O-Minimal Invariants for Discrete-Time Dynamical Systems

Shaull Almagor,James Worrell,Dmitry Chistikov,Joël Ouaknine

doi:10.1145/3501299

Abstract

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.

Highlights

This article is concerned with the existence and algorithmic synthesis of suitable invariants for discrete-time linear dynamical systems (LDS)
To this day, automated invariant synthesis remains a topic of active research; see, e.g., [22], and Section 8 therein
Our focus here is on simple linear loops, of the following form: P : x ← s; while x F do x ← Ax, (1)

Summary

INTRODUCTION

This article is concerned with the existence and algorithmic synthesis of suitable invariants for discrete-time linear dynamical systems (LDS). Given s ∈ Qd , A ∈ Qd×d , and F ⊆ Rd , our main results are the following: if F is a semi-algebraic set, it is decidable whether there exists an o-minimal invariant I containing s and disjoint from F , and in positive instances such an invariant can be defined explicitly in R0. We heavily rely on Baker’s Theorem in order to obtain decidability results These tools are needed here, as opposed to [15, 16], since, intuitively, when F is not a singleton the general termination problem is not known to be decidable, and in particular, the “interaction” of the orbit with F is not limited to a single orbit point, but may require reasoning about the asymptotics of the orbit

PRELIMINARIES AND MAIN DEFINITIONS

FROM THE ORBIT TO TRAJECTORY CONES AND RAYS

CONSTRUCTING INVARIANTS FROM TRAJECTORY CONES

DECIDING THE EXISTENCE OF O-MINIMAL INVARIANTS

Unconditional Decidability for Semi-algebraic Halting Sets

A Note on Complexity