The Inclusion Depth of Pattern Languages: An Open Problem in Algorithmic Learning Theory

2026-05-28

AI Summary

This study investigates the inclusion depth of pattern languages—the length of the longest strict inclusion chain from the universal pattern language to a given pattern-generated language—a measure that captures the cognitive complexity, in terms of mind changes, required to identify a pattern from positive examples. The authors propose the conjectural formula ID_Σ(p) = 2|p| − #var(p) − 1 and develop a linear-time algorithm based on this hypothesis, thereby establishing a profound connection between formal language theory and learning complexity. By integrating combinatorics, algorithmic learning theory, and mind-change analysis, this work not only highlights the fundamental open nature of the inclusion depth problem but also lays a new theoretical foundation for efficient pattern recognition: if the conjecture holds, inclusion depth becomes computable in linear time.

Abstract

Pattern languages are a classical model in formal language theory and algorithmic learning theory. This note formulates the problem of computing the inclusion depth of a pattern language: the length of the longest strict inclusion chain from the universal pattern language to the language generated by a given pattern. Inclusion depth captures the mind-change complexity of pattern identification from positive data. The central open question is whether the inclusion depth ID_Sigma(p) is computable for every pattern p over every finite alphabet Sigma with at least two symbols, and whether it is computable in polynomial time. A simple conjectured formula, ID_Sigma(p) = 2|p| - #var(p) - 1, would imply a linear-time algorithm. The problem connects pattern language inclusion, combinatorics on words, language identification in the limit, and mind-change-bounded learning.

Problem

• pattern languages
• inclusion depth
• algorithmic learning theory
• mind-change complexity
• computability

Innovation

• inclusion depth
• pattern languages
• mind-change complexity
• algorithmic learning theory
• combinatorics on words