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We present stack graphs, an extension of Visser et al.'s scope graphs framework. Stack graphs power Precise Code Navigation at GitHub, allowing users to navigate name binding references both within and across repositories. Like scope graphs, stack graphs encode the name binding information about a program in a graph structure, in which paths represent valid name bindings. Resolving a reference to its definition is then implemented with a simple path-finding search. GitHub hosts millions of repositories, containing petabytes of total code, implemented in hundreds of different programming languages, and receiving thousands of pushes per minute. To support this scale, we ensure that the graph construction and path-finding judgments are file-incremental: for each source file, we create an isolated subgraph without any knowledge of, or visibility into, any other file in the program. This lets us eliminate the storage and compute costs of reanalyzing file versions that we have already seen. Since most commits change a small fraction of the files in a repository, this greatly amortizes the operational costs of indexing large, frequently changed repositories over time. To handle type-directed name lookups (which require "pausing" the current lookup to resolve another name), our name resolution algorithm maintains a stack of the currently paused (but still pending) lookups. Stack graphs can be constructed via a purely syntactic analysis of the program's source code, using a new declarative graph construction language. This means that we can extract name binding information for every repository without any per-package configuration, and without having to invoke an arbitrary, untrusted, package-specific build process.

Safety in the automotive domain is a well-known topic, which has been in constant development in the past years. The complexity of new systems that add more advanced components in each function has opened new trends that have to be covered from the safety perspective. In this case, not only specifications and requirements have to be covered but also scenarios, which cover all relevant information of the vehicle environment. Many of them are not yet still sufficient defined or considered. In this context, Safety of the Intended Functionality (SOTIF) appears to ensure the system when it might fail because of technological shortcomings or misuses by users. An identification of the plausibly insufficiencies of ADAS/ADS functions has to be done to discover the potential triggering conditions that can lead to these unknown scenarios, which might effect a hazardous behaviour. The main goal of this publication is the definition of an use case to identify these triggering conditions that have been applied to the collision avoidance function implemented in our self-developed mobile Hardware-in-Loop (HiL) platform.

A semi-Lagrangian Characteristic Mapping method for the solution of the tracer transport equations on the sphere is presented. The method solves for the solution operator of the equations by approximating the inverse of the diffeomorphism generated by a given velocity field. The evolution of any tracer and mass density can then be computed via pullback with this map. We present a spatial discretization of the manifold-valued map using a projection-based approach with spherical spline interpolation. The numerical scheme yields $C^1$ continuity for the map and global second-order accuracy for the solution of the tracer transport equations. Error estimates are provided and supported by convergence tests involving solid body rotation, moving vortices, deformational, and compressible flows. Additionally, we illustrate some features of computing the solution operator using a numerical mixing test and the transport of a fractal set in a complex flow environment.

We study hidden-action principal-agent problems with multiple agents. These are problems in which a principal commits to an outcome-dependent payment scheme in order to incentivize some agents to take costly, unobservable actions that lead to favorable outcomes. Previous works on multi-agent problems study models where the principal observes a single outcome determined by the actions of all the agents. Such models considerably limit the contracting power of the principal, since payments can only depend on the joint result of all the agents' actions, and there is no way of paying each agent for their individual result. In this paper, we consider a model in which each agent determines their own individual outcome as an effect of their action only, the principal observes all the individual outcomes separately, and they perceive a reward that jointly depends on all these outcomes. This considerably enhances the principal's contracting capabilities, by allowing them to pay each agent on the basis of their individual result. We analyze the computational complexity of finding principal-optimal contracts, revolving around two newly-introduced properties of principal's rewards, which we call IR-supermodularity and DR-submodularity. Intuitively, the former captures settings with increasing returns, where the rewards grow faster as the agents' effort increases, while the latter models the case of diminishing returns, in which rewards grow slower instead. These two properties naturally model two common real-world phenomena, namely diseconomies and economies of scale. In this paper, we first address basic instances in which the principal knows everything about the agents, and, then, more general Bayesian instances where each agent has their own private type determining their features, such as action costs and how actions stochastically determine individual outcomes.

Pandora's Box is a central problem in decision making under uncertainty that can model various real life scenarios. In this problem we are given $n$ boxes, each with a fixed opening cost, and an unknown value drawn from a known distribution, only revealed if we pay the opening cost. Our goal is to find a strategy for opening boxes to minimize the sum of the value selected and the opening cost paid. In this work we revisit Pandora's Box when the value distributions are correlated, first studied in Chawla et al. (arXiv:1911.01632). We show that the optimal algorithm for the independent case, given by Weitzman's rule, directly works for the correlated case. In fact, our algorithm results in significantly improved approximation guarantees compared to the previous work, while also being substantially simpler. We finally show how to implement the rule given only sample access to the correlated distribution of values. Specifically, we find that a number of samples that is polynomial in the number of boxes is sufficient for the algorithm to work.

Motivated by the success of Bayesian optimisation algorithms in the Euclidean space, we propose a novel approach to construct Intrinsic Bayesian optimisation (In-BO) on manifolds with a primary focus on complex constrained domains or irregular-shaped spaces arising as submanifolds of R2, R3 and beyond. Data may be collected in a spatial domain but restricted to a complex or intricately structured region corresponding to a geographic feature, such as lakes. Traditional Bayesian Optimisation (Tra-BO) defined with a Radial basis function (RBF) kernel cannot accommodate these complex constrained conditions. The In-BO uses the Sparse Intrinsic Gaussian Processes (SIn-GP) surrogate model to take into account the geometric structure of the manifold. SInGPs are constructed using the heat kernel of the manifold which is estimated as the transition density of the Brownian Motion on manifolds. The efficiency of In-BO is demonstrated through simulation studies on a U-shaped domain, a Bitten torus, and a real dataset from the Aral sea. Its performance is compared to that of traditional BO, which is defined in Euclidean space.

Characterizing the implicit structure of the computation within neural networks is a foundational problem in the area of deep learning interpretability. Can the inner decision process of neural networks be captured symbolically in some familiar logic? We show that any fixed-precision transformer neural network can be translated into an equivalent fixed-size $\mathsf{FO}(\mathsf{M})$ formula, i.e., a first-order logic formula that, in addition to standard universal and existential quantifiers, may also contain majority-vote quantifiers. The proof idea is to design highly uniform boolean threshold circuits that can simulate transformers, and then leverage known theoretical connections between circuits and logic. Our results reveal a surprisingly simple formalism for capturing the behavior of transformers, show that simple problems like integer division are "transformer-hard", and provide valuable insights for comparing transformers to other models like RNNs. Our results suggest that first-order logic with majority may be a useful language for expressing programs extracted from transformers.

We propose a continuous optimization framework for discovering a latent directed acyclic graph (DAG) from observational data. Our approach optimizes over the polytope of permutation vectors, the so-called Permutahedron, to learn a topological ordering. Edges can be optimized jointly, or learned conditional on the ordering via a non-differentiable subroutine. Compared to existing continuous optimization approaches our formulation has a number of advantages including: 1. validity: optimizes over exact DAGs as opposed to other relaxations optimizing approximate DAGs; 2. modularity: accommodates any edge-optimization procedure, edge structural parameterization, and optimization loss; 3. end-to-end: either alternately iterates between node-ordering and edge-optimization, or optimizes them jointly. We demonstrate, on real-world data problems in protein-signaling and transcriptional network discovery, that our approach lies on the Pareto frontier of two key metrics, the SID and SHD.

We present a novel method for the safety verification of nonlinear dynamical models that uses neural networks to represent abstractions of their dynamics. Neural networks have extensively been used before as approximators; in this work, we make a step further and use them for the first time as abstractions. For a given dynamical model, our method synthesises a neural network that overapproximates its dynamics by ensuring an arbitrarily tight, formally certified bound on the approximation error. For this purpose, we employ a counterexample-guided inductive synthesis procedure. We show that this produces a neural ODE with non-deterministic disturbances that constitutes a formal abstraction of the concrete model under analysis. This guarantees a fundamental property: if the abstract model is safe, i.e., free from any initialised trajectory that reaches an undesirable state, then the concrete model is also safe. By using neural ODEs with ReLU activation functions as abstractions, we cast the safety verification problem for nonlinear dynamical models into that of hybrid automata with affine dynamics, which we verify using SpaceEx. We demonstrate that our approach performs comparably to the mature tool Flow* on existing benchmark nonlinear models. We additionally demonstrate and that it is effective on models that do not exhibit local Lipschitz continuity, which are out of reach to the existing technologies.