How (and when) can you fit examples to logic-based hypothesis classes over infinite structures?
It provides a theoretical complexity analysis of training problems for logically-defined hypothesis classes, which is foundational for understanding the limits of learning in such settings.
The paper studies the computational and descriptive complexity of fitting finite samples to logic-based hypothesis classes over infinite structures like the real ordered field and Presburger arithmetic, isolating the complexity of these problems and identifying cases where queries in a natural language can determine fittability.
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.