Learning Shuffle Ideals with Membership Queries and Contrastive Examples

2026-09-26

AI Summary

This study investigates the learnability of shuffle ideals under membership and counter queries. To address the theoretical bottleneck that certain simple classes cannot be efficiently learned, this work integrates formal language theory with computational learning theory to propose novel structural conditions based on universal words and short lexicographic partitions, alongside corresponding algorithms. The primary contributions lie in delineating the boundaries of unlearnability for specific classes of shuffle ideals while demonstrating that efficient learning is achievable when the proposed structural conditions are satisfied. These findings offer a new perspective on combinatorial problems and deepen the understanding of learning complexity within query-based models.

Abstract

This paper studies learning of shuffle ideals with membership queries as well as with contrastive queries---a form of membership query that reveals not only whether a selected word $w$ is in the target language or not, but also provides a most similar word $w'$ that belongs to the target language if %and only if $w$ does not. For both settings, it is shown that even some very simple classes of shuffle ideals cannot be learned efficiently. By contrast, we obtain positive learnability results for classes of shuffle ideals that meet certain structural conditions. In the case of membership queries, these structural conditions are related to the previously studied notion of universal words, and raise new questions in word combinatorics. In the case of contrastive queries, the structural conditions relate to partitioning sets of words that are listed in shortlex order.

Problem

• Shuffle Ideals
• Membership Queries
• Contrastive Queries
• Learnability
• Formal Languages

Innovation

• Shuffle Ideals
• Membership Queries
• Contrastive Queries
• Universal Words
• Shortlex Order