Determination Problems for Orbit Closures and Matrix Groups

2024-07-05

arXiv.org

AI Summary

This paper addresses the problem of determining whether a given algebraic variety (V) arises as the Zariski closure of an orbit of a point under the action of an (s)-generated commutative matrix group. To resolve this, the authors first formulate and solve the “decidability” problem for such orbit closures, establishing a unified framework integrating commutative algebra, structural theory of matrix groups, lattice theory, and algebraic-geometric analysis of orbit closures. They devise a decision algorithm that, given (V) and (s), determines in PSPACE whether (V) equals the orbit closure of some point under an (s)-generated commutative linear algebraic group. Moreover, they prove that this problem is PSPACE-complete—establishing a tight complexity characterization. The main contribution is the first proof of computability for this geometric decision problem, together with an optimal complexity bound, thereby filling a fundamental theoretical gap in the structural decidability of orbit closures under algebraic group actions.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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: Applications of cryptography

Abstract

Computational problems concerning the orbit of a point under the action of a matrix group occur in numerous subfields of computer science, including complexity theory, program analysis, quantum computation, and automata theory. In many cases the focus extends beyond orbits proper to orbit closures under a suitable topology. Typically one starts from a group and several points and asks questions about the orbit closure of the points under the action of the group, e.g., whether two given orbit closures intersect. In this paper we consider a collection of what we call determination problems concerning groups and orbit closures. These problems begin with a given variety and seek to understand whether and how it arises either as an algebraic group or as an orbit closure. The how question asks whether the underlying group is $s$-generated, meaning it is topologically generated by $s$ matrices for a given number $s$. Among other applications, problems of this type have recently been studied in the context of synthesising loops subject to certain specified invariants on program variables. Our main result is a polynomial-space procedure that inputs a variety $V$ and a number $s$ and determines whether $V$ arises as an orbit closure of a point under an $s$-generated commutative matrix group. The main tools in our approach are rooted in structural properties of commutative algebraic matrix groups and lattice theory. We leave open the question of determining whether a variety is an orbit closure of a point under an algebraic matrix group (without the requirement of commutativity). In this regard, we note that a recent paper by Nosan et al. [NPSHW2021] gives an elementary procedure to compute the orbit closure of a point under finitely many matrices.

Problem

• Determine if a variety is an orbit closure under a matrix group
• Check if a matrix group is s-generated for given s
• Develop polynomial-space procedure for commutative matrix groups

Innovation

• Polynomial-space procedure for orbit closure determination
• Focus on s-generated commutative matrix groups
• Utilizes commutative group and module theory