Deciding subspace reachability problems with application to Skolem's Problem

TL;DR

This paper offers a geometric approach to decide certain instances of the higher-dimensional Orbit Problem, which relates to Skolem's Problem, providing simpler decision procedures and alternative proofs.

Contribution

It introduces a geometric perspective to the Orbit Problem, enabling decision procedures for specific cases and offering new proofs for existing results.

Findings

Derived a simple decision procedure for a class of Orbit Problem instances

Provided alternative geometric proofs for known results

Connected the Orbit Problem to Skolem's Problem through geometric methods

Abstract

The higher-dimensional version of Kannan and Lipton's Orbit Problem asks whether it is decidable if a target subspace can be reached from a starting point under repeated application of a linear transformation. Similarly, the continuous analog of the Orbit Problem asks if a flow induced by a linear system of differential equations ever reaches some specified subspace. The decidability of both problems remains open, and in fact the problems generalize the discrete and continuous versions of Skolem's Problem. The object of this paper is to communicate a geometric perspective of the discrete and continuous Orbit Problems, alternate to the traditional and highly technical algebraic and number-theoretic approaches to the problem. We derive a simple decision procedure capable of deciding a certain class of instances of the Orbit Problem, and, as an application, we obtain alternate proofs to a number of results using elementary geometric arguments.