17 research outputs found
What's Decidable about Discrete Linear Dynamical Systems?
We survey the state of the art on the algorithmic analysis of discrete linear dynamical systems, and outline a number of research directions
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
Model Checking Linear Dynamical Systems under Floating-point Rounding
We consider linear dynamical systems under floating-point rounding. In these
systems, a matrix is repeatedly applied to a vector, but the numbers are
rounded into floating-point representation after each step (i.e., stored as a
fixed-precision mantissa and an exponent). The approach more faithfully models
realistic implementations of linear loops, compared to the exact
arbitrary-precision setting often employed in the study of linear dynamical
systems.
Our results are twofold: We show that for non-negative matrices there is a
special structure to the sequence of vectors generated by the system: the
mantissas are periodic and the exponents grow linearly. We leverage this to
show decidability of -regular temporal model checking against
semialgebraic predicates. This contrasts with the unrounded setting, where even
the non-negative case encompasses the long-standing open Skolem and positivity
problems.
On the other hand, when negative numbers are allowed in the matrix, we show
that the reachability problem is undecidable by encoding a two-counter machine.
Again, this is in contrast to the unrounded setting where point-to-point
reachability is known to be decidable in polynomial time
O-minimal invariants for discrete-time dynamical systems
Termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination relates to deep open problems in number theory, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this article, we introduce the class of o-minimal invariants, which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture is transcendental number theory
Model Checking Linear Dynamical Systems under Floating-point Rounding
We consider linear dynamical systems under floating-point rounding. In these systems, a matrix is repeatedly applied to a vector, but the numbers are rounded into floating-point representation after each step (i.e., stored as a fixed-precision mantissa and an exponent). The approach more faithfully models realistic implementations of linear loops, compared to the exact arbitrary-precision setting often employed in the study of linear dynamical systems
Polynomial Invariants for Affine Programs
We exhibit an algorithm to compute the strongest polynomial (or algebraic)
invariants that hold at each location of a given affine program (i.e., a
program having only non-deterministic (as opposed to conditional) branching and
all of whose assignments are given by affine expressions). Our main tool is an
algebraic result of independent interest: given a finite set of rational square
matrices of the same dimension, we show how to compute the Zariski closure of
the semigroup that they generate
Complete Semialgebraic Invariant Synthesis for the Kannan-Lipton Orbit Problem
International audienceThe Orbit Problem consists of determining, given a matrix A on Q d , together with vectors x and y, whether the orbit of x under repeated applications of A can ever reach y. This problem was famously shown to be decidable by Kannan and Lipton in the 1980s. In this paper, we are concerned with the problem of synthesising suitable invariants P ⊆ R d , i.e., sets that are stable under A and contain x but not y, thereby providing compact and versatile certificates of non-reachability. We show that whether a given instance of the Orbit Problem admits a semialge-braic invariant is decidable, and moreover in positive instances we provide an algorithm to synthesise suitable succinct invariants of polynomial size. Fijalkow et al Our results imply that the class of closed semialgebraic invariants is closure-complete: there exists a closed semialgebraic invariant if and only if y is not in the topological closure of the orbit of x under A