Khalil Ghorbal

National Institute for Research in Computer Science and Control | INRIA

This paper is concerned with a covering problem of smooth manifolds of dimension $n-1$ using a specific arrangement, called Q-arrangement, of $2^{n}$ $n$-simplices which could be degenerate and are allowed to overlap (preventing therefore the immediate use of simplicial homology). We leverage the underlying inductive nature of the problem to give a...

We present two characterizations of positive invariance of sets under the flow of systems of ordinary differential equations. The first characterization uses inward sets which intuitively collect those points from which the flow evolves within the set for a short period of time, whereas the second characterization uses the notion of exit sets, whic...

Set positive invariance is an important concept in the theory of dynamical systems and one which also has practical applications in areas of computer science, such as formal verification, as well as in control theory. Great progress has been made in understanding positively invariant sets in continuous dynamical systems and powerful computational t...

Several BDD variants were designed to exploit special features of Boolean functions to achieve better compression rates.Deciding a priori which variant to use is as hard as constructing the diagrams themselves and the conversion between variants comes in general with a prohibitive cost.This observation leads naturally to a growing interest into whe...

Zero-suppressed binary Decision Diagram (ZDD) is a notable alternative data structure of Reduced Ordered Binary Decision Diagram (ROBDD) that achieves a better size compression rate for Boolean functions that evaluate to zero almost everywhere. Deciding a priori which variant is more suitable to represent a given Boolean function is as hard as cons...

Vector Lyapunov functions are a multi-dimensional extension of the more familiar (scalar) Lyapunov functions, commonly used to prove stability properties in systems of non-linear ordinary differential equations (ODEs). This paper explores an analogous vector extension for so-called barrier certificates used in safety verification. As with vector Ly...

Our objective is to model and simulate Cyber-Physical Systems (CPS) such as robots, vehicles, and power plants. The structure of CPS models may change during simulation due to the desired operation, due to failure situations or due to changes in physical conditions. Corresponding models are called multi-mode. We are interested in multi-domain, comp...

The Next-Generation Airborne Collision Avoidance System (ACAS X) is intended to be installed on all large aircraft to give advice to pilots and prevent mid-air collisions with other aircraft. It is currently being developed by the Federal Aviation Administration (FAA). In this paper we determine the geometric configurations under which the advice g...

In this article we study ways of constructing meaningful operational models of piecewise-smooth systems (PWS). The systems we consider are described by polynomial vector fields defined on non-overlapping semi-algebraic sets, which form a partition of the state space. Our approach is to give meaning to motion in systems of this type by automatically...

We formally verify a hybrid control law designed to perform a station keeping maneuver for a planar vehicle. Such maneuver requires that the vehicle reaches a neighborhood of its station in finite time and remains in it while waiting for further instructions. We model the dynamics as well as the control law as a hybrid program and formally verify b...

Differential Algebraic Equation (DAE) systems constitute the mathematical model supporting physical modeling languages such as Modelica, VHDL-AMS, or Simscape. Unlike ODEs, they exhibit subtle issues because of their implicit latent equations and related differentiation index. Multi-mode DAE (mDAE) systems are much harder to deal with, not only bec...

We investigate decoupling abstractions, by which we seek to simulate (i.e. abstract) a given system of ordinary differential equations (ODEs) by another system that features completely independent (i.e. uncoupled) sub-systems, which can be considered as separate systems in their own right. Beyond a purely mathematical interest as a tool for the qua...

The safety of mobile robots in dynamic environments is predicated on making sure that they do not collide with obstacles. In support of such safety arguments, we analyze and formally verify a series of increasingly powerful safety properties of controllers for avoiding both stationary and moving obstacles: (i) static safety, which ensures that no c...

This paper presents a method for generating semi-algebraic invariants for systems governed by non-linear polynomial ordinary differential equations under semi-algebraic evolution constraints. Based on the notion of discrete abstraction, our method eliminates unsoundness and unnecessary coarseness found in existing approaches for computing abstracti...

This paper studies sound proof rules for checking positive invariance of algebraic and semi-algebraic sets, that is, sets satisfying polynomial equalities and those satisfying finite boolean combinations of polynomial equalities and inequalities, under the flow of polynomial ordinary differential equations. Problems of this nature arise in formal v...

The Next-Generation Airborne Collision Avoidance System (ACAS X) is intended to be installed on all large aircraft to give advice to pilots and prevent mid-air collisions with other aircraft. It is currently being developed by the Federal Aviation Administration (FAA). In this paper we determine the geometric configurations under which the advice g...

This paper presents a theoretical and experimental comparison of sound proof rules for proving invariance of algebraic sets, that is, sets satisfying polynomial equalities, under the flow of polynomial ordinary differential equations. Problems of this nature arise in formal verification of continuous and hybrid dynamical systems, where there is an...

Complex software systems are becoming increasingly prevalent in aerospace applications: in particular, to accomplish critical tasks. Ensuring the safety of these systems is crucial, as they can have subtly different behaviors under slight variations in operating conditions. This paper advocates the use of formal verification techniques and in parti...

In this paper we seek to provide greater automation for formal deductive verification tools working with continuous and hybrid dynamical systems. We present an efficient procedure to check invariance of conjunctions of polynomial equalities under the flow of polynomial ordinary differential equations. The procedure is based on a necessary and suffi...

Systems and methods for model checking of live systems are shown that include learning an interval discrete-time Markov chain (IDTMC) model of a deployed system from system logs; and checking the IDTMC model with a processor to determine a probability of violating one or more probabilistic safety properties. Checking the IDTMC model includes calcul...

A computer implemented program analysis method employing a set of new abstract domains applicable to non-convex invarients. The method analyzes programs statically using abstract interpretation while advantageously considering non-convex structures and in particular those situations in which an internal region of an unreachable state exists within...

We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This so-called differential radical characterization relies on a sound abstraction of the reachable set of solutions by the smallest variety that contains it. The charact...

Nowadays, robots interact more frequently with a dynamic environment outside limited manufacturing sites and in close proximity with humans. Thus, safety of motion and obstacle avoidance are vital safety features of such robots. We formally study two safety properties of avoiding both stationary and moving obstacles: (i) passive safety, which ensur...

We introduce a new technique to model check reachability properties on interval discrete-time Markov chains (IDTMC). We compute a sound over-approximation of the probabilities of satisfying a given property where the accuracy is characterized in terms of error bounds. We leverage affine arithmetic to propagate the first-order error terms. Higher-or...

Program analysis using abstract interpretation has been successfully applied in practice to find runtime bugs or prove software correct. Most abstract domains that are used widely rely on convexity for their scalability. However, the ability to express non-convex properties is sometimes required in order to achieve a precise analysis of some numeri...

We aim at proving automatically the correctness of numerical behavior of a program by inferring invariants on numerical variables. More precisely, we over-approximate in a sound manner the set of reached values. We use Abstract Interpretation-based Static Analysis as a generic framework to de ne and ap- proximate the semantics of a program in a uni...

We define and study a new abstract domain which is a fine-grained combination of zonotopes with polyhedric domains such as the interval, octagon, linear templates or polyhedron domain. While abstract transfer functions are still rather inexpensive and accurate even for interpreting non-linear computations, we are able to also interpret tests (i.e....

Static analysis by abstract interpretation [1] aims at automatically inferring properties on the behaviour of programs. We focus here on a specific kind of numerical invariants: the set of values taken by numerical variables, with a real numbers semantics, at each control point of a program.

This paper reports the results of an ESA funded project on the use of abstract interpretation to validate critical real-time embedded space software. Abstract interpretation is industrially used since several years, especially for the validation of the Ariane 5 launcher. However, the limitations of the tools used so far prevented a wider deployment...