Model-Adaptive Tool Necessity Reveals the Knowing-Doing Gap in LLM Tool Use

Large language models (LLMs) increasingly act as autonomous agents that must decide when to answer directly vs. when to invoke external tools. Prior work studying adaptive tool use has largely treated tool necessity as a model-agnostic property, annotated by human or LLM judge, and mostly cover cases where the answer is obvious (e.g., fetching the weather vs. paraphrasing text). However, tool necessity in the wild is more nuanced due to the divergence of capability boundaries across models: a problem solvable by a strong model on its own may still require tools for a weaker one. In this work, we introduce a model-adaptive definition of tool-necessity, grounded in each model's empirical performance. Following this definition, we compare the necessity against observed tool-call behavior across four models on arithmetic and factual QA dataset, and find substantial mismatches of 26.5-54.0% and 30.8-41.8%, respectively. To diagnose the failure, we decompose tool use into two stages: an internal cognition stage that reflects whether a model believes a tool is necessary, and an execution stage that determines whether the model actually makes a tool-call action. By probing the LLM hidden states, we find that both signals are often linearly decodable, yet their probe directions become nearly orthogonal in the late-layer, last-token regime that drives the next-token action. By tracing the trajectory of samples in the two-stage process, we further discover that the majority of mismatch is concentrated in the cognition-to-action transition, not in cognition itself. These results reveal a knowing-doing gap in LLM tool-use: improving tool-use reliability requires not only better recognition of when tools are needed, but also better translation of that recognition into action.

Accelerating Scientific Research with Gemini: Case Studies and Common Techniques

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

Failing to Explore: Language Models on Interactive Tasks

We evaluate language models on their ability to explore interactive environments under a limited interaction budget. We introduce three parametric tasks with controllable exploration difficulty, spanning continuous and discrete environments. Across state-of-the-art models, we find systematic under-exploration and suboptimal solutions, with performance often significantly worse than simple explore--exploit heuristic baselines and scaling weakly as the budget increases. Finally, we study two lightweight interventions: splitting a fixed budget into parallel executions, which surprisingly improves performance despite a no-gain theoretical result for our tasks, and periodically summarizing the interaction history, which preserves key discoveries and further improves exploration.

Value Iteration with Guessing for Markov Chains and Markov Decision Processes

Two standard models for probabilistic systems are Markov chains (MCs) and Markov decision processes (MDPs). Classic objectives for such probabilistic models for control and planning problems are reachability and stochastic shortest path. The widely studied algorithmic approach for these problems is the Value Iteration (VI) algorithm which iteratively applies local updates called Bellman updates. There are many practical approaches for VI in the literature but they all require exponentially many Bellman updates for MCs in the worst case. A preprocessing step is an algorithm that is discrete, graph-theoretical, and requires linear space. An important open question is whether, after a polynomial-time preprocessing, VI can be achieved with sub-exponentially many Bellman updates. In this work, we present a new approach for VI based on guessing values. Our theoretical contributions are twofold. First, for MCs, we present an almost-linear-time preprocessing algorithm after which, along with guessing values, VI requires only subexponentially many Bellman updates. Second, we present an improved analysis of the speed of convergence of VI for MDPs. Finally, we present a practical algorithm for MDPs based on our new approach. Experimental results show that our approach provides a considerable improvement over existing VI-based approaches on several benchmark examples from the literature.