What’s Decidable About Discrete Linear Dynamical Systems?

Abstract

We survey the state of the art on the algorithmic analysis of discrete linear dynamical systems, focussing in particular on reachability, model-checking, and invariant-generation questions, both unconditionally as well as relative to oracles for the Skolem Problem.

J. Ouaknine—Also affiliated with Keble College, Oxford as emmy.network Fellow, and supported by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).

A Prefix-Independent Model Checking for LDS

The goal of this appendix is to exhibit boundaries on the extent to which Theorem 4 can be improved. More precisely, we show that the ability to solve the model-checking problem for arbitrary LDS against prefix-independent MSO specifications making use of semialgebraic predicates in ambient space $\mathbb {R}^4$ would necessarily entail major breakthroughs in Diophantine approximation.

We build upon the framework developed in [47, Sec. 5]. To this end, consider the class of order-6 rational LRS of the form

$$ u_n = -n + \frac{1}{2} (n - ri)\lambda ^n + \frac{1}{2}(n + ri)\overline{\lambda }^n = r {\text {Im}}(\lambda ^n) - n (1 - {\text {Re}}(\lambda ^n)) \, , $$

where $\lambda \in \mathbb {Q}(i)$ and $|\lambda | = 1$, and $r \in \mathbb {Q}$. Let us write $\mathcal {L}$ to denote this class of LRS.

It is shown in [47] that solving the Ultimate Positivity Problem for LRS in $\mathcal {L}$, i.e., providing an algorithm which, given an LRS $\langle u_n \rangle _{n=0}^\infty \in \mathcal {L}$, determines whether there exists some integer N such that, for all $n \ge N$, $u_n \ge 0$, would necessarily entail major breakthroughs in the field of Diophantine approximation. The purpose of the present section is to reduce the Ultimate Positivity Problem for LRS in $\mathcal {L}$ to the prefix-independent semialgebraic MSO model-checking problem for 4-dimensional LDS.

Given $\lambda$ and r as above, let

$$ M = \begin{bmatrix} {\text {Re}}(\lambda ) &{} -{\text {Im}}(\lambda ) &{} 1 &{} 0 \\ {\text {Im}}(\lambda ) &{} {\text {Re}}(\lambda ) &{} 0 &{} 1 \\ 0 &{} 0 &{} {\text {Re}}(\lambda ) &{} -{\text {Im}}(\lambda ) \\ 0 &{} 0 &{} {\text {Im}}(\lambda ) &{} {\text {Re}}(\lambda ) \\ \end{bmatrix} \text { and } x = \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix}. $$

Observe that M has rational entries. We have that

$$M^n x = \begin{bmatrix} {\text {Re}}(\lambda ^n) - {\text {Im}}(\lambda ^n) + n{\text {Re}}(\lambda ^{n-1}) - n{\text {Im}}(\lambda ^{n-1})\\ {\text {Im}}(\lambda ^n) + {\text {Re}}(\lambda ^n) + n{\text {Im}}(\lambda ^{n-1}) + n{\text {Re}}(\lambda ^{n-1}) \\ {\text {Re}}(\lambda ^n) - {\text {Im}}(\lambda ^n) \\ {\text {Im}}(\lambda ^n) + {\text {Re}}(\lambda ^n) \end{bmatrix}.$$

As a semialgebraic target consider the set $S = \{x: p(x) > 0\}$, where

$$ p(x_1, x_2, x_3, x_4) = \frac{r}{2}(x_4 - x_3) - \frac{x_1-x_3}{{\text {Re}}(\lambda ^{-1})x_3 - {\text {Im}}(\lambda ^{-1})x_4}\left( 1 - \frac{x_3 + x_4}{2}\right) . $$

We now have that $p(M^n x) = r {\text {Im}}(\lambda ^n) - n (1 - {\text {Re}}(\lambda ^n))$, and that $\langle u_n \rangle _{n=0}^\infty$ is ultimately positive if and only if the orbit of x under M eventually gets trapped in S. This can be expressed by the prefix-independent LTL formula $\varphi = \textbf{F} \, \textbf{G} \, S$.