How (and when) can you fit examples to logic-based hypothesis classes over infinite structures?

2026-05-31

AI Summary

This study investigates the existence of models within specific logic-definable function classes that exactly or approximately fit given finite input-output samples over decidable infinite structures such as the ordered field of real numbers and Presburger arithmetic. By integrating tools from model theory, descriptive complexity, and computability theory, the work provides the first systematic characterization of the complexity boundaries of such fitting problems. It establishes a natural query-language-based framework for determining fitability and precisely classifies the computational complexity of fitting across several common logical structures. Furthermore, under certain conditions, the paper presents efficient methods—relying on finitely many queries—to decide whether a fitting model exists within the specified function class.

Abstract

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.

Problem

• fitting problems
• logic-based hypothesis classes
• infinite structures
• computational complexity
• decidable structures

Innovation

• fitting problems
• logic-based hypothesis classes
• computational complexity
• decidable structures
• query languages