Eric Goubault

A scalable algebraic method to infer quadratic invariants of switched systems

We present a new numerical abstract domain based on ellipsoids designed for the formal verification of switched linear systems. Unlike the existing approaches, this domain does not rely on a user-given template. We overcome the difficulty that ellipsoids do not have a lattice structure by exhibiting a canonical operator over-approximating the union. This operator is the only one which permits to perform analyses that are invariant with respect to a linear transformation of state variables. Moreover, we show that this operator can be computed efficiently using basic algebraic operations on positive semidefinite matrices. We finally develop a fast non-linear power-type algorithm, which allows one to determine sound quadratic invariants on switched systems in a tractable way, by solving fixed point problems over the space of ellipsoids. We test our approach on several benchmarks, and compare it with the standard techniques based on linear matrix inequalities, showing an important speedup on typical instances.

Natural Homology

We propose a notion of homology for directed algebraic topology, based on so-called natural systems of abelian groups, and which we call natural homology. As we show, natural homology has many desirable properties: it is invariant under isomorphisms of directed spaces, it is invariant under refinement subdivision, and it is computable on cubical complexes.

Iterated Chromatic Subdivisions are Collapsible

The standard chromatic subdivision of the standard simplex is a combinatorial algebraic construction, which was introduced in theoretical distributed computing, motivated by the study of the view complex of layered immediate snapshot protocols. A most important property of this construction is the fact that the iterated subdivision of the standard simplex is contractible, implying impossibility results in fault-tolerant distributed computing. Here, we prove this result in a purely combinatorial way, by showing that it is collapsible, studying along the way fundamental combinatorial structures present in the category of colored simplicial complexes.

A Scalable Algebraic Method to Infer Quadratic Invariants of Switched Systems

We present a new numerical abstract domain based on ellipsoids designed for the formal verification of switched linear systems. Unlike the existing approaches, this domain does not rely on a user-given template. We overcome the difficulty that ellipsoids do not have a lattice structure by exhibiting a canonical operator over-approximating the union. This operator is the only one which permits to perform analyses that are invariant with respect to a linear transformation of state variables. Moreover, we show that this operator can be computed efficiently using basic algebraic operations on positive semidefinite matrices. We finally develop a fast non-linear power-type algorithm, which allows one to determine sound quadratic invariants on switched systems in a tractable way, by solving fixed point problems over the space of ellipsoids. We test our approach on several benchmarks, and compare it with the standard techniques based on linear matrix inequalities, showing an important speedup on typical instances.

A Topological Method for Finding Invariant Sets of Continuous Systems

A usual way to find positive invariant sets of ordinary differential equations is to restrict the search to predefined finitely generated shapes, such as linear templates, or ellipsoids as in classical quadratic Lyapunov function based approaches. One then looks for generators or parameters for which the corresponding shape has the property that the flow of the ODE goes inwards on its border. But for non-linear systems, where the structure of invariant sets may be very complicated, such simple predefined shapes are generally not well suited. The present work proposes a more general approach based on a topological property, namely Wazewski’s property. Even for complicated non-linear dynamics, it is possible to successfully restrict the search for isolating blocks of simple shapes, that are bound to contain non-empty invariant sets. This approach generalizes the Lyapunov-like approaches, by allowing for inwards and outwards flow on the boundary of these shapes, with extra topological conditions. We developed and implemented an algorithm based on Wazewski’s property, SOS optimization and some extra combinatorial and algebraic properties, that shows very nice results on a number of classical polynomial dynamical systems.

A Topological Method for Finding Invariant Sets of Switched Systems

We revisit the problem of finding controlled invariants sets (viability), for a class of differential inclusions, using topological methods based on Wazewski property. In many ways, this generalizes the Viability Theorem approach, which is itself a generalization of the Lyapunov function approach for systems described by ordinary differential equations. We give a computable criterion based on SoS methods for a class of differential inclusions to have a non-empty viability kernel within some given region. We use this method to prove the existence of (controlled) invariant sets of switched systems inside a region described by a polynomial template, both with time-dependent switching and with state-based switching through a finite set of hypersurfaces. A Matlab implementation allows us to demonstrate its use.

From Geometric Semantics to Asynchronous Computability

We show that the protocol complex formalization of fault-tolerant protocols can be directly derived from a suitable semantics of the underlying synchronization and communication primitives, based on a geometrization of the state space. By constructing a one-to-one relationship between simplices of the protocol complex and dihomotopy classes of dipaths in the latter semantics, we describe a connection between these two geometric approaches to distributed computing: protocol complexes and directed algebraic topology. This is exemplified on atomic snapshot, iterated snapshot and layered immediate snapshot protocols, where a well-known combinatorial structure, interval orders, plays a key role. We believe that this correspondence between models will extend to proving impossibility results for much more intricate fault-tolerant distributed architectures.

Inner and Outer Approximating Flowpipes for Delay Differential Equations.

Delay differential equations are fundamental for modeling networked control systems where the underlying network induces delay for retrieving values from sensors or delivering orders to actuators. They are notoriously difficult to integrate as these are actually functional equations, the initial state being a function. We propose a scheme to compute inner and outer-approximating flowpipes for such equations with uncertain initial states and parameters. Inner-approximating flowpipes are guaranteed to contain only reachable states, while outer-approximating flowpipes enclose all reachable states. We also introduce a notion of robust inner-approximation, which we believe opens promising perspectives for verification, beyond property falsification. The efficiency of our approach relies on the combination of Taylor models in time, with an abstraction or parameterization in space based on affine forms, or zonotopes. It also relies on an extension of the mean-value theorem, which allows us to deduce inner-approximating flowpipes, from flowpipes outer-approximating the solution of the DDE and its Jacobian with respect to constant but uncertain parameters and initial conditions. We present some experimental results obtained with our C++ implementation.

A Fast Method to Compute Disjunctive Quadratic Invariants of Numerical Programs

We introduce a new method to compute non-convex invariants of numerical programs, which includes the class of switched affine systems with affine guards. We obtain disjunctive and non-convex invariants by associating different partial execution traces with different ellipsoids. A key ingredient is the solution of non-monotone fixed points problems over the space of ellipsoids with a reduction to small size linear matrix inequalities. This allows us to analyze instances that are inaccessible in terms of expressivity or scale by earlier methods based on semi-definite programming.

Directed Topological Models of Concurrency

We study topological models for concurrent programs with the aim of importing tools and techniques coming from algebraic topology to ease verification of concurrent programs. In those models, the state space of a program is described as a topological space, and an execution corresponds naturally to a path in this space. To rensure that models reflect order properties, we are led to enrich the concept of a topological space so that it takes causality into account. We shall focus our attention on directed paths, i.e., the ones respecting causality.