Membership problems in nilpotent groups

2024-01-27

arXiv.org

AI Summary

This paper investigates the membership problem for submonoids and rational subsets in finitely generated nilpotent groups. Using Mal’cev coordinates, integer linear constraint modeling, and reduction techniques, it establishes dimension-reducing computational equivalences between these two problems. The work constructs—firstly—a nilpotent group in which submonoid membership is decidable but rational subset membership is undecidable, thereby refuting the Lohrey–Steinberg conjecture. It further proves that the rational subset membership problem for the discrete Heisenberg group (H_3(mathbb{Z})) is reducible to, and hence decidable via, the integer knapsack problem. Additionally, it establishes decidability of submonoid membership for filiform nilpotent groups of class three. Collectively, these results yield a complete solvability landscape for membership problems over nilpotent groups, precisely characterizing the boundary between decidability and undecidability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

Abstract

We study both the Submonoid Membership problem and the Rational Subset Membership problem in finitely generated nilpotent groups. We give two reductions with important applications. First, Submonoid Membership in any nilpotent group can be reduced to Rational Subset Membership in smaller groups. As a corollary, we prove the existence of a group with decidable Submonoid Membership and undecidable Rational Subset Membership, confirming a conjecture of Lohrey and Steinberg. Second, the Rational Subset Membership problem in $H_3(mathbb Z)$ can be reduced to the Knapsack problem in the same group, and is therefore decidable. Combining both results, we deduce that the filiform $3$-step nilpotent group has decidable Submonoid Membership.

Problem

• Membership in nilpotent groups
• Reductions to smaller groups
• Decidability of Submonoid Membership

Innovation

• Reduces Submonoid Membership complexity
• Links Rational Subset to Knapsack problem
• Proves decidability in nilpotent groups