On the Subspace Orbit Problem and the Simultaneous Skolem Problem

2026-01-26

AI Summary

This study addresses the decidability of whether the orbit of a point under a given rational matrix can reach a specified target subspace, with particular focus on cases where the target subspace has high dimension. By integrating tools from algebraic number theory, linear recurrence sequence analysis, and computational complexity theory, the authors establish that when the dimension of the target subspace is logarithmic in the orbit dimension, the problem is decidable over the rationals and lies in the complexity class NP^RP. In contrast, when the target subspace dimension is linear in the orbit dimension, the problem becomes equivalent to the long-standing Skolem problem, thereby inheriting its computational hardness. This work reveals a deep connection between subspace orbit problems and the Skolem problem, and precisely delineates decidability boundaries and complexity characterizations across different dimensional regimes.

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Decentralized Web and Fediverse systems

Abstract

The Orbit Problem asks whether the orbit of a point under a matrix reaches a given target set. When the target is a single point, the problem was shown to be decidable in polynomial time by Kannan and Lipton. This decidability result was later extended by Chonev et al. to targets of dimension 3 (in arbitrary ambient dimension), but decidability remains open for subspaces of dimension 4. At the other extreme, the special case of the Orbit Problem in which the target set is a hyperplane of codimension 1 is equivalent to the Skolem Problem for linear recurrence sequences, whose decidability has been open for many decades. In this paper, we show that the Orbit Problem is decidable if the target subspace has dimension logarithmic in the dimension of the orbit. Over rationals, we moreover obtain a complexity bound NP^RP in this case. On the other hand, we show that the version of the Orbit Problem where the dimension of the target subspace is linear in the dimension of the orbit is as hard as the Skolem Problem.

Problem

• Orbit Problem
• Subspace
• Decidability
• Skolem Problem
• Linear Recurrence Sequences

Innovation

• Orbit Problem
• Skolem Problem
• decidability
• subspace dimension
• computational complexity