Recent advances in reasoning techniques have substantially improved the performance of large language models (LLMs), raising expectations for their ability to provide accurate, truthful, and reliable information. However, emerging evidence suggests that iterative reasoning may foster belief entrenchment, rather than enhancing truth-seeking behavior. In this study, we propose a systematic evaluation framework for belief entrenchment in LLM reasoning by leveraging the Martingale property from Bayesian statistics. This property implies that, under rational belief updating, the expected value of future beliefs should remain equal to the current belief, i.e., belief updates cannot be predicted from solely the current belief. We propose the unsupervised, regression-based Martingale Score to measure violations of this property, signaling a deviation from the Bayesian ability of updating on new evidence. In open-ended problem domains, including event forecasting, value-laden questions, and academic paper review, we found such violations to be widespread across models, reasoning paradigms, problem domains, and system prompts, where the future beliefs are consistently predictable from the model's current belief, a phenomenon which we term belief entrenchment. Through comprehensive experiments, we identify the models (e.g., GPT-4o), reasoning techniques (e.g., chain of thought), and domains (e.g., forecasting) more prone to belief entrenchment. Finally, we validate the Martingale Score by showing that it predicts ground-truth accuracy on problem domains where ground truth labels are available. This indicates that, while designed as an unsupervised metric that operates even in domains without access to ground truth, the Martingale Score is a useful proxy of the truth-seeking ability of the LLM reasoning process.

We introduce Neural Green's Function, a neural solution operator for linear partial dif sitions.

Reasoning based on Large Language Models (LLMs) has garnered increasing attention due to outstanding performance of these models in mathematical and complex logical tasks. Beginning with the Chain-of-Thought (CoT) prompting technique, numerous reasoning methods have emerged that decompose problems into smaller, sequential steps (or thoughts). However, existing reasoning models focus on conventional problem-solving and do not necessarily generate creative solutions by ``creative reasoning''. In domains where the solution space is expansive and conventional solutions are suboptimal, such as drug discovery or business strategization, creative reasoning to discover innovative solutions is crucial. To address this gap, first we introduce a computational framework for creative reasoning inspired by established cognitive science principles. With this framework, we propose three core creative reasoning paradigms, namely, \textit{combinational}, \textit{exploratory}, and \textit{transformative} reasoning, where each offers specific directions for systematic exploration of the universe of thoughts to generate creative solutions. Next, to materialize this framework using LLMs, we introduce the \textit{Universe of Thoughts} (or \textit{UoT}, for short), a novel set of methods to implement the aforementioned three creative processes. Finally, we introduce three novel tasks that necessitate creative problem-solving, along with an evaluation benchmark to assess creativity from three orthogonal perspectives: feasibility as constraint, and utility and novelty as metrics. With a comparative analysis against the state-of-the-art (SOTA) reasoning techniques as well as representative commercial models with reasoning capability, we show that UoT demonstrates superior performance in creative reasoning.

We introduce Neural Green's Function, a neural solution operator for linear partial differential equations (PDEs) whose differential operators admit eigendecompositions. Inspired by Green's functions, the solution operators of linear PDEs that depend exclusively on the domain geometry, we design Neural Green's Function to imitate their behavior, achieving superior generalization across diverse irregular geometries and source and boundary functions. Specifically, Neural Green's Function extracts per-point features from a volumetric point cloud representing the problem domain and uses them to predict a decomposition of the solution operator, which is subsequently applied to evaluate solutions via numerical integration. Unlike recent learning-based solution operators, which often struggle to generalize to unseen source or boundary functions, our framework is, by design, agnostic to the specific functions used during training, enabling robust and efficient generalization. In the steady-state thermal analysis of mechanical part geometries from the MCB dataset, Neural Green's Function outperforms state-of-the-art neural operators, achieving an average error reduction of 13.9\% across five shape categories, while being up to 350 times faster than a numerical solver that requires computationally expensive meshing.

Despite huge advances, LLMs still lack convenient and reliable methods to quantify the uncertainty in their responses, making them difficult to trust in high-stakes applications. One of the simplest approaches to eliciting more accurate answers is to select the mode of many responses, a technique known as ensembling. In this work, we expand on typical ensembling approaches by looking at ensembles with a variable voting threshold. We introduce a theoretical framework for question answering and show that, by permitting ensembles to "abstain" from providing an answer when the dominant response falls short of the threshold, it is possible to dramatically increase the trustworthiness of the remaining answers. From this framework, we derive theoretical results as well as report experimental results on two problem domains: arithmetic problem solving and clinical-note question-answering. In both domains, we observe that large gains in answer trustworthiness can be achieved using highly restrictive voting ensembles, while incurring relatively modest reductions in response yield and accuracy. Due to this quality, voting ensembles may be particularly useful in applications - such as healthcare and data annotation - that require a high degree of certainty but which may not require that every question receive an automated answer.

Physics Informed Neural Networks offer a mesh free framework for solving PDEs but are highly sensitive to loss weight selection. We propose two dimensional analysis based weighting schemes, one based on quantifiable terms, and another also incorporating unquantifiable terms for more balanced training. Benchmarks on heat conduction, convection diffusion, and lid driven cavity flows show that the second scheme consistently improves stability and accuracy over equal weighting. Notably, in high Peclet number convection diffusion, where traditional solvers fail, PINNs with our scheme achieve stable, accurate predictions, highlighting their robustness and generalizability in CFD problems.

-- While much research explores improving robot capabilities, there is a deficit in researching how robots are expected to perform tasks safely, especially in high-risk problem domains. Robots must earn the trust of human operators in order to be effective collaborators in safety-critical tasks, specifically those where robots operate in human environments. We propose the human-robot red teaming paradigm for safety-aware reasoning . We expect humans and robots to work together to challenge assumptions about an environment and explore the space of hazards that may arise. This exploration will enable robots to perform safety-aware reasoning, specifically hazard identification, risk assessment, risk mitigation, and safety reporting. We demonstrate that: (a) human-robot red teaming allows human-robot teams to plan to perform tasks safely in a variety of domains, and (b) robots with different embodiments can learn to operate safely in two different environments--a lunar habitat and a household--with varying definitions of safety. T aken together, our work on human-robot red teaming for safety-aware reasoning demonstrates the feasibility of this approach for safely operating and promoting trust on human-robot teams in safety-critical problem domains. I. INTRODUCTION Enabling robots to reason over risks is a crucial capability of performing collaborative assistive tasks in safety-critical domains.

Over the years, genetic programming (GP) has evolved, with many proposed variations, especially in how they represent a solution. Being essentially a program synthesis algorithm, it is capable of tackling multiple problem domains. Current benchmarking initiatives are fragmented, as the different representations are not compared with each other and their performance is not measured across the different domains. In this work, we propose a unified framework, dubbed TinyverseGP (inspired by tinyGP), which provides support to multiple representations and problem domains, including symbolic regression, logic synthesis and policy search.

We thank the reviewers for detailed and helpful reviews. This is incremental overall ... low technical novelty..." We indeed adopt a common CNN architecture (specifically VGG) as is now typical in vision tasks. However, we adopt the network to a new problem domain: generating segmentation object proposals. For this problem domain, all previous approaches operate by merging or grouping superpixels or edges (with no or only minor data-driven components). Instead, we formulate the problem directly as a learning problem.