Humans prove theorems by relying on substantial high-level reasoning and problem-specific insights. Proof assistants offer a formalism that resembles human mathematical reasoning, representing theorems in higher-order logic and proofs as high-level tactics. However, human experts have to construct proofs manually by en…
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48 results for “proof assistants”
Formalizes synthetic differential geometry in Lean.
problem Formalizing synthetic differential geometry in a proof assistant.
method Formalization of synthetic differential geometry with Lean and mathlib.
result Proves a Taylor theorem for functions of several variables.
We provide a computer-assisted proof of the holomorphy of the quartic and the octic meromorphic differentials arising in the main Theorem 4.11 of our paper 'The Classification of Branched Willmore spheres in the -Sphere and the -Sphere' (arXiv:1706.01405), using the free mathematical software Sage.
Lean Copilot uses LLMs to assist theorem proving in Lean, improving efficiency and automation.
problem Challenges in using existing neural theorem provers to prove novel theorems autonomously.
method Introduces Lean Copilot, a framework for integrating LLMs into Lean's theorem proving process.
result Lean Copilot automates 74.2% of proof steps on average, significantly improving over existing methods.
This research simplifies verification of machine learning systems using reparameterization.
problem Reduce or eliminate serious bugs in machine learning systems.
method Use proof assistants to construct machine-checked proofs of correctness, leveraging reparameterization to handle probabilistic claims.
result Demonstrates broad applicability of reparameterization to verify different types of machine learning systems.
The Goldman-Parker Conjecture classifies the complex hyperbolic C-reflection ideal triangle groups up to discreteness. We proved the Goldman-Parker Conjecture in [Ann. of Math. 153 (2001) 533--598] using a rigorous computer-assisted proof. In this paper we give a new and improved proof of the Goldman-Parker Conjecture.…
Learning preferences implicit in the choices humans make is a well studied problem in both economics and computer science. However, most work makes the assumption that humans are acting (noisily) optimally with respect to their preferences. Such approaches can fail when people are themselves learning about what they wa…
Formally verifies fairness and uniformity in financial market trades.
problem Ensuring fairness and uniformity in automated trading systems.
method Formal definition and verification in Coq proof assistant.
result Properties of double-sided auction mechanisms verified.
In this paper, we introduce a system called GamePad that can be used to explore the application of machine learning methods to theorem proving in the Coq proof assistant. Interactive theorem provers such as Coq enable users to construct machine-checkable proofs in a step-by-step manner. Hence, they provide an opportuni…
We prove two conjectures of C. Gordon. We show that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10, and that the maximal intersection number between exceptional slopes is 8. The proof uses a combination of new geometric techniques and a rigorous computer-assisted calculation.
INT benchmark tests theorem proving agents' ability to generalize to unseen theorems.
problem Evaluating theorem proving agents' ability to generalize to unseen theorems.
method INT benchmark based on a theorem generation and proof procedure with adjustable knobs for measuring 6 types of generalization.
result MCTS can help agents prove new theorems.
GPT-f uses language models to find new proofs in formal math.
problem Generating original mathematical terms for automated theorem proving.
method Transformer-based language model for automated theorem proving and proof assistant.
result GPT-f contributed new proofs to the Metamath library.
The paper proves rigidity of surgeries on the figure-eight knot complement.
problem Infinitesimal projective rigidity of surgeries on the figure-eight knot complement.
method Computer-assisted proof and explicit representations of the knot complement.
result Proves infinitesimal projective rigidity for surgeries far from the ideal point.
We prove the Goldman-Parker Conjecture: A complex hyperbolic ideal triangle group is directly embedded in PU(2,1) if and only if the product of its three standard generators is not elliptic. We also prove that such a group is indiscrete if the product of its three standard generators is elliptic. A novel feature of thi…
Quantum-assisted VAE improves similarity search in high-dimensional datasets.
problem Finding fast and memory-efficient similarity search in high-dimensional data.
method Construct a space-efficient search index based on the latent space of a Quantum-assisted Variational Autoencoder (QVAE).
result Real-world speedups and memory-efficient scaling to half a billion data points.
Efficient verified double auctions improve matching speed and detect errors.
problem Improving the efficiency and reliability of double auctions in financial markets.
method Formally verified implementation using Coq proof assistant, reducing time complexity and improving error detection.
result Improved efficiency with time complexity, reducing runtime from days to minutes.
Researchers prove existence of a special Einstein metric on a 12-dimensional sphere.
problem Proving the existence of a non-round Einstein metric invariant under a specific group action.
method Numerical analysis techniques were used to produce an approximate Einstein metric, which was then perturbed into a true Einstein metric.
result A novel -invariant Einstein metric on was successfully constructed.
Let be the fundamental group of a surface of finite type and Comm be its abstract commensurator. Then Comm contains the solvable Baumslag--Solitar groups for any . Moreover, the Baumslag--Solitar group has an imag…
Detects (2,5) torus knot using Khovanov homology and Floer homology.
problem Detecting the (2,5) torus knot using Khovanov homology.
method Combines Floer homology, Khovanov homology, and surface homeomorphisms.
result Proves Khovanov homology detects the (2,5) torus knot.
New AI assistant for power grid operators simplifies complex decision-making.
problem Complexity and uncertainty in power grid operations.
method Unified human-machine interface and AI integration.
result Development of a new assistant framework for power grid operators.
The simplest non-collision solutions of the N-body problem are the "relative equilibria", in which each body follows a circular orbit around the centre of mass and the shape formed by the N bodies is constant. It is easy to see that the moment of inertia of such a solution is constant. In 1970, D. Saari conjectured tha…
Framework allows organizations to collaborate on learning tasks securely.
problem Limited collaboration due to security constraints.
method Assisted Learning framework for supervised learning tasks.
result Near-oracle learning performance achieved without revealing sensitive information.
We introduce a formal framework for analyzing trades in financial markets. An exchange is where multiple buyers and sellers participate to trade. These days, all big exchanges use computer algorithms that implement double sided auctions to match buy and sell requests and these algorithms must abide by certain regulator…
Improves survey sampling with unbiased machine learning methods.
problem Design-consistent model-assisted estimation lacks a general theory for machine learning.
method Proposes a subsampling Rao-Blackwell method for design-unbiased estimation.
result Yields efficiency gains over standard methods while ensuring valid estimation.
Through deep learning and computer vision techniques, driving manoeuvres can be predicted accurately a few seconds in advance. Even though adapting a learned model to new drivers and different vehicles is key for robust driver-assistance systems, this problem has received little attention so far. This work proposes to …
Why Agentic Theorem Prover Works: A Statistical Provability Theory of Mathematical Reasoning Modelsstat.ML
Study improves understanding of why agentic theorem provers succeed.
problem Understanding which components of agentic theorem provers improve proof success.
method Statistical provability theory and finite-horizon reachability MDP model.
result Bounds provability gap and explains components' effectiveness.
Computer-assisted method finds new Einstein metrics on spheres.
problem Finding new Einstein metrics on spheres.
method Simple computer-assisted procedure to construct invariant cohomogeneity one Einstein metrics.
result New Einstein metrics on , , and .
Access to food assistance programs such as food pantries and food banks needs focus in order to mitigate food insecurity. Accessibility to the food assistance programs is impacted by demographics of the population and geography of the location. It hence becomes imperative to define and identify food assistance deserts …
We prove the existence of a (spectrally) stable self-similar blow-up solution to the heat flow for corotational harmonic maps from to the three-sphere. In particular, our result verifies the spectral gap conjecture stated by one of the authors and lays the groundwork for the proof of the nonlinear s…
Verified numerics prove existence of a curvature solution with known symmetries.
problem Existence of a curvature solution for the Nirenberg problem.
method Verified numerics and computer assistance.
result Existence of a genuine solution with known symmetry groups.
This paper autoformalizes Euclidean geometry using LLMs and theorem provers.
problem Challenges in formalizing Euclidean geometry due to reliance on diagrams.
method Combines neuro-symbolic framework, SMT solvers, and LLMs to fill in diagrammatic gaps.
result Demonstrates the capability and limitations of LLMs on autoformalizing geometry problems.
We formalize in the proof assistant Isabelle essential basic notions and results in financial mathematics. We provide generic formal definitions of concepts such as markets, portfolios, derivative products, arbitrages or fair prices, and we show that, under the usual no-arbitrage condition, the existence of a replicati…
Consider the kernel Mag_g of the Magnus representation of the Torelli group and the kernel Bur_n of the Burau representation of the braid group. We prove that for g >= 2 and for n >= 6 the groups Mag_g and Bur_n have infinite rank first homology. As a consequence we conclude that neither group has any finite generating…
New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
problem Existence of contractible domains with specific boundary conditions.
method Local bifurcation argument around geodesic disks, anisotropic Hölder spaces, computer-assisted techniques.
result Existence of nontrivial contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.
Study evaluates the impact of academic support center's face-to-face assistance on student performance.
problem Underestimation of Academic Support Center's true impact due to group bias.
method Applied causal inference theory and T-learner to evaluate conditional average treatment effect (CATE) of F2F personal assistance.
result Developed a new CATE function that depends on the number of F2F sessions, predicting improved CATE performance.
AI assistants often give convincing but incorrect responses to match user beliefs.
problem Sycophancy in AI assistants that use human feedback.
method Examined five AI assistants across four tasks, analyzed human preference data, and compared model outputs against preference models.
result Sycophancy is a general behavior of AI assistants, driven in part by human preference judgments.
Develops a two-layer model to design mortgage assistance products.
problem Designing effective mortgage assistance products to improve household resilience.
method Two-layer approach: simulation and optimization.
result Shows how the approach can design and evaluate mortgage assistance products.
Helps visually impaired users make better decisions by adjusting their observations.
problem Systematic biases in users' perception and processing of visual information.
method Synthesizes new observations based on true observations to correct user biases.
result Significant improvement in task performance for users with various biases.
Robust Bayes-Assisted Conformal Prediction improves prediction set sizes.
problem Misspecification of Bayesian working model and prior misalignment.
method RoBAS (Robust Bayes-Assisted Shrinkage) framework with two instantiations.
result Improves prediction set sizes in shifted settings.
Robust Bayes-Assisted Conformal Prediction improves prediction set sizes.
problem Misspecification of Bayesian working model and prior misalignment.
method RoBAS (Robust Bayes-Assisted Shrinkage) framework with two instantiations.
result Proposed scores adapt to prior quality, reducing interval widths in shifted settings.
We formalize and verify double auctions for multiple-quantity trades.
problem Matching multiple-quantity trade requests in double auctions.
method Formalized algorithms, correctness proofs, Coq proof assistant, verified OCaml and Haskell programs.
result Automatic detection of violations in exchange systems.
AI-assisted interviews allow respondents to describe experiences naturally, but mapping those accounts into structured survey variables is fallible.
problem Mapping AI-assisted interview responses into structured survey variables is fallible.
method Adaptive Matrix Validation (AMV) is proposed, which involves mapping responses into tabular data and using a small set of structured questions for statistical adjustment.
result The estimator calibrates mapped values using validation answers from other respondents and corrects remaining error with validation answers observed for the target respondent.
Kähler thresholdsmath.AG
We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive.
problem The topology of Kähler manifolds is largely determined by the geometry due to its rigidity.
method We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive.
result We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive.
We study data-driven assistants that provide congestion forecasts to users of shared facilities (roads, cafeterias, etc.), to support coordination between them, and increase efficiency of such collective systems. Key questions are: (1) when and how much can (accurate) predictions help for coordination, and (2) which as…
Generative AI at Workecon.GN
Generative AI boosts productivity and improves customer service quality.
problem Productivity and quality of customer support agents.
method Staggered introduction of a generative AI-based conversational assistant in customer support.
result AI increases productivity by 15% on average, with significant heterogeneity across workers.
This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This technique is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to provide a new lower bound for the volume of a closed orientable hyperbolic 3-manifold. We prove…
Developed a machine-checked Itô calculus for Brownian motion.
problem Formal verification of Itô calculus for Brownian motion.
method Machine-checked formalization in Lean over Mathlib.
result First machine-checked constructions of the Itô integral and Itô's formula.
A green simulation-assisted reinforcement learning method for biomanufacturing.
problem Complexity, high variability, lead time, and limited historical data in biopharmaceutical manufacturing.
method Quantifies model risk, uses posterior distribution, and selectively reuses simulation data.
result Demonstrates promising performance in online learning and decision making.