Conjecturing and theorem proving are activities at the center of mathematical practice and are difficult to separate. In this paper, we propose a framework for completing incomplete conjectures and incomplete proofs. The framework can turn a conjecture with missing assumptions and with an under-specified goal into a proper theorem. Also, the proposed framework can help in completing a proof sketch into a human-readable and machine-checkable proof. Our approach is focused on synthetic geometry, and uses coherent logic and constraint solving. The proposed approach is uniform for all three kinds of tasks, flexible and, to our knowledge, unique such approach.
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Quantum computing promises to help humanity solve problems that would otherwise be intractable on classical computers. Unlike today's machines, quantum computers use a novel computing process that leverages the foundational quantum mechanical laws of nature. This unlocks unparalleled compute power for certain applications and promises to help solve some of our generation's gravest challenges, including the climate crisis, food insecurity, and widespread disease. No one entity will be able to realize this end state alone. Developing a fault-tolerant quantum supercomputer and a vibrant ecosystem around it will require deep partnerships between industry, governments, and academia. It will also require collective action to enable and promote positive applications of quantum computing and ensure that the safe and responsible use of the technology is at the center of its development and deployment. Achieving these objectives will require focusing on three priorities: 1. Impact. Ensure quantum computing benefits all of humankind by developing quantum solutions to solve critical, global problems. 2. Use. Protect against malicious use by accelerating the deployment of quantum-safe cryptography and developing governance processes and controls for the responsible use of quantum machines. 3. Access. Democratize the potential for economic growth across all of society through skilling, workforce and ecosystem development, and digital infrastructure. This paper discusses each in turn.
We present AlloyInEcore, a tool for specifying metamodels with their static semantics to facilitate automated, formal reasoning on models. Software development projects require that software systems be specified in various models (e.g., requirements models, architecture models, test models, and source code). It is crucial to reason about those models to ensure the correct and complete system specifications. AlloyInEcore allows the user to specify metamodels with their static semantics, while, using the semantics, it automatically detects inconsistent models, and completes partial models. It has been evaluated on three industrial case studies in the automotive domain (//modelwriter.github.io/AlloyInEcore/).
Statistical mechanics has made significant contributions to the study of biological neural systems by modeling them as recurrent networks of interconnected units with adjustable interactions. Several algorithms have been proposed to optimize the neural connections to enable network tasks such as information storage (i.e. associative memory) and learning probability distributions from data (i.e. generative modeling). Among these methods, the Unlearning algorithm, aligned with emerging theories of synaptic plasticity, was introduced by John Hopfield and collaborators. The primary objective of this thesis is to understand the effectiveness of Unlearning in both associative memory models and generative models. Initially, we demonstrate that the Unlearning algorithm can be simplified to a linear perceptron model which learns from noisy examples featuring specific internal correlations. The selection of structured training data enables an associative memory model to retrieve concepts as attractors of a neural dynamics with considerable basins of attraction. Subsequently, a novel regularization technique for Boltzmann Machines is presented, proving to outperform previously developed methods in learning hidden probability distributions from data-sets. The Unlearning rule is derived from this new regularized algorithm and is showed to be comparable, in terms of inferential performance, to traditional Boltzmann-Machine learning.
Laboratory experiments have shown that communication plays an important role in solving social dilemmas. Here, by extending the AI-Economist, a mixed motive multi-agent reinforcement learning environment, I intend to find an answer to the following descriptive question: which governing system does facilitate the emergence and evolution of communication and teaching among agents? To answer this question, the AI-Economist is extended by a voting mechanism to simulate three different governing systems across individualistic-collectivistic axis, from full-libertarian to Full-Utilitarian governing systems. Moreover, the AI-Economist is further extended to include communication with possible misalignment, a variant of signalling game, by letting agents to build houses together if they are able to name mutually complement material resources by the same letter. Moreover, another extension is made to the AI-Economist to include teaching with possible misalignment, again a variant of signalling game, by letting half the agents as teachers who know how to use mutually complement material resources to build houses but are not capable of building actual houses, and the other half as students who do not have this information but are able to actually build those houses if teachers teach them. I found a strong evidence that collectivistic environment such as Full-Utilitarian system is more favourable for the emergence of communication and teaching, or more precisely, evolution of language alignment. Moreover, I found some evidence that evolution of language alignment through communication and teaching under collectivistic governing systems makes individuals more advantageously inequity averse. As a result, there is a positive correlation between evolution of language alignment and equality in the society.
As cyber attacks continue to increase in frequency and sophistication, detecting malware has become a critical task for maintaining the security of computer systems. Traditional signature-based methods of malware detection have limitations in detecting complex and evolving threats. In recent years, machine learning (ML) has emerged as a promising solution to detect malware effectively. ML algorithms are capable of analyzing large datasets and identifying patterns that are difficult for humans to identify. This paper presents a comprehensive review of the state-of-the-art ML techniques used in malware detection, including supervised and unsupervised learning, deep learning, and reinforcement learning. We also examine the challenges and limitations of ML-based malware detection, such as the potential for adversarial attacks and the need for large amounts of labeled data. Furthermore, we discuss future directions in ML-based malware detection, including the integration of multiple ML algorithms and the use of explainable AI techniques to enhance the interpret ability of ML-based detection systems. Our research highlights the potential of ML-based techniques to improve the speed and accuracy of malware detection, and contribute to enhancing cybersecurity
The Causal Roadmap outlines a systematic approach to asking and answering questions of cause-and-effect: define quantity of interest, evaluate needed assumptions, conduct statistical estimation, and carefully interpret results. It is paramount that the algorithm for statistical estimation and inference be carefully pre-specified to optimize its expected performance for the specific real-data application. Simulations that realistically reflect the application, including key characteristics such as strong confounding and dependent or missing outcomes, can help us gain a better understanding of an estimator's applied performance. We illustrate this with two examples, using the Causal Roadmap and realistic simulations to inform estimator selection and full specification of the Statistical Analysis Plan. First, in an observational longitudinal study, outcome-blind simulations are used to inform nuisance parameter estimation and variance estimation for longitudinal targeted maximum likelihood estimation (TMLE). Second, in a cluster-randomized controlled trial with missing outcomes, treatment-blind simulations are used to ensure control for Type-I error in Two-Stage TMLE. In both examples, realistic simulations empower us to pre-specify an estimator that is expected to have strong finite sample performance and also yield quality-controlled computing code for the actual analysis. Together, this process helps to improve the rigor and reproducibility of our research.
Hex-dominant mesh generation has received significant attention in recent research due to its superior robustness compared to pure hex-mesh generation techniques. In this work, we introduce the first structure for analyzing hex-dominant meshes. This structure builds on the base complex of pure hex-meshes but incorporates the non-hex elements for a more comprehensive and complete representation. We provide its definition and describe its construction steps. Based on this structure, we present an extraction and categorization of sheets using advanced graph matching techniques to handle the non-hex elements. This enables us to develop an enhanced visual analysis of the structure for any hex-dominant meshes.We apply this structure-based visual analysis to compare hex-dominant meshes generated by different methods to study their advantages and disadvantages. This complements the standard quality metric based on the non-hex element percentage for hex-dominant meshes. Moreover, we propose a strategy to extract a cleaned (optimized) valence-based singularity graph wireframe to analyze the structure for both mesh and sheets. Our results demonstrate that the proposed hybrid base complex provides a coarse representation for mesh element, and the proposed valence singularity graph wireframe provides a better internal visualization of hex-dominant meshes.
While there exists a rich array of matrix column subset selection problem (CSSP) algorithms for use with interpolative and CUR-type decompositions, their use can often become prohibitive as the size of the input matrix increases. In an effort to address these issues, the authors in \cite{emelianenko2024adaptive} developed a general framework that pairs a column-partitioning routine with a column-selection algorithm. Two of the four algorithms presented in that work paired the Centroidal Voronoi Orthogonal Decomposition (\textsf{CVOD}) and an adaptive variant (\textsf{adaptCVOD}) with the Discrete Empirical Interpolation Method (\textsf{DEIM}) \cite{sorensen2016deim}. In this work, we extend this framework and pair the \textsf{CVOD}-type algorithms with any CSSP algorithm that returns linearly independent columns. Our results include detailed error bounds for the solutions provided by these paired algorithms, as well as expressions that explicitly characterize how the quality of the selected column partition affects the resulting CSSP solution.
Graph Neural Networks (GNNs) have been studied from the lens of expressive power and generalization. However, their optimization properties are less well understood. We take the first step towards analyzing GNN training by studying the gradient dynamics of GNNs. First, we analyze linearized GNNs and prove that despite the non-convexity of training, convergence to a global minimum at a linear rate is guaranteed under mild assumptions that we validate on real-world graphs. Second, we study what may affect the GNNs' training speed. Our results show that the training of GNNs is implicitly accelerated by skip connections, more depth, and/or a good label distribution. Empirical results confirm that our theoretical results for linearized GNNs align with the training behavior of nonlinear GNNs. Our results provide the first theoretical support for the success of GNNs with skip connections in terms of optimization, and suggest that deep GNNs with skip connections would be promising in practice.
Current deep learning research is dominated by benchmark evaluation. A method is regarded as favorable if it empirically performs well on the dedicated test set. This mentality is seamlessly reflected in the resurfacing area of continual learning, where consecutively arriving sets of benchmark data are investigated. The core challenge is framed as protecting previously acquired representations from being catastrophically forgotten due to the iterative parameter updates. However, comparison of individual methods is nevertheless treated in isolation from real world application and typically judged by monitoring accumulated test set performance. The closed world assumption remains predominant. It is assumed that during deployment a model is guaranteed to encounter data that stems from the same distribution as used for training. This poses a massive challenge as neural networks are well known to provide overconfident false predictions on unknown instances and break down in the face of corrupted data. In this work we argue that notable lessons from open set recognition, the identification of statistically deviating data outside of the observed dataset, and the adjacent field of active learning, where data is incrementally queried such that the expected performance gain is maximized, are frequently overlooked in the deep learning era. Based on these forgotten lessons, we propose a consolidated view to bridge continual learning, active learning and open set recognition in deep neural networks. Our results show that this not only benefits each individual paradigm, but highlights the natural synergies in a common framework. We empirically demonstrate improvements when alleviating catastrophic forgetting, querying data in active learning, selecting task orders, while exhibiting robust open world application where previously proposed methods fail.
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