O-Minimal Invariants for Linear Loops

The termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Such deceptively simple questions also relate to a number of deep open problems, 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 paper, 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

Optimal Analysis of Discrete-time Affine Systems

Our very first concern is the resolution of the verification problem for the class of discrete-time affine dynamical systems. This verification problem is turned into an optimization problem where the constraint set is the reachable values set of the dynamical system. To solve this optimization problem, we truncate the infinite sequences belonging to the reachable values set at some step which is uniform with respect to the initial conditions. In theory, the best possible uniform step is the optimal solution of a non-convex semi-definite program. In practice, we propose a methodology to compute a uniform step that over-approximate the best solution.Comment: 16 page

On strongest algebraic program invariants

A polynomial program is one in which all assignments are given by polynomial expressions and in which all branching is nondeterministic (as opposed to conditional). Given such a program, an algebraic invariant is one that is defined by polynomial equations over the program variables at each program location. Müller-Olm and Seidl have posed the question of whether one can compute the strongest algebraic invariant of a given polynomial program. In this article, we show that, while strongest algebraic invariants are not computable in general, they can be computed in the special case of affine programs, that is, programs with exclusively linear assignments. For the latter result, 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