We study fitting problems, sometimes called ``training problems'', where we have a finite sample consisting of inputs and outputs, and we want to know whether there is a function in a certain class that could produce these outputs, exactly or approximately, on the given inputs. We focus on the computational and descriptive complexity of fitting for logically-defined classes in common decidable structures, like the real ordered field and Presburger arithmetic, and also for broader classes defined via combinatorial or model-theoretic properties. We isolate the complexity of these fitting problems, with particular attention to cases where we can use queries in a natural query language over the sample to determine whether a sample is fittable.

翻译：我们研究拟合问题（有时称为“训练问题”），其中给定一个由输入和输出组成的有限样本，我们希望确定是否存在某个类中的函数能够在给定输入上精确或近似地产生这些输出。我们重点关注在常见可判定结构（如实闭域和普雷斯伯格算术）中，由逻辑定义的类的拟合问题的计算复杂性与描述复杂性，同时也关注通过组合性质或模型论性质定义的更广泛类。我们解析了这些拟合问题的复杂性，特别关注那些我们可以利用样本上的自然查询语言中的查询来确定样本是否可拟合的情况。