In the logical framework introduced by Grohe and Tur\'an (TOCS 2004) for Boolean classification problems, the instances to classify are tuples from a logical structure, and Boolean classifiers are described by parametric models based on logical formulas. This is a specific scenario for supervised passive learning, where classifiers should be learned based on labelled examples. Existing results in this scenario focus on Boolean classification. This paper presents learnability results beyond Boolean classification. We focus on multiclass classification problems where the task is to assign input tuples to arbitrary integers. To represent such integer-valued classifiers, we use aggregate queries specified by an extension of first-order logic with counting terms called FOC1. Our main result shows the following: given a database of polylogarithmic degree, within quasi-linear time, we can build an index structure that makes it possible to learn FOC1-definable integer-valued classifiers in time polylogarithmic in the size of the database and polynomial in the number of training examples.

翻译：在Grohe和Turán（TOCS 2004）为布尔分类问题引入的逻辑框架中，待分类的实例来自逻辑结构中的元组，布尔分类器由基于逻辑公式的参数化模型描述。这是监督式被动学习的一个特定场景，分类器需要根据标注样本进行学习。该场景下的现有研究成果主要集中于布尔分类。本文提出了超越布尔分类的可学习性结果。我们聚焦于多类别分类问题，其任务是将输入元组映射到任意整数值。为表示此类整数值分类器，我们采用通过一阶计数逻辑扩展（称为FOC1）定义的聚合查询。我们的主要结果表明：给定一个多对数级度的数据库，在拟线性时间内可以构建索引结构，使得学习FOC1可定义的整数值分类器的时间复杂度达到数据库规模的多对数级和训练样本数量的多项式级。