On LTL Model Checking for Low-Dimensional Discrete Linear Dynamical Systems

What’s decidable about linear loops?

“…Recall that we are interested in understanding Z(T ) for T contained inside such a subspace. To this end, we will combine Theorem 4.5 with the following result about three-dimensional dynamical systems from [Karimov et al 2020].…”

“…In recent years, motivated in part by verification problems for stochastic systems and linear loops, researchers have begun investigating more sophisticated specification formalisms than mere reachability: for example, the paper [Agrawal et al 2015] studies approximate LTL model checking of Markov chains (which themselves can be viewed as particular kinds of linear dynamical systems), whereas [Karimov et al 2020] focuses on LTL model checking of low-dimensional linear dynamical systems with semialgebraic predicates. In [Almagor et al 2021a], the authors investigate the model-checking problem for diagonalisable linear dynamical systems in arbitrary dimension against prefix-independent MSO properties; both are significant restrictionsÐin particular, reachability queries are not prefix-independent and therefore do not fall within the scope of the problems considered in [Almagor et al 2021a].…”

What's Decidable about Discrete Linear Dynamical Systems?

“…Recall that a semialgebraic target T ⊆ R d is called admissible if it is either contained in a three-dimensional subspace of R d , or has intrinsic dimension at most one. 5 The focus on target sets of this type has origins in the results of [18,32,6,9]. A common theme is that for admissible targets, the proofs that establish how to decide reachability also provide us with a means of representing, in a finite manner, all the time steps at which the orbit is in a particular target set T .…”

“…In recent years, motivated in part by verification problems for stochastic systems and linear loops, researchers have begun investigating more sophisticated specification formalisms than mere reachability: for example, the paper [1] studies approximate LTL model checking of Markov chains (which themselves can be viewed as particular kinds of linear dynamical systems), whereas [32] focuses on LTL model checking of low-dimensional linear dynamical systems with semialgebraic predicates. 3 In [4], the authors solve the semialgebraic modelchecking problem for diagonalisable linear dynamical systems in arbitrary dimension against prefix-independent MSO 4 properties, whereas [31] investigates semialgebraic MSO model checking of linear dynamical systems in which the dimensions of predicates are constrained.…”