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A sequence of random variables is called exchangeable if its joint distribution is invariant under permutations. The original formulation of de Finetti's theorem says that any exchangeable sequence of $\{0,1\}$-valued random variables can be thought of as a mixture of independent and identically distributed sequences in a certain precise mathematical sense. Interpreting this statement from a convex analytic perspective, Hewitt and Savage obtained the same conclusion for more general state spaces under some topological conditions. The main contribution of this paper is in providing a new framework that explains the theorem purely as a consequence of the underlying distribution of the random variables, with no topological conditions (beyond Hausdorffness) on the state space being necessary if the distribution is Radon. We also show that it is consistent with the axioms of ZFC that de Finetti's theorem holds for all sequences of exchangeable random variables taking values in any complete metric space. The framework we use is based on nonstandard analysis. We have provided a self-contained introduction to nonstandard analysis as an appendix, thus rendering measure theoretic probability and point-set topology as the only prerequisites for this paper. Our introduction aims to develop some new ideologies that might be of interest to mathematicians, philosophers, and mathematics educators alike. Our technical tools come from nonstandard topological measure theory, in which a highlight is a new generalization of Prokhorov's theorem. Modulo such technical tools, our proof relies on properties of the empirical measures induced by hyperfinitely many identically distributed random variables -- a feature that allows us to establish de Finetti's theorem in the generality that we seek while still retaining the combinatorial intuition of proofs of simpler versions of de Finetti's theorem.

The one-dimensional PDE model of the wave equation with a state feedback controller at its boundary, which describes wave dynamics of a wide-range of controlled mechanical systems, has exponentially stable solutions. However, it is known that the reduced models of the wave equation by the standard Finite Differences and Finite Elements suffer from the lack of exponential stability (and exact observability without a state feedback controller) uniformly as the discretization parameter tends to zero. This is due to the loss of uniform gap among the high-frequency eigenvalues as the discretization parameter tends to zero. One common remedy to overcome this discrepancy is the direct Fourier filtering of the reduced models, where the high-frequency spurious eigenvalues are filtered out. After filtering, besides from the strong convergency, the exponential decay rate, mimicking the one for the partial differential equation counterpart, can be retained uniformly. However, the existing results in the literature are solely based on an observability inequality of the control-free model, to which the filtering is implemented. Moreover, the decay rate as a function of the filtering parameter is implicit. In this paper, exponential stability results for both filtered Finite Difference and Finite Element reduced models are established directly by a Lyapunov-based approach and a thorough eigenvalue estimation.The maximal decay rate is explicitly provided as a function of the feedback gain and filtering parameter. Our results, expectedly, mimic the ones of the PDE counterpart uniformly as the discretization parameter tends to zero. Several numerical tests are provided to support our results.

Graphics Processing Units (GPUs) are over-stressed to accelerate High-Performance Computing applications and are used to accelerate Deep Neural Networks in several domains where they have a life expectancy of many years. These conditions expose the GPUs hardware to (premature) aging, causing permanent faults to arise after the usual end-of-manufacturing test. Techniques to assess the impact of permanent faults in GPUs are then strongly required, thus allowing to estimate the reliability risk and to possibly mitigate it. In this paper, we present a method to evaluate the effects of permanent faults affecting the GPU scheduler and control units, which are the most peculiar and stressed resources, along with the first figures that allow quantifying these effects. We characterize over 5.83x10^5 permanent fault effects in the scheduler and controllers of a gate-level GPU model. Then, we map the observed error categories in software by instrumenting the code of 13 applications and two convolutional neural networks, injecting more than 1.65x10^5 permanent errors. Our two-level fault injection strategy reduces the evaluation time from hundreds of years of gate-level evaluation to hundreds of hours.We found that faults in the GPU parallelism management units can modify the opcode, the addresses, and the status of thread(s) and warp(s). The large majority (up to 99%) of these hardware permanent errors impacts the running software execution. Errors affecting the instruction operation or resource management hang the code, while 45% of errors in the parallelism management or control-flow induce silent data corruptions.

Large language models (LLMs), like ChatGPT, have shown some human-like cognitive abilities. For comparing these abilities of different models, several benchmarks (i.e. sets of standard test questions) from different fields (e.g., Literature, Biology and Psychology) are often adopted and the test results under traditional metrics such as accuracy, recall and F1, are reported. However, such way for evaluating LLMs can be inefficient and inaccurate from the cognitive science perspective. Inspired by Computerized Adaptive Testing (CAT) used in psychometrics, we propose an adaptive testing framework for LLM evaluation. Rather than using a standard test set and simply reporting accuracy, this approach dynamically adjusts the characteristics of the test questions, such as difficulty, based on the model's performance. This allows for a more accurate estimation of the model's abilities, using fewer questions. More importantly, it allows LLMs to be compared with humans easily, which is essential for NLP models that aim for human-level ability. Our diagnostic reports have found that ChatGPT often behaves like a ``careless student'', prone to slip and occasionally guessing the questions. We conduct a fine-grained diagnosis and rank the latest 6 instruction-tuned LLMs from three aspects of Subject Knowledge, Mathematical Reasoning, and Programming, where GPT4 can outperform other models significantly and reach the cognitive ability of middle-level students. Different tests for different models using efficient adaptive testing -- we believe this has the potential to become a new norm in evaluating large language models.

Given a Boolean formula $\phi$ over $n$ variables, the problem of model counting is to compute the number of solutions of $\phi$. Model counting is a fundamental problem in computer science with wide-ranging applications. Owing to the \#P-hardness of the problems, Stockmeyer initiated the study of the complexity of approximate counting. Stockmeyer showed that $\log n$ calls to an NP oracle are necessary and sufficient to achieve $(\varepsilon,\delta)$ guarantees. The hashing-based framework proposed by Stockmeyer has been very influential in designing practical counters over the past decade, wherein the SAT solver substitutes the NP oracle calls in practice. It is well known that an NP oracle does not fully capture the behavior of SAT solvers, as SAT solvers are also designed to provide satisfying assignments when a formula is satisfiable, without additional overhead. Accordingly, the notion of SAT oracle has been proposed to capture the behavior of SAT solver wherein given a Boolean formula, an SAT oracle returns a satisfying assignment if the formula is satisfiable or returns unsatisfiable otherwise. Since the practical state-of-the-art approximate counting techniques use SAT solvers, a natural question is whether an SAT oracle is more powerful than an NP oracle in the context of approximate model counting. The primary contribution of this work is to study the relative power of the NP oracle and SAT oracle in the context of approximate model counting. The previous techniques proposed in the context of an NP oracle are weak to provide strong bounds in the context of SAT oracle since, in contrast to an NP oracle that provides only one bit of information, a SAT oracle can provide $n$ bits of information. We therefore develop a new methodology to achieve the main result: a SAT oracle is no more powerful than an NP oracle in the context of approximate model counting.

Quantifier elimination (qelim) is used in many automated reasoning tasks including program synthesis, exist-forall solving, quantified SMT, Model Checking, and solving Constrained Horn Clauses (CHCs). Exact qelim is computationally expensive. Hence, it is often approximated. For example, Z3 uses "light" pre-processing to reduce the number of quantified variables. CHC-solver Spacer uses model-based projection (MBP) to under-approximate qelim relative to a given model, and over-approximations of qelim can be used as abstractions. In this paper, we present the QEL framework for fast approximations of qelim. QEL provides a uniform interface for both quantifier reduction and model-based projection. QEL builds on the egraph data structure -- the core of the EUF decision procedure in SMT -- by casting quantifier reduction as a problem of choosing ground (i.e., variable-free) representatives for equivalence classes. We have used QEL to implement MBP for the theories of Arrays and Algebraic Data Types (ADTs). We integrated QEL and our new MBP in Z3 and evaluated it within several tasks that rely on quantifier approximations, outperforming state-of-the-art.

The problem Power Dominating Set (PDS) is motivated by the placement of phasor measurement units to monitor electrical networks. It asks for a minimum set of vertices in a graph that observes all remaining vertices by exhaustively applying two observation rules. Our contribution is twofold. First, we determine the parameterized complexity of PDS by proving it is $W[P]$-complete when parameterized with respect to the solution size. We note that it was only known to be $W[2]$-hard before. Our second and main contribution is a new algorithm for PDS that efficiently solves practical instances. Our algorithm consists of two complementary parts. The first is a set of reduction rules for PDS that can also be used in conjunction with previously existing algorithms. The second is an algorithm for solving the remaining kernel based on the implicit hitting set approach. Our evaluation on a set of power grid instances from the literature shows that our solver outperforms previous state-of-the-art solvers for PDS by more than one order of magnitude on average. Furthermore, our algorithm can solve previously unsolved instances of continental scale within a few minutes.

The Information Bottleneck (IB) is a method of lossy compression. Its rate-distortion (RD) curve describes the fundamental tradeoff between input compression and the preservation of relevant information. However, it conceals the underlying dynamics of optimal input encodings. We argue that these typically follow a piecewise smooth trajectory as the input information is being compressed, as recently shown in RD. These smooth dynamics are interrupted when an optimal encoding changes qualitatively, at a bifurcation. By leveraging the IB's intimate relations with RD, sub-optimal solutions can be seen to collide or exchange optimality there. Despite the acceptance of the IB and its applications, there are surprisingly few techniques to solve it numerically, even for finite problems whose distribution is known. We derive anew the IB's first-order Ordinary Differential Equation, which describes the dynamics underlying its optimal tradeoff curve. To exploit these dynamics, one needs not only to detect IB bifurcations but also to identify their type in order to handle them accordingly. Rather than approaching the optimal IB curve from sub-optimal directions, the latter allows us to follow a solution's trajectory along the optimal curve, under mild assumptions. Thereby, translating an understanding of IB bifurcations into a surprisingly accurate numerical algorithm.

Graph Neural Networks (GNNs) have been successfully used in many problems involving graph-structured data, achieving state-of-the-art performance. GNNs typically employ a message-passing scheme, in which every node aggregates information from its neighbors using a permutation-invariant aggregation function. Standard well-examined choices such as the mean or sum aggregation functions have limited capabilities, as they are not able to capture interactions among neighbors. In this work, we formalize these interactions using an information-theoretic framework that notably includes synergistic information. Driven by this definition, we introduce the Graph Ordering Attention (GOAT) layer, a novel GNN component that captures interactions between nodes in a neighborhood. This is achieved by learning local node orderings via an attention mechanism and processing the ordered representations using a recurrent neural network aggregator. This design allows us to make use of a permutation-sensitive aggregator while maintaining the permutation-equivariance of the proposed GOAT layer. The GOAT model demonstrates its increased performance in modeling graph metrics that capture complex information, such as the betweenness centrality and the effective size of a node. In practical use-cases, its superior modeling capability is confirmed through its success in several real-world node classification benchmarks.

Graph Neural Networks (GNNs) for representation learning of graphs broadly follow a neighborhood aggregation framework, where the representation vector of a node is computed by recursively aggregating and transforming feature vectors of its neighboring nodes. Many GNN variants have been proposed and have achieved state-of-the-art results on both node and graph classification tasks. However, despite GNNs revolutionizing graph representation learning, there is limited understanding of their representational properties and limitations. Here, we present a theoretical framework for analyzing the expressive power of GNNs in capturing different graph structures. Our results characterize the discriminative power of popular GNN variants, such as Graph Convolutional Networks and GraphSAGE, and show that they cannot learn to distinguish certain simple graph structures. We then develop a simple architecture that is provably the most expressive among the class of GNNs and is as powerful as the Weisfeiler-Lehman graph isomorphism test. We empirically validate our theoretical findings on a number of graph classification benchmarks, and demonstrate that our model achieves state-of-the-art performance.