With the increasing ubiquity of safety-critical autonomous systems operating in uncertain environments, there is a need for mathematical methods for formal verification of stochastic models. Towards formally verifying properties of stochastic systems, methods based on discrete, finite Markov approximations - abstractions - thereof have surged in recent years. These are found in contexts where: either a) one only has partial, discrete observations of the underlying continuous stochastic process, or b) the original system is too complex to analyze, so one partitions the continuous state-space of the original system to construct a handleable, finite-state model thereof. In both cases, the abstraction is an approximation of the discrete stochastic process that arises precisely from the discretization of the underlying continuous process. The fact that the abstraction is Markov and the discrete process is not (even though the original one is) leads to approximation errors. Towards accounting for non-Markovianity, we introduce memory-dependent abstractions for stochastic systems, capturing dynamics with memory effects. Our contribution is twofold. First, we provide a formalism for memory-dependent abstractions based on transfer operators. Second, we quantify the approximation error by upper bounding the total variation distance between the true continuous state distribution and its discrete approximation.

Neural-based, data-driven analysis and control of dynamical systems have been recently investigated and have shown great promise, e.g. for safety verification or stability analysis. Indeed, not only do neural networks allow for an entirely model-free, data-driven approach, but also for handling arbitrary complex functions via their power of representation (as opposed to, e.g. algebraic optimization techniques that are restricted to polynomial functions). Whilst classical Lyapunov techniques allow to provide a formal and robust guarantee of stability of a switched dynamical system, very little is yet known about correctness guarantees for Neural Lyapunov functions, nor about their performance (amount of data needed for a certain accuracy). We formally introduce Neural Lyapunov functions for the stability analysis of switched linear systems: we benchmark them on this paradigmatic problem, which is notoriously difficult (and in general Turing-undecidable), but which admits existing recently-developed technologies and theoretical results. Inspired by switched systems theory, we provide theoretical guarantees on the representative power of neural networks, leveraging recent results from the ML community. We additionally experimentally display how Neural Lyapunov functions compete with state-of-the-art results and techniques, while admitting a wide range of improvement, both in theory and in practice. This study intends to improve our understanding of the opportunities and current limitations of neural-based data-driven analysis and control of complex dynamical systems.

We introduce the concept of memoryless concretization relation ( <?TeX $\operatorname{MCR}$?> Math 1 ) to describe abstraction within the context of controller synthesis. This relation is a specific instance of alternating simulation relation ( <?TeX $\operatorname{ASR}$?> Math 2 ), where it is possible to simplify the controller architecture. In the case of <?TeX $\operatorname{ASR}$?> Math 3 , the concretized controller needs to simulate the concurrent evolution of two systems, the original and abstract systems, while for <?TeX $\operatorname{MCR}$?> Math 4 , the designed controllers only need knowledge of the current concrete state. We demonstrate that the distinction between <?TeX $\operatorname{ASR}$?> Math 5 and <?TeX $\operatorname{MCR}$?> Math 6 becomes significant only when a non-deterministic quantizer is involved, such as in cases where the state space discretization consists of overlapping cells. We also show that any abstraction of a system that alternatingly simulates a system can be completed to satisfy <?TeX $\operatorname{MCR}$?> Math 7 at the expense of increasing the non-determinism in the abstraction. We clarify the difference between the <?TeX $\operatorname{MCR}$?> Math 8 and the feedback refinement relation ( <?TeX $\operatorname{FRR}$?> Math 9 ), showing in particular that the former allows for non-constant controllers within cells. This provides greater flexibility in constructing a practical abstraction, for instance, by reducing non-determinism in the abstraction. Finally, we prove that this relation is not only sufficient, but also necessary, for ensuring the above properties.

Meta-learning is a line of research that develops the ability to leverage past experiences to efficiently solve new learning problems.In the context of Reinforcement Learning (RL), meta-RL methods demonstrate a capability to learn behaviors that efficiently acquire and exploit information on a set of related tasks.The Alchemy benchmark has been proposed in [1] to test such methods.Alchemy features a rich structured latent space that is challenging for state-of-the-art model-free RL methods.These methods fail to learn to properly explore then exploit.We develop a model-based algorithm.We train a model whose principal block is a Transformer Decoder to fit the symbolic Alchemy environment dynamics.Then we define an online planner with the learned model using a tree search method.This algorithm significantly outperforms previously applied methods on the symbolic Alchemy problem.Our results reveal the relevance of model-based approaches with online planning to perform exploration and exploitation successfully in meta-RL.

As part of the development of Lyapunov techniques for cyber-physical systems, we study and compare graph-based stability certificates with respect to their conservatism. Previous work have highlighted the dependence of this ordering with respect to the properties of the chosen template of candidate Lyapunov functions. We extend here previous results from the literature to the case of templates closed under addition, as for instance the set of quadratic functions. In this context, we provide a characterization of the ordering, using an approach based on abstract operations on graphs, called lifts, which encode in a combinatorial way the algebraic properties of the chosen template. We finally provide a numerical method to algorithmically check the ordering relation.

This poster presents a new technique to synthesize a reactive hybrid controller which actuates a non-linear control system in response to external logical inputs to fulfill an omega-regular specification over a finite set of logical input and observation predicates.

In the context of discrete-time switched systems, we study the comparison of stability certificates based on path-complete Lyapunov methods. A characterization of this general ordering has been provided recently, but we show here that this characterization is too strong when a particular template is considered, as it is the case in practice. In the present work we provide a characterization for templates that are closed under pointwise minimum/maximum, which covers several templates that are often used in practice. We use an approach based on abstract operations on graphs, called lifts, to highlight the dependence of the ordering with respect to the analytical properties of the template. We finally provide more preliminary results on another family of templates: those that are closed under addition, as for instance the set of quadratic functions.

A set of nonnegative matrices is called primitive if there exists a product of these matrices that is entrywise positive. Motivated by recent results relating synchronizing automata and primitive sets, we study the length of the shortest product of a primitive set having a column or a row with k positive entries, called its k-rendezvous time (k-RT), in the case of sets of matrices having no zero rows and no zero columns. We prove that the k-RT is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We provide two upper bounds on the k-RT: the second is an improvement of the first one, although the latter can be written in closed form. We then report numerical results comparing our upper bounds on the k-RT with heuristic approximation methods.

We introduce and study the concept of worst-case topological entropy of switched linear systems under arbitrary switching. It is shown that this quantity is equal to the minimal data rate (number of bits per second) required for the state observation of the switched linear system with any switching signal. A computable closed-form expression is presented for the worst-case topological entropy of switched linear systems. Finally, a practical coder-decoder, operating at a data rate arbitrarily close to the worst-case topological entropy, is described.

Particle filtering is a powerful tool for target tracking. When the budget for observations is restricted, it is necessary to reduce the measurements to a limited amount of samples carefully selected. A discrete stochastic nonlinear dynamical system is studied over a finite time horizon. The problem of selecting the optimal measurement times for particle filtering is formalized as a combinatorial optimization problem. We propose an approximated solution based on the nesting of a genetic algorithm, a Monte Carlo algorithm and a particle filter. Firstly, an example demonstrates that the genetic algorithm outperforms a random trial optimization. Then, the interest of non-regular measurements versus measurements performed at regular time intervals is illustrated and the efficiency of our proposed solution is quantified: better filtering performances are obtained in 87.5% of the cases and on average, the relative improvement is 27.7%.

We study criteria allowing to compare the conservativeness of stability certificates for switching systems. The stability certificates under consideration are Path-Complete Lyapunov functions (PCLFs), which are multiple Lyapunov functions with an underlying combinatorial structure.

Motivated by the randomized generation of slowly synchronizing automata, we study automata made of permutation letters and a merging letter of rank n-1 . We present a constructive randomized procedure to generate synchronizing automata of that kind with (potentially) large alphabet size based on recent results on primitive sets of matrices. We report numerical results showing that our algorithm finds automata with much larger reset threshold than a mere uniform random generation and we present new families of automata with reset threshold of Omega(n^2/4) . We finally report theoretical results on randomized generation of primitive sets of matrices: a set of permutation matrices with a 0 entry changed into a 1 is primitive and has exponent of O(n log n) with high probability in case of uniform random distribution and the same holds for a random set of binary matrices where each entry is set, independently, equal to 1 with probability p and equal to 0 with probability 1-p , when np-log n - > infty as n - > infty .

A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate for discrete-time arbitrary switching systems. In this paper, we prove that the satisfiability of such a criterion implies the existence of a Common Lyapunov Function, expressed as the composition of minima and maxima of the pieces of the Path-Complete Lyapunov function. the converse however is not true even for discrete-time linear systems: we present such a system where a max-of-2 quadratics Lyapunov function exists while no corresponding Path-Complete Lyapunov function with 2 quadratic pieces exists. In light of this, we investigate when it is possible to decide if a Path- Complete Lyapunov function is less conservative than another. By analyzing the combinatorial and algebraic structure of the graph and the pieces respectively, we provide simple tools to decide when the existence of such a Lyapunov function implies that of another.

We characterize and compute the maximal admissible positively invariant set for asymptotically stable constrained switching linear systems. Motivated by practical problems found, e.g., in obstacle avoidance, power electronics and nonlinear switching systems, in our setting the constraint set is formed by a finite number of polynomial inequalities. First, we observe that the so-called Veronese lifting allows to represent the constraint set as a polyhedral set. Next, by exploiting the fact that the lifted system dynamics remains linear, we establish a method based on reachability computations to characterize and compute the maximal admissible invariant set, which coincides with the domain of attraction when the system is asymptotically stable. After developing the necessary theoretical background, we propose algorithmic procedures for its exact computation, based on linear or semidefinite programs. The approach is illustrated in several numerical examples.

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. Many algorithms exist for estimating the JSR but not much is known about how to generate an infinite sequence of matrices with an optimal asymptotic growth rate. To the best of our knowledge, the currently known algorithms select a small sequence with large spectral radius using brute force (or branch-and-bound variants) and repeats this sequence infinitely.

The Černý conjecture is a longstanding open problem in automata theory. We study two different concepts, which allow us to approach it from a new angle. The first one is the triple rendezvous time, i.e., the length of the shortest word mapping three states onto a single one. The second one is the synchronizing probability function of an automaton, a recently introduced tool which reinterprets the synchronizing phenomenon as a two-player game and allows us to obtain optimal strategies through a linear program. Our contribution is twofold. First, by coupling two different novel approaches based on the synchronizing probability function and properties of linear programming, we obtain a new upper bound on the triple rendezvous time. Second, by exhibiting a family of counterexamples, we disprove a conjecture on the growth of the synchronizing probability function. We then suggest natural follow-ups toward the Černý conjecture.

We study the boundedness of products of matrices associated with words in a regular language. This question naturally arises in the stability analysis of switching systems with constrained switching sequences.

We present a toolbox for computing the Joint Spectral Radius of a set of matrices, i.e., the maximal asymptotic growth rate of products of matrices taken in that set. The Joint Spectral Radius has a wide range of applications, including switched and hybrid systems, combinatorial words theory, or the study of wavelets. However, it is notoriously difficult to compute or approximate; it is actually uncomputable, and its approximation is NP-hard. The toolbox compiles several recent computation and approximation methods, and also contains an automatic blackbox method for inexperienced users, selecting the most appropriate methods based on an automatic study of the matrix set provided. The tool is implemented in Matlab and is freely downloadable (with documentation and demos) from Matlab Central1.

We study the synchronization phenomenon for deterministic finite state automata and the related longstanding Černý conjecture. We formulate this conjecture in the setting of a two-player probabilistic game. Our goal is twofold. On the one hand, the probabilistic interpretation is of interest in its own right and can be applied to real-world situations. On the other hand, our formulation makes use of standard convex optimization techniques, which appear powerful to shed light on Černý's conjecture. We analyze the synchronization phenomenon through this particular point of view. Among other properties, we prove that the synchronization process cannot stagnate too long in a certain sense. We propose a new conjecture and demonstrate that its validity would imply Černý's conjecture. We show numerical evidence for the pertinence of the approach.

We study the problem of approximating the joint spectral radius (JSR) of a finite set of matrices. Our approach is based on the analysis of the underlying switched linear system via inequalities imposed between multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, maximum/minimum-of-quadratics Lyapunov functions. We characterize all path-complete graphs consisting of two nodes on an alphabet of two matrices and compare their performance. For the general case of any set of n x n matrices we propose semidefinite programs of modest size that approximate the JSR within a multiplicative factor of 1/4√n of the true value. We establish a notion of duality among path-complete graphs and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.

We revisit the well-known problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set $P$, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of $P$. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (STOC 1992) showed that there exists a polynomial-time algorithm for the problem achieving this bound up to a constant factor. Their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it impractical.

We consider the problem of partial order production: arrange the elements of an unknown totally ordered set T into a target partially ordered set S by comparing a minimum number of pairs in T. Special cases include sorting by comparisons, selection, multiple selection, and heap construction. We give an algorithm performing $ITLB+o(ITLB)+O(n)$ comparisons in the worst case. Here, n denotes the size of the ground sets, and $ITLB$ denotes a natural information-theoretic lower bound on the number of comparisons needed to produce the target partial order. Our approach is to replace the target partial order by a weak order (that is, a partial order with a layered structure) extending it, without increasing the information-theoretic lower bound too much. We then solve the problem by applying an efficient multiple selection algorithm. The overall complexity of our algorithm is polynomial. This answers a question of Yao [SIAM J. Comput., 18 (1989), pp. 679–689]. We base our analysis on the entropy of the target partial order, a quantity that can be efficiently computed and provides a good estimate of the information-theoretic lower bound.

Proceedings of the 41st annual ACM Symposium on Theory of Computing STOC 2009, Bethesda, Maryland, 31 mai–2 juin 2009

Some questions related to the computation of the capacity of codes that avoid forbidden difference patterns are analysed. The maximal number of n-bit sequences whose pairwise differences do not contain some given forbidden difference patterns is known to increase exponentially with n; the coefficient of the exponent is the capacity of the forbidden patterns. In this paper, new inequalities for the capacity are given that allow for the approximation of the capacity with arbitrary high accuracy. The computational cost of the algorithm derived from these inequalities is fixed once the desired accuracy is given. Subsequently, a polynomial time algorithm is given for determining if the capacity of a set is positive while the same problem is shown to be NP-hard when the sets of forbidden patterns are defined over an extended set of symbols. Finally, the existence of extremal norms is proved for any set of matrices arising in the capacity computation. Based on this result, a second capacity approximating algorithm is proposed. The usefulness of this algorithm is illustrated by computing exactly the capacity of particular codes that were only known approximately