What's Decidable about Discrete Linear Dynamical Systems?

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

The Pseudo-Skolem Problem is Decidable

We study fundamental decision problems on linear dynamical systems in discrete time. We focus on pseudo-orbits, the collection of trajectories of the dynamical system for which there is an arbitrarily small perturbation at each step. Pseudo-orbits are generalizations of orbits in the topological theory of dynamical systems. We study the pseudo-orbit problem, whether a state belongs to the pseudo-orbit of another state, and the pseudo-Skolem problem, whether a hyperplane is reachable by an ?-pseudo-orbit for every ?. These problems are analogous to the well-studied orbit problem and Skolem problem on unperturbed dynamical systems. Our main results show that the pseudo-orbit problem is decidable in polynomial time and the Skolem problem on pseudo-orbits is decidable. The former extends the seminal result of Kannan and Lipton from orbits to pseudo-orbits. The latter is in contrast to the Skolem problem for linear dynamical systems, which remains open for proper orbits

Reachability in Dynamical Systems with Rounding

We consider reachability in dynamical systems with discrete linear updates, but with fixed digital precision, i.e., such that values of the system are rounded at each step. Given a matrix $M \in \mathbb{Q}^{d \times d}$, an initial vector $x\in\mathbb{Q}^{d}$, a granularity $g\in \mathbb{Q}_+$ and a rounding operation $[\cdot]$ projecting a vector of $\mathbb{Q}^{d}$ onto another vector whose every entry is a multiple of $g$, we are interested in the behaviour of the orbit $\mathcal{O}={}$, i.e., the trajectory of a linear dynamical system in which the state is rounded after each step. For arbitrary rounding functions with bounded effect, we show that the complexity of deciding point-to-point reachability---whether a given target $y \in\mathbb{Q}^{d}$ belongs to $\mathcal{O}$---is PSPACE-complete for hyperbolic systems (when no eigenvalue of $M$ has modulus one). We also establish decidability without any restrictions on eigenvalues for several natural classes of rounding functions.Comment: To appear at FSTTCS'2

The Pseudo-Reachability Problem for Diagonalisable Linear Dynamical Systems

We study fundamental reachability problems on pseudo-orbits of linear dynamical systems. Pseudo-orbits can be viewed as a model of computation with limited precision and pseudo-reachability can be thought of as a robust version of classical reachability. Using an approach based on o-minimality of ?_exp we prove decidability of the discrete-time pseudo-reachability problem with arbitrary semialgebraic targets for diagonalisable linear dynamical systems. We also show that our method can be used to reduce the continuous-time pseudo-reachability problem to the (classical) time-bounded reachability problem, which is known to be conditionally decidable

The Power of Positivity

The Positivity Problem for linear recurrence sequences over a ring R of real algebraic numbers is to determine, given an LRS (un)n∈N over R, whether u n ≥ 0 for all n. It is known to be Turing-equivalent to the following reachability problem: given a linear dynamical system (M, s) R d×d ×R d and a halfspace H ⊆ ℝ d , determine whether the orbit (Mns)n∈N ever enters H. The more general model-checking problem for LDS is to determine, given (M, s) and an ω-regular property φ over semialgebraic predicates T 1 ,…, T ℓ ⊆ ℝ d , whether the orbit of (M, s) satisfies φ.In this paper, we establish the following1)The Positivity Problem for LRS over real algebraic numbers reduces to the Positivity Problem for LRS over the integers; and2)The model-checking problem for LDS with diagonalisable M is decidable subject to a Positivity oracle for simple LRS over the integers.In other words, the full semialgebraic model-checking problem for diagonalisable linear dynamical systems is no harder than the Positivity Problem for simple integer linear recurrence sequences. This is in sharp contrast with the situation for arbitrary (not necessarily diagonalisable) LDS and arbitrary (not necessarily simple) integer LRS, for which no such correspondence is expected to hold

What’s decidable about linear loops?

International audienceWe consider the MSO model-checking problem for simple linear loops, or equivalently discrete-time linear dynamical systems, with semialgebraic predicates (i.e., Boolean combinations of polynomial inequalities on the variables). We place no restrictions on the number of program variables, or equivalently the ambient dimension. We establish decidability of the model-checking problem provided that each semialgebraic predicate either has intrinsic dimension at most 1, or is contained within some three-dimensional subspace. We also note that lifting either of these restrictions and retaining decidability would necessarily require major breakthroughs in number theory

What’s decidable about linear loops?

We consider the MSO model-checking problem for simple linear loops, or equivalently discrete-time linear dynamical systems, with semialgebraic predicates (i.e., Boolean combinations of polynomial inequalities on the variables). We place no restrictions on the number of program variables, or equivalently the ambient dimension. We establish decidability of the model-checking problem provided that each semialgebraic predicate either has intrinsic dimension at most 1, or is contained within some three-dimensional subspace. We also note that lifting either of these restrictions and retaining decidability would necessarily require major breakthroughs in number theory. </jats:p

Parameter Synthesis for Parametric Probabilistic Dynamical Systems and Prefix-Independent Specifications

International audienceWe consider the model-checking problem for parametric probabilistic dynamical systems, formalised as Markov chains with parametric transition functions, analysed under the distribution-transformer semantics (in which a Markov chain induces a sequence of distributions over states). We examine the problem of synthesising the set of parameter valuations of a parametric Markov chain such that the orbits of induced state distributions satisfy a prefix-independent ω-regular property. Our main result establishes that in all non-degenerate instances, the feasible set of parameters is (up to a null set) semialgebraic, and can moreover be computed (in polynomial time assuming that the ambient dimension, corresponding to the number of states of the Markov chain, is fixed)