On the Subspace Orbit Problem and the Simultaneous Skolem Problem

TL;DR

This work studies the Subspace Orbit Problem (SOP) for linear dynamical systems and its connection to the Skolem problem via the Simultaneous Skolem Problem (SimSkolem). It introduces the Krylov subspace, stable and inherent dimensions, and shows SOP reduces to SimSkolem through duality with respect to the target subspace, enabling new decidability bounds. The authors prove SOP is decidable when the target dimension $t$ is at most $2\log_3 d$ (over $\overline{\mathbb{Q}}$) with an $\text{NP}^{\text{RP}}$ complexity bound, and they develop the MSTV-reducibility framework to handle many cases by reducing to non-degenerate LRS in the MSTV class. Conversely, they establish hardness results showing that SOP becomes as hard as the Skolem problem when $t$ grows linearly with $d$, and they provide an improved reduction from Skolem to SOP. Together, these results delineate the boundary between decidability and hardness for SOP and its Skolem-related variants, with implications for reachability in linear loops and related program-analysis problems.

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.

Figures (1)

• Figure 1: A table depicting the state of the art, and our contributions for $\textnormal{Orbit}_{{\overline{\mathbb{Q}}}}(d,t)$. Green: shown decidable over $\mathbb{Q}$ by chonev_complex2016, and extended in the present article to ${\overline{\mathbb{Q}}}$. Blue: instances newly shown decidable in the present paper over ${\overline{\mathbb{Q}}}$; we show $\textnormal{Orbit}_\mathbb{Q}(d,t)$ for them to be in $\NP^\RP$, and in $\coRP$ when $d$ is fixed. White: instances which are provably not MSTV-reducible (cf. \ref{['def:MSTV-reducible']}), as shown by \ref{['ex:notmstv']}. Red: instances which are $\textnormal{Skolem}_\mathbb{Q}(k)$-hard or $\textnormal{Skolem}(k)$-hard for some $k$ (see Appendix \ref{['sec:table_hard']}).

Theorems & Definitions (76)

• theorem 1: Skolem-Mahler-Lech
• theorem 2
• lemma 1
• proof
• Remark 3
• proposition 1
• definition 1
• proposition 2
• proposition 3
• proof