We propose a many-sorted modal logic for reasoning about knowledge in multi-agent systems. Our logic introduces a clear distinction between participating agents and the environment. This allows to express local properties of agents and global properties of worlds in a uniform way, as well as to talk about the presence or absence of agents in a world. The logic subsumes the standard epistemic logic and is a conservative extension of it. The semantics is given in chromatic hypergraphs, a generalization of chromatic simplicial complexes, which were recently used to model knowledge in distributed systems. We show that the logic is sound and complete with respect to the intended semantics. We also show a further connection of chromatic hypergraphs with neighborhood frames.

Abstract The reliability and usefulness of verification depend on the ability to represent appropriately the uncertainty. Most existing work on neural network verification relies on the hypothesis of either set-based or probabilistic information on the inputs. In this work, we rely on the framework of imprecise probabilities, specifically p-boxes, to propose a quantitative verification of ReLU neural networks, which can account for both probabilistic information and epistemic uncertainty on inputs. On classical benchmarks, including the ACAS Xu examples, we demonstrate that our approach improves the tradeoff between tightness and efficiency compared to related work on probabilistic network verification, while handling much more general classes of uncertainties on the inputs and providing fully guaranteed results.

We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such as robustness, motion planning or controllers comparison. We propose an interval-based method which allows for tractable but tight approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation.

In recent years, a new class of models for multi-agent epistemic logic has emerged, based on simplicial complexes. Since then, many variants of these simplicial models have been investigated, giving rise to different logics and axiomatizations. In this paper, we present a further generalization, which encompasses all previously studied variants of simplicial models. Geometrically, this is achieved by generalizing beyond simplicial complexes, and considering instead semi-simplicial sets. By doing so, we define a new semantics for epistemic logic with distributed knowledge, where a group of agents may distinguish two worlds, even though each individual agent in the group is unable to distinguish them. As it turns out, these models are the geometric counterpart of a generalization of Kripke models, called "pseudo-models". We show how to recover the previously defined variants of simplicial models as sub-classes of our models; and give a sound and complete axiomatization for each of them.

Abstract We present a unified approach, implemented in the RINO tool, for the computation of inner and outer-approximations of reachable sets of discrete-time and continuous-time dynamical systems, possibly controlled by neural networks with differentiable activation functions. RINO combines a zonotopic set representation with generalized mean-value AE extensions to compute under and over-approximations of the robust range of differentiable functions, and applies these techniques to the particular case of learning-enabled dynamical systems. The AE extensions require an efficient and accurate evaluation of the function and its Jacobian with respect to the inputs and initial conditions. For continuous-time systems, possibly controlled by neural networks, the function to evaluate is the solution of the dynamical system. It is over-approximated in RINO using Taylor methods in time coupled with a set-based evaluation with zonotopes. We demonstrate the good performances of RINO compared to state-of-the art tools Verisig 2.0 and ReachNN* on a set of classical benchmark examples of neural network controlled closed loop systems. For generally comparable precision to Verisig 2.0 and higher precision than ReachNN*, RINO is always at least one order of magnitude faster, while also computing the more involved inner-approximations that the other tools do not compute.

Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every forward evaluation of a NODE requires numerical integration of the neural network used to capture the system dynamics, making their training prohibitively expensive. Existing works rely on off-the-shelf adaptive step-size numerical integration schemes, which often require an excessive number of evaluations of the underlying dynamics network to obtain sufficient accuracy for training. By contrast, we accelerate the evaluation and the training of NODEs by proposing a data-driven approach to their numerical integration. The proposed Taylor-Lagrange NODEs (TL-NODEs) use a fixed-order Taylor expansion for numerical integration, while also learning to estimate the expansion's approximation error. As a result, the proposed approach achieves the same accuracy as adaptive step-size schemes while employing only low-order Taylor expansions, thus greatly reducing the computational cost necessary to integrate the NODE. A suite of numerical experiments, including modeling dynamical systems, image classification, and density estimation, demonstrate that TL-NODEs can be trained more than an order of magnitude faster than state-of-the-art approaches, without any loss in performance.

The standard semantics of multi-agent epistemic logic S5_n is based on Kripke models whose accessibility relations are reflexive, symmetric and transitive. This one dimensional structure contains implicit higher-dimensional information beyond pairwise interactions, that we formalized as pure simplicial models in a previous work in Information and Computation 2021 [Éric Goubault et al., 2021]. Here we extend the theory to encompass simplicial models that are not necessarily pure. The corresponding class of Kripke models are those where the accessibility relation is symmetric and transitive, but might not be reflexive. Such models correspond to the epistemic logic KB4_n. Impure simplicial models arise in situations where two possible worlds may not have the same set of agents. We illustrate it with distributed computing examples of synchronous systems where processes may crash.

We extend the formalisation of confluence results in Kleene algebras to a formalisation of coherent confluence proofs. For this, we introduce the structure of higher globular Kleene algebra, a higher-dimensional generalisation of modal and concurrent Kleene algebra. We calculate a coherent Church-Rosser theorem and a coherent Newman's lemma in higher Kleene algebras by equational reasoning. We instantiate these results in the context of higher rewriting systems modelled by polygraphs.

In the context of developing safe air transportation, our work is focused on understanding how Reinforcement Learning methods can improve the state of the art in traditional control, in nominal as well as non-nominal cases. The end goal is to train provably safe controllers, by improving both training and verification methods. In this paper, we explore this path for controlling the attitude of a quadcopter: we discuss theoretical as well as practical aspects of training neural nets for controlling a crazyflie 2.0 drone. In particular we describe thoroughly the choices in training algorithms, neural net architecture, hyperparameters, observation space etc. We also discuss the robustness of the obtained controllers, both to partial loss of power for one rotor and to wind gusts. Finally, we measure the performance of the approach by using a robust form of a signal temporal logic to quantitatively evaluate the vehicle's behavior.

In this paper, we propose polynomial forms to represent distributions of state variables over time for discrete-time stochastic dynamical systems. This problem arises in a variety of applications in areas ranging from biology to robotics. Our approach allows us to rigorously represent the probability distribution of state variables over time, and provide guaranteed bounds on the expectations, moments and probabilities of tail events involving the state variables. First, we recall ideas from interval arithmetic, and use them to rigorously represent the state variables at time t as a function of the initial state variables and noise symbols that model the random exogenous inputs encountered before time t. Next, we show how concentration of measure inequalities can be employed to prove rigorous bounds on the tail probabilities of these state variables. We demonstrate interesting applications that demonstrate how our approach can be useful in some situations to establish mathematically guaranteed bounds that are of a different nature from those obtained through simulations with pseudo-random numbers.

We investigate the information and guarantees provided by different inner and outer approximated reachability analyses, for proving properties of dynamical systems. We explore the connection of these approximated sets with the maximal and minimal reachable sets of Mitchell [31], with an additional notion of robustness to disturbance. We demonstrate the practical use of a specific computation of these approximated reachable sets. We revisit in particular the reachavoid properties.

We introduce a family of tasks for n processes, as a generalization of the two process equality negation task of Lo and Hadzilacos (SICOMP 2000). Each process starts the computation with a private input value taken from a finite set of possible inputs. After communicating with the other processes using immediate snapshots, the process must decide on a binary output value, 0 or 1. The specification of the task is the following: in an execution, if the set of input values is large enough, the processes should agree on the same output; if the set of inputs is small enough, the processes should disagree; and in-between these two cases, any output is allowed. Formally, this specification depends on two threshold parameters k and l, with k<l, indicating when the cardinality of the set of inputs becomes "small" or "large", respectively. We study the solvability of this task depending on those two parameters. First, we show that the task is solvable whenever k+2 <= l. For the remaining cases (l = k+1), we use various combinatorial topology techniques to obtain two impossibility results: the task is unsolvable if either k <= n/2 or n-k is odd. The remaining cases are still open.

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.

Linearizability is a widely accepted notion of correctness for concurrent objects. Recent research has investigated redefining linearizability for particular hardware weak memory models, in particular for TSO. In this paper, we provide an overview of this research and show that such redefinitions of linearizability are not required: under an interpretation of specification behaviour which abstracts from weak memory effects, the standard definition of linearizability is sound and complete on all hardware weak memory models. We prove our result with respect to a definition of object refinement which takes a weak memory model as a parameter. The main consequence of our findings is that we can leverage the range of existing techniques and tools for standard linearizability when verifying concurrent objects running on hardware weak memory models.

We identify a general principle of distributed computing: one cannot force two processes running in parallel to see each other. This principle is formally stated in the context of asynchronous processes communicating through shared objects, using trace-based semantics. We prove that it holds in a reasonable computational model, and then study the class of concurrent specifications which satisfy this property. This allows us to derive a Galois connection theorem for different variants of linearizability.

Numerical uncertainties from noisy inputs or finite-precision roundoff errors are unavoidable on resource-constrained systems. While techniques exist to compute worst-case bounds on these errors for arithmetic operations, these approaches do not generalize to programs which take discrete decisions. In this case, the more interesting quantity is the probability of the program making the wrong decision. In this paper, we study two approaches to compute a guaranteed bound on this probability: 1) exact probabilistic inference and 2) probabilistic static analysis. By themselves, they provide accuracy and scalability, respectively, but unfortunately not at the same time. We propose an extension to the latter approach which allows us to bound the probability tightly and fully automatically while scaling to small but interesting embedded examples.

We propose an approach for computing inner-approximations (also called under-approximations) of reachable sets of dynamical systems defined by non-linear, uncertain, ordinary differential equations. This is a notoriously difficult problem, much more intricate than outer-approximations (also called over-approximations), for which there exist well known solutions, mostly based on Taylor models. The few methods developed recently for inner-approximation mostly rely on backward flowmaps, and extra ingredients, either coming from optimization, or involving topological criteria, are required. Our solution, in comparison, builds on rather inexpensive set-based methods, namely a generalized mean-value theorem combined with Taylor models outer-approximations of the flow and its Jacobian with respect to the uncertain inputs and parameters. We demonstrate with a C/C++ prototype implementation that our method is both efficient and precise on classical examples. The combination of such forward inner and outer Taylor-model based approximations can be used as a basis for the verification and falsification of properties of cyber-physical systems.

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.

In this paper, we propose a categorical framework for bisimulations and unfoldings that unifies the classical approach from Joyal and al. via open maps and unfoldings. This is based on a notion of categories accessible with respect to a subcategory of path shapes, i.e., for which one can define a nice notion of trees as glueing of paths. We prove that transitions systems and pre sheaf models are a particular case of our framework. We also prove that in our framework, several characterizations of bisimulation coincide, in particular an "operational one" akin to the standard definition in transition systems. Also, accessibility is preserved by coreflexions. We then design a notion of unfolding, which has good properties in the accessible case: its is a right adjoint and is a universal covering, i.e., initial among the morphisms that have the unique lifting property with respect to path shapes. As an application, we prove that the universal covering of a groupoid, a standard construction in algebraic topology, coincides with an unfolding, when the category of path shapes is well chosen.

The homotopy hypothesis was originally stated by Grothendieck: topological spaces should be "equivalent" to (weak) infinite-groupoids, which give algebraic representatives of homotopy types. Much later, several authors developed geometrizations of computational models, e.g., for rewriting, distributed systems, (homotopy) type theory etc. But an essential feature in the work set up in concurrency theory, is that time should be considered irreversible, giving rise to the field of directed algebraic topology. Following the path proposed by Porter, we state here a directed homotopy hypothesis: Grandis' directed topological spaces should be "equivalent" to a weak form of topologically enriched categories, still very close to (infinite,1)-categories. We develop, as in ordinary algebraic topology, a directed homotopy equivalence and a weak equivalence, and show invariance of a form of directed homology.

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.

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 overapproximating the union. This operator is the only one that permits the performance of analyses that are invariant with respect to a linear transformation of state variables. It provides the minimum volume ellipsoid enclosing two given ellipsoids. We show that it can be computed in O ( n 3 ) elementary algebraic operations. We finally develop a fast nonlinear 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.

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.

Computing a tight inner approximation of the range of a function over some set is notoriously difficult, way beyond obtaining outer approximations. We propose here a new method to compute a tight inner approximation of the set of reachable states of non-linear dynamical systems on a bounded time interval. This approach involves affine forms and Kaucher arithmetic, plus a number of extra ingredients from set-based methods. An implementation of the method is discussed, and illustrated on representative numerical schemes, discrete-time and continuous-time dynamical systems.

editorial Free Access Share on Editorial: Special Section VCPSS’09 Guest Editors: Georgios Fainekos Arizona State University Arizona State UniversityView Profile , Eric Goubault CEA LIST CEA LISTView Profile , Franjo Ivančić Nec Laboratories America Nec Laboratories AmericaView Profile , Sriram Sankaranarayanan University of Colorado Boulder University of Colorado BoulderView Profile Authors Info & Claims ACM Transactions on Embedded Computing SystemsVolume 11Issue S2Article No.: 52pp 1–3https://doi.org/10.1145/2331147.2331162Published:01 August 2012Publication History 0citation126DownloadsMetricsTotal Citations0Total Downloads126Last 12 Months6Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is a characterization of the extreme points of a polyhedron in terms of a system of constraints which define it. We show that checking the extremality of a point reduces to checking whether there is only one minimal strongly connected component in an hypergraph. The latter problem can be solved in almost linear time, which allows us to eliminate quickly redundant generators. We report extensive tests (including benchmarks from an application to static analysis) showing that the method outperforms experimentally the previous ones by orders of magnitude. The present tools also lead to worst case bounds which improve the ones provided by previous methods.

Article Free Access Share on Schedulers as abstract interpretations of higher-dimensional automata Author: Eric Goubault LIENS, École Normale Supérieure, 45 rue d'Ulm, 75230 Paris, Cedex 05, FRANCE LIENS, École Normale Supérieure, 45 rue d'Ulm, 75230 Paris, Cedex 05, FRANCEView Profile Authors Info & Claims PEPM '95: Proceedings of the 1995 ACM SIGPLAN symposium on Partial evaluation and semantics-based program manipulationJune 1995 Pages 134–145https://doi.org/10.1145/215465.215577Published:23 June 1995Publication History 12citation226DownloadsMetricsTotal Citations12Total Downloads226Last 12 Months12Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

An investigation is conducted of the problem of providing low-cost compression of video sequences to be used in applications like video sessions between workstations. An evaluation of a hybrid coder for very low bit-rate (64 Kb/s) coding, based on vector quantization of the motion-compensated prediction where only the regions that exhibit noticeable changes between successive frames are coded and transmitted, is presented. Motion compensation (MC) is performed in three steps, over successive subband decompositions of the subsequent frames. Such a hierarchical MC technique substantially reduces the complexity with respect to classical MC, due to the subsampling of the subbands. Vector quantization is applied to the coding of the motion-compensation prediction error blocks. Fast search VQ algorithms are used to achieve a low complexity coding.>