1},\ldots,x_{i_k})$ with $\phi_i \in \Phi$; the question is
whether this input is satisfied by any partial order on $x_1,\ldots,x_n$ or
not. We show that every such problem is NP-complete or can be solved in
polynomial time, depending on $\Phi$. All Poset-SAT problems can be formalized
as constraint satisfaction problems on reducts of the random partial order. We
use model-theoretic concepts and techniques from universal algebra to study
these reducts. In the course of this analysis we establish a dichotomy that we
believe is of independent interest in universal algebra and model theory.

本文研究了扩展时序约束问题的复杂性，并针对一组量词自由的 “小于等于（leq）” 公式所形成的 Poset-SAT 问题进行了讨论。其问题是对于 $x_1$，$x_2$，…，$x_n$ 成为偏序集时是否存在满足给定公式的赋值。最后，作者使用模型理论概念和通用代数技术研究了这些问题的约束满足问题，并提出了独立于通用代数和模型理论的分裂定理。