Daily Papers

Auto-formalization (AF) translates natural-language reasoning problems into solver-executable programs, enabling symbolic solvers to perform sound logical deduction. In practice, however, AF pipelines are currently brittle: programs may fail to execute, or execute but encode incorrect semantics. While prior work largely mitigates syntactic failures via repairs based on solver feedback, reducing semantics failures remains a major bottleneck. We propose Draft-and-Prune (D&P), an inference-time framework that improves AF-based logical reasoning via diversity and verification. D&P first drafts multiple natural-language plans and conditions program generation on them. It further prunes executable but contradictory or ambiguous formalizations, and aggregates predictions from surviving paths via majority voting. Across four representative benchmarks (AR-LSAT, ProofWriter, PrOntoQA, LogicalDeduction), D&P substantially strengthens AF-based reasoning without extra supervision. On AR-LSAT, in the AF-only setting, D&P achieves 78.43% accuracy with GPT-4 and 78.00% accuracy with GPT-4o, significantly outperforming the strongest AF baselines MAD-LOGIC and CLOVER. D&P then attains near-ceiling performance on the other benchmarks, including 100% on PrOntoQA and LogicalDeduction.

Optimization modeling enables critical decisions across industries but remains difficult to automate: informal language must be mapped to precise mathematical formulations and executable solver code. Prior LLM approaches either rely on brittle prompting or costly retraining with limited generalization. We present AlphaOPT, a self-improving experience library that enables an LLM to learn from limited demonstrations (even answers alone, without gold-standard programs) and solver feedback - without annotated reasoning traces or parameter updates. AlphaOPT operates in a continual two-phase cycle: (i) a Library Learning phase that reflects on failed attempts, extracting solver-verified, structured insights as {taxonomy, condition, explanation, example}; and (ii) a Library Evolution phase that diagnoses retrieval misalignments and refines the applicability conditions of stored insights, improving transfer across tasks. This design (1) learns efficiently from limited demonstrations without curated rationales, (2) expands continually without costly retraining by updating the library rather than model weights, and (3) makes knowledge explicit and interpretable for human inspection and intervention. Experiments show that AlphaOPT steadily improves with more data (65% to 72% from 100 to 300 training items) and surpasses the strongest baseline by 7.7% on the out-of-distribution OptiBench dataset when trained only on answers. Code and data are available at: https://github.com/Minw913/AlphaOPT.

Existing math datasets evaluate the reasoning abilities of large language models (LLMs) by either using the final answer or the intermediate reasoning steps derived from static examples. However, the former approach fails to surface model's uses of shortcuts and wrong reasoning while the later poses challenges in accommodating alternative solutions. In this work, we seek to use symbolic programs as a means for automated evaluation if a model can consistently produce correct final answers across various inputs to the program. We begin by extracting programs for popular math datasets (GSM8K and MATH) using GPT4-o. For those executable programs verified using the original input-output pairs, they are found to encapsulate the proper reasoning required to solve the original text questions. We then prompt GPT4-o to generate new questions using alternative input-output pairs based the extracted program. We apply the resulting datasets to evaluate a collection of LLMs. In our experiments, we observe significant accuracy drops using our proposed evaluation compared with original static examples, suggesting the fragility of math reasoning in state-of-the-art LLMs.

We introduce FrontierCS, a benchmark of 156 open-ended problems across diverse areas of computer science, designed and reviewed by experts, including CS PhDs and top-tier competitive programming participants and problem setters. Unlike existing benchmarks that focus on tasks with known optimal solutions, FrontierCS targets problems where the optimal solution is unknown, but the quality of a solution can be objectively evaluated. Models solve these tasks by implementing executable programs rather than outputting a direct answer. FrontierCS includes algorithmic problems, which are often NP-hard variants of competitive programming problems with objective partial scoring, and research problems with the same property. For each problem we provide an expert reference solution and an automatic evaluator. Combining open-ended design, measurable progress, and expert curation, FrontierCS provides a benchmark at the frontier of computer-science difficulty. Empirically, we find that frontier reasoning models still lag far behind human experts on both the algorithmic and research tracks, that increasing reasoning budgets alone does not close this gap, and that models often over-optimize for generating merely workable code instead of discovering high-quality algorithms and system designs.

Scientists often infer abstract procedures from specific instances of problems and use the abstractions to generate new, related instances. For example, programs encoding the formal rules and properties of a system have been useful in fields ranging from RL (procedural environments) to physics (simulation engines). These programs can be seen as functions which execute to different outputs based on their parameterizations (e.g., gridworld configuration or initial physical conditions). We introduce the term EFA (Executable Functional Abstraction) to denote such programs for math problems. EFA-like constructs have been shown to be useful for math reasoning as problem generators for stress-testing models. However, prior work has been limited to abstractions for grade-school math (whose simple rules are easy to encode in programs), while generating EFAs for advanced math has thus far required human engineering. We explore the automatic construction of EFAs for advanced math problems. We operationalize the task of automatically constructing EFAs as a program synthesis task, and develop EFAGen, which conditions an LLM on a seed math problem and its step-by-step solution to generate candidate EFA programs that are faithful to the generalized problem and solution class underlying the seed problem. Furthermore, we formalize properties any valid EFA must possess in terms of executable unit tests, and show how the tests can be used as verifiable rewards to train LLMs to become better writers of EFAs. We demonstrate that EFAs constructed by EFAGen behave rationally by remaining faithful to seed problems, produce learnable problem variations, and that EFAGen can infer EFAs across multiple diverse sources of competition-level math problems. Finally, we show downstream uses of model-written EFAs e.g. finding problem variations that are harder or easier for a learner to solve, as well as data generation.

Large linear systems play an important role in high-energy theory, appearing in amplitude bootstraps and during integral reduction. This paper introduces FiniteFieldSolve, a general-purpose toolkit for exactly solving large linear systems over the rationals. The solver interfaces directly with Mathematica, is straightforward to install, and seamlessly replaces Mathematica's native solvers. In testing, FiniteFieldSolve is approximately two orders of magnitude faster than Mathematica and uses an order of magnitude less memory. The package also compares favorably against other public solvers in FiniteFieldSolve's intended use cases. As the name of the package suggests, solutions are obtained via well-known finite field methods. These methods suffer from introducing an inordinate number of modulo (or integer division) operations with respect to different primes. By automatically recompiling itself for each prime, FiniteFieldSolve converts the division operations into much faster combinations of instructions, dramatically improving performance. The technique of compiling the prime can be applied to any finite field solver, where the time savings will be solver dependent. The operation of the package is illustrated through a detailed example of an amplitude bootstrap.

Solving algebraic word problems requires executing a series of arithmetic operations---a program---to obtain a final answer. However, since programs can be arbitrarily complicated, inducing them directly from question-answer pairs is a formidable challenge. To make this task more feasible, we solve these problems by generating answer rationales, sequences of natural language and human-readable mathematical expressions that derive the final answer through a series of small steps. Although rationales do not explicitly specify programs, they provide a scaffolding for their structure via intermediate milestones. To evaluate our approach, we have created a new 100,000-sample dataset of questions, answers and rationales. Experimental results show that indirect supervision of program learning via answer rationales is a promising strategy for inducing arithmetic programs.

Optimization problems are pervasive in sectors from manufacturing and distribution to healthcare. However, most such problems are still solved heuristically by hand rather than optimally by state-of-the-art solvers because the expertise required to formulate and solve these problems limits the widespread adoption of optimization tools and techniques. This paper introduces OptiMUS, a Large Language Model (LLM)-based agent designed to formulate and solve (mixed integer) linear programming problems from their natural language descriptions. OptiMUS can develop mathematical models, write and debug solver code, evaluate the generated solutions, and improve its model and code based on these evaluations. OptiMUS utilizes a modular structure to process problems, allowing it to handle problems with long descriptions and complex data without long prompts. Experiments demonstrate that OptiMUS outperforms existing state-of-the-art methods on easy datasets by more than 20% and on hard datasets (including a new dataset, NLP4LP, released with this paper that features long and complex problems) by more than 30%.

Algorithmic reasoning refers to the ability to understand the complex patterns behind the problem and decompose them into a sequence of reasoning steps towards the solution. Such nature of algorithmic reasoning makes it a challenge for large language models (LLMs), even though they have demonstrated promising performance in other reasoning tasks. Within this context, some recent studies use programming languages (e.g., Python) to express the necessary logic for solving a given instance/question (e.g., Program-of-Thought) as inspired by their strict and precise syntaxes. However, it is non-trivial to write an executable code that expresses the correct logic on the fly within a single inference call. Also, the code generated specifically for an instance cannot be reused for others, even if they are from the same task and might require identical logic to solve. This paper presents Think-and-Execute, a novel framework that decomposes the reasoning process of language models into two steps. (1) In Think, we discover a task-level logic that is shared across all instances for solving a given task and then express the logic with pseudocode; (2) In Execute, we further tailor the generated pseudocode to each instance and simulate the execution of the code. With extensive experiments on seven algorithmic reasoning tasks, we demonstrate the effectiveness of Think-and-Execute. Our approach better improves LMs' reasoning compared to several strong baselines performing instance-specific reasoning (e.g., CoT and PoT), suggesting the helpfulness of discovering task-level logic. Also, we show that compared to natural language, pseudocode can better guide the reasoning of LMs, even though they are trained to follow natural language instructions.

We introduce a new type of programming challenge called programming puzzles, as an objective and comprehensive evaluation of program synthesis, and release an open-source dataset of Python Programming Puzzles (P3). Each puzzle is defined by a short Python program f, and the goal is to find an input which makes f return True. The puzzles are objective in that each one is specified entirely by the source code of its verifier f, so evaluating f is all that is needed to test a candidate solution. They do not require an answer key or input/output examples, nor do they depend on natural language understanding. The dataset is comprehensive in that it spans problems of a range of difficulties and domains, ranging from trivial string manipulation problems, to classic programming puzzles (e.g., Tower of Hanoi), to interview/competitive-programming problems (e.g., dynamic programming), to longstanding open problems in algorithms and mathematics (e.g., factoring). We develop baseline enumerative program synthesis, GPT-3 and Codex solvers that are capable of solving puzzles -- even without access to any reference solutions -- by learning from their own past solutions. Codex performs best, solving up to 18% of 397 test problems with a single try and 80% of the problems with 1,000 tries per problem. In a small user study, we find a positive correlation between puzzle-solving performance and coding experience, and between the puzzle difficulty for humans and AI solvers. Therefore, further improvements on P3 could have a significant impact on many program synthesis areas.

Model-predictive control (MPC) is a powerful framework for controlling dynamic systems under constraints, but it remains challenging to deploy on resource-constrained platforms, especially for problems involving conic constraints. To address this, we extend recent work developing fast, structure-exploiting, cached ADMM solvers for embedded applications, to provide support for second-order cones, as well as C++ code generation from Python, MATLAB, and Julia for easy deployment. Microcontroller benchmarks show that our solver provides up to a two-order-of-magnitude speedup, ranging from 10.6x to 142.7x, over state-of-the-art embedded solvers on QP and SOCP problems, and enables us to fit order-of-magnitude larger problems in memory. We validate our solver's deployed performance through simulation and hardware experiments, including conically-constrained trajectory tracking on a 27g Crazyflie quadrotor. To get started with Conic-TinyMPC, visit our documentation, examples, and the open-source codebase at https://tinympc.org.

Code generation models have increasingly become integral to aiding software development, offering assistance in tasks such as code completion, debugging, and code translation. Although current research has thoroughly examined the correctness of code produced by code generation models, a vital aspect, i.e., the efficiency of the generated code, has often been neglected. This paper presents EffiBench, a benchmark with 1,000 efficiency-critical coding problems for assessing the efficiency of code generated by code generation models. EffiBench contains a diverse set of LeetCode coding problems. Each problem is paired with an executable human-written canonical solution. With EffiBench, we empirically examine the capability of 21 Large Language Models (13 open-sourced and 8 closed-sourced) in generating efficient code. The results demonstrate that GPT-4-turbo generates the most efficient code, significantly outperforming Palm-2-chat-bison, Claude-instant-1, Gemini-pro, GPT-4, and GPT-3.5. Nevertheless, its code efficiency is still worse than the efficiency of human-written canonical solutions. In particular, the average and worst execution time of GPT-4-turbo generated code is 1.69 and 45.49 times that of the canonical solutions.

The correctness of computations remains a significant challenge in computer science, with traditional approaches relying on automated testing or formal verification. Self-testing/correcting programs introduce an alternative paradigm, allowing a program to verify and correct its own outputs via randomized reductions, a concept that previously required manual derivation. In this paper, we present Bitween, a method and tool for automated learning of randomized (self)-reductions and program properties in numerical programs. Bitween combines symbolic analysis and machine learning, with a surprising finding: polynomial-time linear regression, a basic optimization method, is not only sufficient but also highly effective for deriving complex randomized self-reductions and program invariants, often outperforming sophisticated mixed-integer linear programming solvers. We establish a theoretical framework for learning these reductions and introduce RSR-Bench, a benchmark suite for evaluating Bitween's capabilities on scientific and machine learning functions. Our empirical results show that Bitween surpasses state-of-the-art tools in scalability, stability, and sample efficiency when evaluated on nonlinear invariant benchmarks like NLA-DigBench. Bitween is open-source as a Python package and accessible via a web interface that supports C language programs.

Large vision language models exhibit notable limitations on Geometry Problem Solving (GPS) because of their unreliable diagram interpretation and pure natural-language reasoning. A recent line of work mitigates this by using symbolic solvers: the model directly generates a formal program that a geometry solver can execute. However, this direct program generation lacks intermediate reasoning, making the decision process opaque and prone to errors. In this work, we explore a new approach that integrates Chain-of-Thought (CoT) with formal language. The model interleaves natural language reasoning with incremental emission of solver-executable code, producing a hybrid reasoning trace in which critical derivations are expressed in formal language. To teach this behavior at scale, we combine (1) supervised fine-tuning on an 11K newly developed synthetic dataset with interleaved natural language reasoning and automatic formalization, and (2) solver-in-the-loop reinforcement learning that jointly optimizes both the CoT narrative and the resulting program through outcome-based rewards. Built on Qwen2.5-VL-7B, our new model, named GF-Reasoner, achieves up to 15% accuracy improvements on standard GPS benchmarks, surpassing both 7B-scale peers and the much larger model Qwen2.5-VL-72B. By exploiting high-order geometric knowledge and offloading symbolic computation to the solver, the generated reasoning traces are noticeably shorter and cleaner. Furthermore, we present a comprehensive analysis of method design choices (e.g., reasoning paradigms, data synthesis, training epochs, etc.), providing actionable insights for future research.

While large language models (LLMs) have shown strong performance in math and logic reasoning, their ability to handle combinatorial optimization (CO) -- searching high-dimensional solution spaces under hard constraints -- remains underexplored. To bridge the gap, we introduce NLCO, a Natural Language Combinatorial Optimization benchmark that evaluates LLMs on end-to-end CO reasoning: given a language-described decision-making scenario, the model must output a discrete solution without writing code or calling external solvers. NLCO covers 43 CO problems and is organized using a four-layer taxonomy of variable types, constraint families, global patterns, and objective classes, enabling fine-grained evaluation. We provide solver-annotated solutions and comprehensively evaluate LLMs by feasibility, solution optimality, and reasoning efficiency. Experiments across a wide range of modern LLMs show that high-performing models achieve strong feasibility and solution quality on small instances, but both degrade as instance size grows, even if more tokens are used for reasoning. We also observe systematic effects across the taxonomy: set-based tasks are relatively easy, whereas graph-structured problems and bottleneck objectives lead to more frequent failures.

Enhancing the mathematical reasoning of large language models (LLMs) demands high-quality training data, yet conventional methods face critical challenges in scalability, cost, and data reliability. To address these limitations, we propose a novel program-assisted synthesis framework that systematically generates a high-quality mathematical corpus with guaranteed diversity, complexity, and correctness. This framework integrates mathematical knowledge systems and domain-specific tools to create executable programs. These programs are then translated into natural language problem-solution pairs and vetted by a bilateral validation mechanism that verifies solution correctness against program outputs and ensures program-problem consistency. We have generated 12.3 million such problem-solving triples. Experiments demonstrate that models fine-tuned on our data significantly improve their inference capabilities, achieving state-of-the-art performance on several benchmark datasets and showcasing the effectiveness of our synthesis approach.

As a seemingly self-explanatory task, problem-solving has been a significant component of science and engineering. However, a general yet concrete formulation of problem-solving itself is missing. With the recent development of AI-based problem-solving agents, the demand for process-level verifiability is rapidly increasing yet underexplored. To fill these gaps, we present a principled formulation of problem-solving as a deterministic Markov decision process; a novel framework, FPS (Formal Problem-Solving), which utilizes existing FTP (formal theorem proving) environments to perform process-verified problem-solving; and D-FPS (Deductive FPS), decoupling solving and answer verification for better human-alignment. The expressiveness, soundness and completeness of the frameworks are proven. We construct three benchmarks on problem-solving: FormalMath500, a formalization of a subset of the MATH500 benchmark; MiniF2F-Solving and PutnamBench-Solving, adaptations of FTP benchmarks MiniF2F and PutnamBench. For faithful, interpretable, and human-aligned evaluation, we propose RPE (Restricted Propositional Equivalence), a symbolic approach to determine the correctness of answers by formal verification. We evaluate four prevalent FTP models and two prompting methods as baselines, solving at most 23.77% of FormalMath500, 27.47% of MiniF2F-Solving, and 0.31% of PutnamBench-Solving.

Optimization problems are pervasive across various sectors, from manufacturing and distribution to healthcare. However, most such problems are still solved heuristically by hand rather than optimally by state-of-the-art solvers, as the expertise required to formulate and solve these problems limits the widespread adoption of optimization tools and techniques. We introduce OptiMUS, a Large Language Model (LLM)-based agent designed to formulate and solve MILP problems from their natural language descriptions. OptiMUS is capable of developing mathematical models, writing and debugging solver code, developing tests, and checking the validity of generated solutions. To benchmark our agent, we present NLP4LP, a novel dataset of linear programming (LP) and mixed integer linear programming (MILP) problems. Our experiments demonstrate that OptiMUS solves nearly twice as many problems as a basic LLM prompting strategy. OptiMUS code and NLP4LP dataset are available at https://github.com/teshnizi/OptiMUS{https://github.com/teshnizi/OptiMUS}

Combinatorial problems are present in a wide range of industries. Constraint Programming (CP) is a well-suited problem-solving paradigm, but its core process, namely constraint modelling, is a bottleneck for wider adoption. Aiming to alleviate this bottleneck, recent studies have explored using Large Language Models (LLMs) as modelling assistants, transforming combinatorial problem descriptions to executable constraint models, similar to coding assistants. However, the existing evaluation datasets for constraint modelling are often limited to small, homogeneous, or domain-specific instances, which do not capture the diversity of real-world scenarios. This work addresses this gap by introducing CP-Bench, a novel benchmark dataset that includes a diverse set of well-known combinatorial problem classes sourced from the CP community, structured explicitly for evaluating LLM-driven CP modelling. With this dataset, and given the variety of constraint modelling frameworks, we compare and evaluate the modelling capabilities of LLMs for three distinct constraint modelling systems, which vary in abstraction level and underlying syntax: the high-level MiniZinc language and Python-based CPMpy library, and the lower-level Python interface of the OR-Tools CP-SAT solver. In order to enhance the ability of LLMs to produce valid constraint models, we systematically evaluate the use of prompt-based and inference-time compute methods adapted from existing LLM-based code generation research. Our results underscore the modelling convenience provided by Python-based frameworks, as well as the effectiveness of documentation-rich system prompts, which, augmented with repeated sampling and self-verification, achieve further improvements, reaching up to 70\% accuracy on this new, highly challenging benchmark.

Recent Language Models (LMs) achieve breakthrough performance in code generation when trained on human-authored problems, even solving some competitive-programming problems. Self-play has proven useful in games such as Go, and thus it is natural to ask whether LMs can generate their own instructive programming problems to improve their performance. We show that it is possible for an LM to synthesize programming problems and solutions, which are filtered for correctness by a Python interpreter. The LM's performance is then seen to improve when it is fine-tuned on its own synthetic problems and verified solutions; thus the model 'improves itself' using the Python interpreter. Problems are specified formally as programming puzzles [Schuster et al., 2021], a code-based problem format where solutions can easily be verified for correctness by execution. In experiments on publicly-available LMs, test accuracy more than doubles. This work demonstrates the potential for code LMs, with an interpreter, to generate instructive problems and improve their own performance.

Program errors can occur in any type of programming, and can manifest in a variety of ways, such as unexpected output, crashes, or performance issues. And program error diagnosis can often be too abstract or technical for developers to understand, especially for beginners. The goal of this paper is to present a novel machine-learning approach for Multi-task Program Error Repair and Explanatory Diagnosis (mPRED). A pre-trained language model is used to encode the source code, and a downstream model is specifically designed to identify and repair errors. Programs and test cases will be augmented and optimized from several perspectives. Additionally, our approach incorporates a "chain of thoughts" method, which enables the models to produce intermediate reasoning explanations before providing the final correction. To aid in visualizing and analyzing the program structure, we use a graph neural network for program structure visualization. Overall, our approach offers a promising approach for repairing program errors across different programming languages and providing helpful explanations to programmers.

A fundamental skill among human developers is the ability to understand and reason about program execution. As an example, a programmer can mentally simulate code execution in natural language to debug and repair code (aka. rubber duck debugging). However, large language models (LLMs) of code are typically trained on the surface textual form of programs, thus may lack a semantic understanding of how programs execute at run-time. To address this issue, we propose NExT, a method to teach LLMs to inspect the execution traces of programs (variable states of executed lines) and reason about their run-time behavior through chain-of-thought (CoT) rationales. Specifically, NExT uses self-training to bootstrap a synthetic training set of execution-aware rationales that lead to correct task solutions (e.g., fixed programs) without laborious manual annotation. Experiments on program repair tasks based on MBPP and HumanEval demonstrate that NExT improves the fix rate of a PaLM 2 model, by 26.1% and 14.3% absolute, respectively, with significantly improved rationale quality as verified by automated metrics and human raters. Our model can also generalize to scenarios where program traces are absent at test-time.

When writing programs, people have the ability to tackle a new complex task by decomposing it into smaller and more familiar subtasks. While it is difficult to measure whether neural program synthesis methods have similar capabilities, we can measure whether they compositionally generalize, that is, whether a model that has been trained on the simpler subtasks is subsequently able to solve more complex tasks. In this paper, we characterize several different forms of compositional generalization that are desirable in program synthesis, forming a meta-benchmark which we use to create generalization tasks for two popular datasets, RobustFill and DeepCoder. We then propose ExeDec, a novel decomposition-based synthesis strategy that predicts execution subgoals to solve problems step-by-step informed by program execution at each step. ExeDec has better synthesis performance and greatly improved compositional generalization ability compared to baselines.

Large Language Models (LLMs) increasingly succeed on competitive programming problems, yet existing evaluations conflate algorithmic reasoning with code-level implementation. We argue that competitive programming is fundamentally a problem-solving task and propose centering natural-language editorials in both solution generation and evaluation. Generating an editorial prior to code improves solve rates for some LLMs, with substantially larger gains when using expertly written gold editorials. However, even with gold editorials, models continue to struggle with implementation, while the gap between generated and gold editorials reveals a persistent problem-solving bottleneck in specifying correct and complete algorithms. Beyond pass/fail metrics, we diagnose reasoning errors by comparing model-generated editorials to gold standards using expert annotations and validate an LLM-as-a-judge protocol for scalable evaluation. We introduce a dataset of 83 ICPC-style problems with gold editorials and full test suites, and evaluate 19 LLMs, arguing that future benchmarks should explicitly separate problem solving from implementation.

Despite impressive progress on complex reasoning, current large language models (LLMs) typically operate in isolation - treating each problem as an independent attempt, without accumulating or integrating experiential knowledge. In contrast, expert problem solvers - such as Olympiad or programming contest teams - leverage a rich tapestry of experiences: absorbing mentorship from coaches, developing intuition from past problems, leveraging knowledge of tool usage and library functionality, adapting strategies based on the expertise and experiences of peers, continuously refining their reasoning through trial and error, and learning from other related problems even during competition. We introduce Xolver, a training-free multi-agent reasoning framework that equips a black-box LLM with a persistent, evolving memory of holistic experience. Xolver integrates diverse experience modalities, including external and self-retrieval, tool use, collaborative interactions, agent-driven evaluation, and iterative refinement. By learning from relevant strategies, code fragments, and abstract reasoning patterns at inference time, Xolver avoids generating solutions from scratch - marking a transition from isolated inference toward experience-aware language agents. Built on both open-weight and proprietary models, Xolver consistently outperforms specialized reasoning agents. Even with lightweight backbones (e.g., QWQ-32B), it often surpasses advanced models including Qwen3-235B, Gemini 2.5 Pro, o3, and o4-mini-high. With o3-mini-high, it achieves new best results on GSM8K (98.1%), AIME'24 (94.4%), AIME'25 (93.7%), Math-500 (99.8%), and LiveCodeBench-V5 (91.6%) - highlighting holistic experience learning as a key step toward generalist agents capable of expert-level reasoning. Code and data are available at https://kagnlp.github.io/xolver.github.io/.

Expert problem-solving is driven by powerful languages for thinking about problems and their solutions. Acquiring expertise means learning these languages -- systems of concepts, alongside the skills to use them. We present DreamCoder, a system that learns to solve problems by writing programs. It builds expertise by creating programming languages for expressing domain concepts, together with neural networks to guide the search for programs within these languages. A ``wake-sleep'' learning algorithm alternately extends the language with new symbolic abstractions and trains the neural network on imagined and replayed problems. DreamCoder solves both classic inductive programming tasks and creative tasks such as drawing pictures and building scenes. It rediscovers the basics of modern functional programming, vector algebra and classical physics, including Newton's and Coulomb's laws. Concepts are built compositionally from those learned earlier, yielding multi-layered symbolic representations that are interpretable and transferrable to new tasks, while still growing scalably and flexibly with experience.

Visual programs are executable code generated by large language models to address visual reasoning problems. They decompose complex questions into multiple reasoning steps and invoke specialized models for each step to solve the problems. However, these programs are prone to logic errors, with our preliminary evaluation showing that 58% of the total errors are caused by program logic errors. Debugging complex visual programs remains a major bottleneck for visual reasoning. To address this, we introduce VDebugger, a novel critic-refiner framework trained to localize and debug visual programs by tracking execution step by step. VDebugger identifies and corrects program errors leveraging detailed execution feedback, improving interpretability and accuracy. The training data is generated through an automated pipeline that injects errors into correct visual programs using a novel mask-best decoding technique. Evaluations on six datasets demonstrate VDebugger's effectiveness, showing performance improvements of up to 3.2% in downstream task accuracy. Further studies show VDebugger's ability to generalize to unseen tasks, bringing a notable improvement of 2.3% on the unseen COVR task. Code, data and models are made publicly available at https://github.com/shirley-wu/vdebugger/

Mixed integer linear programs are commonly solved by Branch and Bound algorithms. A key factor of the efficiency of the most successful commercial solvers is their fine-tuned heuristics. In this paper, we leverage patterns in real-world instances to learn from scratch a new branching strategy optimised for a given problem and compare it with a commercial solver. We propose FMSTS, a novel Reinforcement Learning approach specifically designed for this task. The strength of our method lies in the consistency between a local value function and a global metric of interest. In addition, we provide insights for adapting known RL techniques to the Branch and Bound setting, and present a new neural network architecture inspired from the literature. To our knowledge, it is the first time Reinforcement Learning has been used to fully optimise the branching strategy. Computational experiments show that our method is appropriate and able to generalise well to new instances.

Constraint Programming (CP) is a declarative programming paradigm that allows for modeling and solving combinatorial optimization problems, such as the Job-Shop Scheduling Problem (JSSP). While CP solvers manage to find optimal or near-optimal solutions for small instances, they do not scale well to large ones, i.e., they require long computation times or yield low-quality solutions. Therefore, real-world scheduling applications often resort to fast, handcrafted, priority-based dispatching heuristics to find a good initial solution and then refine it using optimization methods. This paper proposes a novel end-to-end approach to solving scheduling problems by means of CP and Reinforcement Learning (RL). In contrast to previous RL methods, tailored for a given problem by including procedural simulation algorithms, complex feature engineering, or handcrafted reward functions, our neural-network architecture and training algorithm merely require a generic CP encoding of some scheduling problem along with a set of small instances. Our approach leverages existing CP solvers to train an agent learning a Priority Dispatching Rule (PDR) that generalizes well to large instances, even from separate datasets. We evaluate our method on seven JSSP datasets from the literature, showing its ability to find higher-quality solutions for very large instances than obtained by static PDRs and by a CP solver within the same time limit.

Programming is a powerful and ubiquitous problem-solving tool. Developing systems that can assist programmers or even generate programs independently could make programming more productive and accessible, yet so far incorporating innovations in AI has proven challenging. Recent large-scale language models have demonstrated an impressive ability to generate code, and are now able to complete simple programming tasks. However, these models still perform poorly when evaluated on more complex, unseen problems that require problem-solving skills beyond simply translating instructions into code. For example, competitive programming problems which require an understanding of algorithms and complex natural language remain extremely challenging. To address this gap, we introduce AlphaCode, a system for code generation that can create novel solutions to these problems that require deeper reasoning. In simulated evaluations on recent programming competitions on the Codeforces platform, AlphaCode achieved on average a ranking of top 54.3% in competitions with more than 5,000 participants. We found that three key components were critical to achieve good and reliable performance: (1) an extensive and clean competitive programming dataset for training and evaluation, (2) large and efficient-to-sample transformer-based architectures, and (3) large-scale model sampling to explore the search space, followed by filtering based on program behavior to a small set of submissions.

Large language models (LLMs) have exhibited great potential in mathematical reasoning. However, there remains a performance gap in this area between existing open-source models and closed-source models such as GPT-4. In this paper, we introduce MathGenie, a novel method for generating diverse and reliable math problems from a small-scale problem-solution dataset (denoted as seed data). We augment the ground-truth solutions of our seed data and train a back-translation model to translate the augmented solutions back into new questions. Subsequently, we generate code-integrated solutions for the new questions. To ensure the correctness of the code-integrated solutions, we employ rationale-based strategy for solution verification. Various pretrained models, ranging from 7B to 70B, are trained on the newly curated data to test the effectiveness of the proposed augmentation technique, resulting in a family of models known as MathGenieLM. These models consistently outperform previous open-source models across five representative mathematical reasoning datasets, achieving state-of-the-art performance. In particular, MathGenieLM-InternLM2 achieves an accuracy of 87.7% on GSM8K and 55.7% on MATH, securing the best overall score among open-source language models.

Recent releases such as o3 highlight human-like "thinking with images" reasoning that combines structured tool use with stepwise verification, yet most open-source approaches still rely on text-only chains, rigid visual schemas, or single-step pipelines, limiting flexibility, interpretability, and transferability on complex tasks. We introduce CodeDance, which explores executable code as a general solver for visual reasoning. Unlike fixed-schema calls (e.g., only predicting bounding-box coordinates), CodeDance defines, composes, and executes code to orchestrate multiple tools, compute intermediate results, and render visual artifacts (e.g., boxes, lines, plots) that support transparent, self-checkable reasoning. To guide this process, we introduce a reward for balanced and adaptive tool-call, which balances exploration with efficiency and mitigates tool overuse. Interestingly, beyond the expected capabilities taught by atomic supervision, we empirically observe novel emergent behaviors during RL training: CodeDance demonstrates novel tool invocations, unseen compositions, and cross-task transfer. These behaviors arise without task-specific fine-tuning, suggesting a general and scalable mechanism of executable visual reasoning. Extensive experiments across reasoning benchmarks (e.g., visual search, math, chart QA) show that CodeDance not only consistently outperforms schema-driven and text-only baselines, but also surpasses advanced closed models such as GPT-4o and larger open-source models.

An AI system can create and maintain knowledge only to the extent that it can verify that knowledge itself. Recent work on long Chain-of-Thought reasoning has demonstrated great potential of LLMs on solving competitive problems, but their verification ability remains to be weak and not sufficiently investigated. In this paper, we propose Heimdall, the long CoT verification LLM that can accurately judge the correctness of solutions. With pure reinforcement learning, we boost the verification accuracy from 62.5% to 94.5% on competitive math problems. By scaling with repeated sampling, the accuracy further increases to 97.5%. Through human evaluation, Heimdall demonstrates impressive generalization capabilities, successfully detecting most issues in challenging math proofs, the type of which is not included during training. Furthermore, we propose Pessimistic Verification to extend the functionality of Heimdall to scaling up the problem solving. It calls Heimdall to judge the solutions from a solver model and based on the pessimistic principle, selects the most likely correct solution with the least uncertainty. Taking DeepSeek-R1-Distill-Qwen-32B as the solver model, Pessimistic Verification improves the solution accuracy on AIME2025 from 54.2% to 70.0% with 16x compute budget and to 83.3% with more compute budget. With the stronger solver Gemini 2.5 Pro, the score reaches 93.0%. Finally, we prototype an automatic knowledge discovery system, a ternary system where one poses questions, another provides solutions, and the third verifies the solutions. Using the data synthesis work NuminaMath for the first two components, Heimdall effectively identifies problematic records within the dataset and reveals that nearly half of the data is flawed, which interestingly aligns with the recent ablation studies from NuminaMath.

Twenty-seven years ago, E. Freuder highlighted that "Constraint programming represents one of the closest approaches computer science has yet made to the Holy Grail of programming: the user states the problem, the computer solves it". Nowadays, CP users have great modeling tools available (like Minizinc and CPMpy), allowing them to formulate the problem and then let a solver do the rest of the job, getting closer to the stated goal. However, this still requires the CP user to know the formalism and respect it. Another significant challenge lies in the expertise required to effectively model combinatorial problems. All this limits the wider adoption of CP. In this position paper, we investigate a possible approach to leverage pre-trained Large Language Models to extract models from textual problem descriptions. More specifically, we take inspiration from the Natural Language Processing for Optimization (NL4OPT) challenge and present early results with a decomposition-based prompting approach to GPT Models.

Programming-by-Examples (PBE) aims to generate an algorithm from input-output examples. Such systems are practically and theoretically important: from an end-user perspective, they are deployed to millions of people, and from an AI perspective, PBE corresponds to a very general form of few-shot inductive inference. Given the success of Large Language Models (LLMs) in code-generation tasks, we investigate here the extent to which LLMs can be said to have `solved' PBE. We experiment on classic domains such as lists and strings, and an uncommon graphics programming domain not well represented in typical pretraining data. We find that pretrained models are not effective at PBE, but that they can be fine-tuned for much higher performance, provided the test problems are in-distribution. We analyze empirically what causes these models to succeed and fail, and take steps toward understanding how to achieve better out-of-distribution generalization. Collectively these results suggest that LLMs make strong progress toward solving the typical suite of PBE tasks, potentially increasing the flexibility and applicability of PBE systems, while also identifying ways in which LLMs still fall short.

Code Language Models have been trained to generate accurate solutions, typically with no regard for runtime. On the other hand, previous works that explored execution optimisation have observed corresponding drops in functional correctness. To that end, we introduce Code-Optimise, a framework that incorporates both correctness (passed, failed) and runtime (quick, slow) as learning signals via self-generated preference data. Our framework is both lightweight and robust as it dynamically selects solutions to reduce overfitting while avoiding a reliance on larger models for learning signals. Code-Optimise achieves significant improvements in pass@k while decreasing the competitive baseline runtimes by an additional 6% for in-domain data and up to 3% for out-of-domain data. As a byproduct, the average length of the generated solutions is reduced by up to 48% on MBPP and 23% on HumanEval, resulting in faster and cheaper inference. The generated data and codebase will be open-sourced at www.open-source.link.

Large language models (LLMs) can translate natural language into optimization code, but silent failures pose a critical risk: code that executes and returns solver-feasible solutions may encode semantically incorrect formulations, creating a feasibility-correctness gap of up to 90 percentage points on compositional problems. We introduce ReLoop, addressing silent failures from two complementary directions. Structured generation decomposes code production into a four-stage reasoning chain (understand, formalize, synthesize, verify) that mirrors expert modeling practice, with explicit variable-type reasoning and self-verification to prevent formulation errors at their source. Behavioral verification detects errors that survive generation by testing whether the formulation responds correctly to solver-based parameter perturbation, without requiring ground truth -- an external semantic signal that bypasses the self-consistency problem inherent in LLM-based code review. The two mechanisms are complementary: structured generation dominates on complex compositional problems, while behavioral verification becomes the largest single contributor on problems with localized formulation defects. Together with execution recovery via IIS-enhanced diagnostics, ReLoop raises correctness from 22.6% to 31.1% and execution from 72.1% to 100.0% on the strongest model, with consistent gains across five models spanning three paradigms (foundation, SFT, RL) and three benchmarks. We additionally release RetailOpt-190, 190 compositional retail optimization scenarios targeting the multi-constraint interactions where LLMs most frequently fail.

Large Language Model (LLM) reasoning for complex tasks inherently involves a trade-off between solution accuracy and computational efficiency. The subsequent step of verification, while intended to improve performance, further complicates this landscape by introducing its own challenging trade-off: sophisticated Generative Reward Models (GenRMs) can be computationally prohibitive if naively integrated with LLMs at test-time, while simpler, faster methods may lack reliability. To overcome these challenges, we introduce FlexiVe, a novel generative verifier that flexibly balances computational resources between rapid, reliable fast thinking and meticulous slow thinking using a Flexible Allocation of Verification Budget strategy. We further propose the Solve-Detect-Verify pipeline, an efficient inference-time scaling framework that intelligently integrates FlexiVe, proactively identifying solution completion points to trigger targeted verification and provide focused solver feedback. Experiments show FlexiVe achieves superior accuracy in pinpointing errors within reasoning traces on ProcessBench. Furthermore, on challenging mathematical reasoning benchmarks (AIME 2024, AIME 2025, and CNMO), our full approach outperforms baselines like self-consistency in reasoning accuracy and inference efficiency. Our system offers a scalable and effective solution to enhance LLM reasoning at test time.

A reliable executable environment is the foundation for ensuring that large language models solve software engineering tasks. Due to the complex and tedious construction process, large-scale configuration is relatively inefficient. However, most methods always overlook fine-grained analysis of the actions performed by the agent, making it difficult to handle complex errors and resulting in configuration failures. To address this bottleneck, we propose EvoConfig, an efficient environment configuration framework that optimizes multi-agent collaboration to build correct runtime environments. EvoConfig features an expert diagnosis module for fine-grained post-execution analysis, and a self-evolving mechanism that lets expert agents self-feedback and dynamically adjust error-fixing priorities in real time. Empirically, EvoConfig matches the previous state-of-the-art Repo2Run on Repo2Run's 420 repositories, while delivering clear gains on harder cases: on the more challenging Envbench, EvoConfig achieves a 78.1% success rate, outperforming Repo2Run by 7.1%. Beyond end-to-end success, EvoConfig also demonstrates stronger debugging competence, achieving higher accuracy in error identification and producing more effective repair recommendations than existing methods.

Formulating optimization problems for industrial applications demands significant manual effort and domain expertise. While Large Language Models (LLMs) show promise in automating this process, evaluating their performance remains difficult due to the absence of robust metrics. Existing solver-based approaches often face inconsistency, infeasibility issues, and high computational costs. To address these issues, we propose ORGEval, a graph-theoretic evaluation framework for assessing LLMs' capabilities in formulating linear and mixed-integer linear programs. ORGEval represents optimization models as graphs, reducing equivalence detection to graph isomorphism testing. We identify and prove a sufficient condition, when the tested graphs are symmetric decomposable (SD), under which the Weisfeiler-Lehman (WL) test is guaranteed to correctly detect isomorphism. Building on this, ORGEval integrates a tailored variant of the WL-test with an SD detection algorithm to evaluate model equivalence. By focusing on structural equivalence rather than instance-level configurations, ORGEval is robust to numerical variations. Experimental results show that our method can successfully detect model equivalence and produce 100\% consistent results across random parameter configurations, while significantly outperforming solver-based methods in runtime, especially on difficult problems. Leveraging ORGEval, we construct the Bench4Opt dataset and benchmark state-of-the-art LLMs on optimization modeling. Our results reveal that although optimization modeling remains challenging for all LLMs, DeepSeek-V3 and Claude-Opus-4 achieve the highest accuracies under direct prompting, outperforming even leading reasoning models.

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.

Code efficiency is a fundamental aspect of software quality, yet how to harness large language models (LLMs) to optimize programs remains challenging. Prior approaches have sought for one-shot rewriting, retrieved exemplars, or prompt-based search, but they do not explicitly distill reusable optimization knowledge, which limits generalization beyond individual instances. In this paper, we present EffiSkill, a framework for code-efficiency optimization that builds a portable optimization toolbox for LLM-based agents. The key idea is to model recurring slow-to-fast transformations as reusable agent skills that capture both concrete transformation mechanisms and higher-level optimization strategies. EffiSkill adopts a two-stage design: Stage I mines Operator and Meta Skills from large-scale slow/fast program pairs to build a skill library; Stage II applies this library to unseen programs through execution-free diagnosis, skill retrieval, plan composition, and candidate generation, without runtime feedback. Results on EffiBench-X show that EffiSkill achieves higher optimization success rates, improving over the strongest baseline by 3.69 to 12.52 percentage points across model and language settings. These findings suggest that mechanism-level skill reuse provides a useful foundation for execution-free code optimization, and that the resulting skill library can serve as a reusable resource for broader agent workflows.

Effective code generation with language models hinges on two critical factors: accurately understanding the intent of the prompt and generating code that applies algorithmic reasoning to produce correct solutions capable of passing diverse test cases while adhering to the syntax of the target programming language. Unlike other language tasks, code generation requires more than accurate token prediction; it demands comprehension of solution-level and structural relationships rather than merely generating the most likely tokens. very large language model (VLLM) are capable of generating detailed steps toward the correct solution of complex tasks where reasoning is crucial in solving the problem. Such reasoning capabilities may be absent in smaller language models. Therefore, in this work, we distill the reasoning capabilities of a VLLM into a smaller, more efficient model that is faster and cheaper to deploy. Our approach trains the model to emulate the reasoning and problem-solving abilities of the VLLM by learning to identify correct solution pathways and establishing a structural correspondence between problem definitions and potential solutions through a novel method of structure-aware loss optimization. This enables the model to transcend token-level generation and to deeply grasp the overarching structure of solutions for given problems. Experimental results show that our fine-tuned model, developed through a cheap and simple to implement process, significantly outperforms our baseline model in terms of pass@1, average data flow, and average syntax match metrics across the MBPP, MBPP Plus, and HumanEval benchmarks.

Satisfiability Modulo Counting (SMC) is a recently proposed general language to reason about problems integrating statistical and symbolic Artificial Intelligence. An SMC problem is an extended SAT problem in which the truth values of a few Boolean variables are determined by probabilistic inference. Approximate solvers may return solutions that violate constraints. Directly integrating available SAT solvers and probabilistic inference solvers gives exact solutions but results in slow performance because of many back-and-forth invocations of both solvers. We propose KOCO-SMC, an integrated exact SMC solver that efficiently tracks lower and upper bounds in the probabilistic inference process. It enhances computational efficiency by enabling early estimation of probabilistic inference using only partial variable assignments, whereas existing methods require full variable assignments. In the experiment, we compare KOCO-SMC with currently available approximate and exact SMC solvers on large-scale datasets and real-world applications. The proposed KOCO-SMC finds exact solutions with much less time.

We propose Step-by-Step Coding (SBSC): a multi-turn math reasoning framework that enables Large Language Models (LLMs) to generate sequence of programs for solving Olympiad level math problems. At each step/turn, by leveraging the code execution outputs and programs of previous steps, the model generates the next sub-task and the corresponding program to solve it. This way, SBSC, sequentially navigates to reach the final answer. SBSC allows more granular, flexible and precise approach to problem-solving compared to existing methods. Extensive experiments highlight the effectiveness of SBSC in tackling competition and Olympiad-level math problems. For Claude-3.5-Sonnet, we observe SBSC (greedy decoding) surpasses existing state-of-the-art (SOTA) program generation based reasoning strategies by absolute 10.7% on AMC12, 8% on AIME and 12.6% on MathOdyssey. Given SBSC is multi-turn in nature, we also benchmark SBSC's greedy decoding against self-consistency decoding results of existing SOTA math reasoning strategies and observe performance gain by absolute 6.2% on AMC, 6.7% on AIME and 7.4% on MathOdyssey.

Proof-oriented programs mix computational content with proofs of program correctness. However, the human effort involved in programming and proving is still substantial, despite the use of Satisfiability Modulo Theories (SMT) solvers to automate proofs in languages such as F*. Seeking to spur research on using AI to automate the construction of proof-oriented programs, we curate a dataset of 600K lines of open-source F* programs and proofs, including software used in production systems ranging from Windows and Linux, to Python and Firefox. Our dataset includes around 32K top-level F* definitions, each representing a type-directed program and proof synthesis problem -- producing a definition given a formal specification expressed as an F* type. We provide a program-fragment checker that queries F* to check the correctness of candidate solutions. We believe this is the largest corpus of SMT-assisted program proofs coupled with a reproducible program-fragment checker. Grounded in this dataset, we investigate the use of AI to synthesize programs and their proofs in F*, with promising results. Our main finding in that the performance of fine-tuned smaller language models (such as Phi-2 or StarCoder) compare favorably with large language models (such as GPT-4), at a much lower computational cost. We also identify various type-based retrieval augmentation techniques and find that they boost performance significantly. With detailed error analysis and case studies, we identify potential strengths and weaknesses of models and techniques and suggest directions for future improvements.

Mathematical programming -- the task of expressing operations and decision-making problems in precise mathematical language -- is fundamental across domains, yet remains a skill-intensive process requiring operations research expertise. Recent advances in large language models for complex reasoning have spurred interest in automating this task, translating natural language into executable optimization models. Current approaches, however, achieve limited accuracy, hindered by scarce and noisy training data without leveraging domain knowledge. In this work, we systematically integrate optimization expertise to improve formulation accuracy for mixed-integer linear programming, a key family of mathematical programs. Our approach first cleans training data through class-based error analysis to explicitly prevent common mistakes within each optimization class. We then develop multi-turn inference strategies that guide LLMs with class-specific error summaries and solver feedback, enabling iterative refinement. Experiments across multiple base LLMs demonstrate that combining cleaned data with domain-informed prompting and feedback improves formulation accuracy by 14 percentage points on average, enabling further progress toward robust LLM-assisted optimization formulation.

This note describes the development of an exact solver for Minimal Directed Feedback Vertex Set as part of the PACE 2022 competition. The solver is powered largely by aggressively trying to reduce the DFVS problem to a Minimal Cover problem, and applying reduction rules adapted from Vertex Cover literature. The resulting problem is solved as an Integer Linear Program (ILP) using SCIP. The resulting solver performed the second-best in the competition, although a bug at submission time disqualified it. As an additional note, we describe a new vertex cover reduction generalizing the Desk reduction rule.

Tool-augmented LLMs are increasingly deployed as agents that interleave natural-language reasoning with executable Python actions, as in CodeAct-style frameworks. In deployment, these agents rely on runtime state that persists across steps. By contrast, common training pipelines treat agent traces as token sequences, with execution semantics left implicit. This raises a data-centric question: Is state persistence merely an inference-time scaffold, or can models learn to exploit it when training data exposes the corresponding execution semantics? We isolate state persistence as a training-time variable. We introduce Opaque Knapsack, a procedurally generated family of partially observable optimization tasks designed to prevent one-shot solutions. Item attributes and constraints are hidden behind budgeted tool calls, forcing multi-turn control flow and iterative state revision. Holding task instances, prompts, tools, model, and supervision fixed, we generate paired trajectories differing only in whether interpreter state persists across steps or resets after each action. We then fine-tune identical base models (Qwen3-8B) on each trace variant and evaluate all four train-runtime combinations. Our 2x2 cross-evaluation shows that execution semantics primarily affect how agents reach solutions, not whether they do: solution quality is statistically indistinguishable across conditions, but token cost and stability differ substantially. A persistent-trained model in a stateless runtime triggers missing-variable errors in roughly 80% of episodes; a stateless-trained model in a persistent runtime redundantly re-derives retained state, using roughly 3.5x more tokens. Interpreter persistence should be treated as a first-class semantic of agent traces. Aligning fine-tuning data with deployment runtimes improves efficiency and reduces brittle train-runtime mismatches.

Replicating AI research is a crucial yet challenging task for large language model (LLM) agents. Existing approaches often struggle to generate executable code, primarily due to insufficient background knowledge and the limitations of retrieval-augmented generation (RAG) methods, which fail to capture latent technical details hidden in referenced papers. Furthermore, previous approaches tend to overlook valuable implementation-level code signals and lack structured knowledge representations that support multi-granular retrieval and reuse. To overcome these challenges, we propose Executable Knowledge Graphs (xKG), a modular and pluggable knowledge base that automatically integrates technical insights, code snippets, and domain-specific knowledge extracted from scientific literature. When integrated into three agent frameworks with two different LLMs, xKG shows substantial performance gains (10.9% with o3-mini) on PaperBench, demonstrating its effectiveness as a general and extensible solution for automated AI research replication. Code will released at https://github.com/zjunlp/xKG.

Program synthesis with language models (LMs) has unlocked a large set of reasoning abilities; code-tuned LMs have proven adept at generating programs that solve a wide variety of algorithmic symbolic manipulation tasks (e.g. word concatenation). However, not all reasoning tasks are easily expressible as code, e.g. tasks involving commonsense reasoning, moral decision-making, and sarcasm understanding. Our goal is to extend an LM's program synthesis skills to such tasks and evaluate the results via pseudo-programs, namely Python programs where some leaf function calls are left undefined. To that end, we propose, Code Generation and Emulated EXecution (CoGEX). CoGEX works by (1) training LMs to generate their own pseudo-programs, (2) teaching them to emulate their generated program's execution, including those leaf functions, allowing the LM's knowledge to fill in the execution gaps; and (3) using them to search over many programs to find an optimal one. To adapt the CoGEX model to a new task, we introduce a method for performing program search to find a single program whose pseudo-execution yields optimal performance when applied to all the instances of a given dataset. We show that our approach yields large improvements compared to standard in-context learning approaches on a battery of tasks, both algorithmic and soft reasoning. This result thus demonstrates that code synthesis can be applied to a much broader class of problems than previously considered. Our released dataset, fine-tuned models, and implementation can be found at https://github.com/nweir127/CoGEX.

Large Language Models (LLMs) prompted to generate chain-of-thought (CoT) exhibit impressive reasoning capabilities. Recent attempts at prompt decomposition toward solving complex, multi-step reasoning problems depend on the ability of the LLM to simultaneously decompose and solve the problem. A significant disadvantage is that foundational LLMs are typically not available for fine-tuning, making adaptation computationally prohibitive. We believe (and demonstrate) that problem decomposition and solution generation are distinct capabilites, better addressed in separate modules, than by one monolithic LLM. We introduce DaSLaM, which uses a decomposition generator to decompose complex problems into subproblems that require fewer reasoning steps. These subproblems are answered by a solver. We use a relatively small (13B parameters) LM as the decomposition generator, which we train using policy gradient optimization to interact with a solver LM (regarded as black-box) and guide it through subproblems, thereby rendering our method solver-agnostic. Evaluation on multiple different reasoning datasets reveal that with our method, a 175 billion parameter LM (text-davinci-003) can produce competitive or even better performance, compared to its orders-of-magnitude larger successor, GPT-4. Additionally, we show that DaSLaM is not limited by the solver's capabilities as a function of scale; e.g., solver LMs with diverse sizes give significant performance improvement with our solver-agnostic decomposition technique. Exhaustive ablation studies evince the superiority of our modular finetuning technique over exorbitantly large decomposer LLMs, based on prompting alone.

This paper presents a detailed analysis of the scalability and parallelization of local search algorithms for the Satisfiability problem. We propose a framework to estimate the parallel performance of a given algorithm by analyzing the runtime behavior of its sequential version. Indeed, by approximating the runtime distribution of the sequential process with statistical methods, the runtime behavior of the parallel process can be predicted by a model based on order statistics. We apply this approach to study the parallel performance of two SAT local search solvers, namely Sparrow and CCASAT, and compare the predicted performances to the results of an actual experimentation on parallel hardware up to 384 cores. We show that the model is accurate and predicts performance close to the empirical data. Moreover, as we study different types of instances (random and crafted), we observe that the local search solvers exhibit different behaviors and that their runtime distributions can be approximated by two types of distributions: exponential (shifted and non-shifted) and lognormal.

Optimization modeling is fundamental to decision-making across diverse domains.Despite progress in automating optimization formulation from natural language descriptions, Large Language Models (LLMs) often struggle to generate formally correct and usable models due to hallucinations, posing a challenge for reliable automation. Inspired by the success of Reinforcement Learning (RL) in enhancing Large Reasoning Models, we present Solver-Informed Reinforcement Learning (SIRL).This novel framework leverages external optimization solvers as verifiable reward mechanisms to significantly improve the authenticity of LLMs for optimization modeling.Acting as precise verifiers, these solvers automatically assess the executable code and the instance-level mathematical model represented by the associated LP file, yielding precise and comprehensive feedback signals -- including syntax, feasibility, and solution quality that directly inform the RL process. This automated verification process, powered by classic optimization solvers, also underpins our instance-enhanced self-consistency method to synthesize high-quality training data. Extensive experiments on diverse public benchmarks demonstrate that SIRL achieves state-of-the-art performance, substantially outperforming existing methods in generating accurate and executable optimization models.

Large language models (LLMs) have demonstrated an impressive ability to generate code for various programming tasks. In many instances, LLMs can generate a correct program for a task when given numerous trials. Consequently, a recent trend is to do large scale sampling of programs using a model and then filtering/ranking the programs based on the program execution on a small number of known unit tests to select one candidate solution. However, these approaches assume that the unit tests are given and assume the ability to safely execute the generated programs (which can do arbitrary dangerous operations such as file manipulations). Both of the above assumptions are impractical in real-world software development. In this paper, we propose CodeRanker, a neural ranker that can predict the correctness of a sampled program without executing it. Our CodeRanker is fault-aware i.e., it is trained to predict different kinds of execution information such as predicting the exact compile/runtime error type (e.g., an IndexError or a TypeError). We show that CodeRanker can significantly increase the pass@1 accuracy of various code generation models (including Codex, GPT-Neo, GPT-J) on APPS, HumanEval and MBPP datasets.

Writing competitive programming problems is exacting. Authors must: set constraints, input distributions, and edge cases that rule out shortcuts; target specific algorithms (e.g., max-flow, dynamic programming, data structures); and calibrate complexity beyond the reach of most competitors. We argue that this makes for an ideal test of general large language model capabilities and study whether they can do this reliably. We introduce AutoCode, which uses multiple rounds of validation to yield competition-grade problem statements and test cases. On held-out problems, AutoCode test suites approach 99% consistency with official judgments, a significant improvement over current state-of-the-art methods like HardTests, which achieve less than 81%. Furthermore, starting with a random seed problem, AutoCode can create novel variants with reference and brute-force solutions. By cross-verifying these generated solutions against test cases, we can further filter out malformed problems. Our system ensures high correctness, as verified by human experts. AutoCode successfully produces novel problems judged by Grandmaster-level (top 0.3%) competitive programmers to be of contest quality.

Quantified formulas with Uninterpreted Functions (UFs) over non-linear real arithmetic pose fundamental challenges for Satisfiability Modulo Theories (SMT) solving. Traditional quantifier instantiation methods struggle because they lack semantic understanding of UF constraints, forcing them to search through unbounded solution spaces with limited guidance. We present AquaForte, a framework that leverages Large Language Models to provide semantic guidance for UF instantiation by generating instantiated candidates for function definitions that satisfy the constraints, thereby significantly reducing the search space and complexity for solvers. Our approach preprocesses formulas through constraint separation, uses structured prompts to extract mathematical reasoning from LLMs, and integrates the results with traditional SMT algorithms through adaptive instantiation. AquaForte maintains soundness through systematic validation: LLM-guided instantiations yielding SAT solve the original problem, while UNSAT results generate exclusion clauses for iterative refinement. Completeness is preserved by fallback to traditional solvers augmented with learned constraints. Experimental evaluation on SMT-COMP benchmarks demonstrates that AquaForte solves numerous instances where state-of-the-art solvers like Z3 and CVC5 timeout, with particular effectiveness on satisfiable formulas. Our work shows that LLMs can provide valuable mathematical intuition for symbolic reasoning, establishing a new paradigm for SMT constraint solving.

Multimodal Large Language Models (MLLMs) have significantly advanced vision-language understanding. However, even state-of-the-art models struggle with geometric reasoning, revealing a critical bottleneck: the extreme scarcity of high-quality image-text pairs. Human annotation is prohibitively expensive, while automated methods fail to ensure fidelity and training effectiveness. Existing approaches either passively adapt to available images or employ inefficient random exploration with filtering, decoupling generation from learning needs. We propose Socratic-Geo, a fully autonomous framework that dynamically couples data synthesis with model learning through multi-agent interaction. The Teacher agent generates parameterized Python scripts with reflective feedback (Reflect for solvability, RePI for visual validity), ensuring image-text pair purity. The Solver agent optimizes reasoning through preference learning, with failure paths guiding Teacher's targeted augmentation. Independently, the Generator learns image generation capabilities on accumulated "image-code-instruction" triplets, distilling programmatic drawing intelligence into visual generation. Starting from only 108 seed problems, Socratic-Solver achieves 49.11 on six benchmarks using one-quarter of baseline data, surpassing strong baselines by 2.43 points. Socratic-Generator achieves 42.4% on GenExam, establishing new state-of-the-art for open-source models, surpassing Seedream-4.0 (39.8%) and approaching Gemini-2.5-Flash-Image (43.1%).

Test-time scaling has enabled Large Language Models (LLMs) with remarkable reasoning capabilities, particularly in mathematical domains, through intermediate chain-of-thought (CoT) reasoning before generating final answers. However, the specific sources and mechanisms underlying these reasoning capabilities remain insufficiently understood. Optimization reasoning, i.e. finding extrema under constraints, represents a fundamental abstraction that underpins critical applications in planning, control, resource allocation, and prompt search. To systematically evaluate this capability, we introduce ExtremBench, a benchmark dataset for solving mathematical extremal problems, curated from inequality exercises used for Chinese Mathematical Olympiad and transformed into 93 standardized extrema-finding problems. We conduct extensive evaluations across various state-of-the-art open-source model families, including the Qwen3, GPT-OSS, and DeepSeek. Our results reveal that LLMs' extremal-solving reasoning capabilities do not always align with those of current mathematical benchmarks such as AIME25 and MATH-500, with some models showing strong general mathematical reasoning but poor extremal-solving skills, and vice versa. This discrepancy highlights a critical gap in current evaluation practices and suggests that existing benchmarks may not comprehensively capture the full spectrum of mathematical reasoning abilities.

Selecting the best code solution from multiple generated ones is an essential task in code generation, which can be achieved by using some reliable validators (e.g., developer-written test cases) for assistance. Since reliable test cases are not always available and can be expensive to build in practice, researchers propose to automatically generate test cases to assess code solutions. However, when both code solutions and test cases are plausible and not reliable, selecting the best solution becomes challenging. Although some heuristic strategies have been proposed to tackle this problem, they lack a strong theoretical guarantee and it is still an open question whether an optimal selection strategy exists. Our work contributes in two ways. First, we show that within a Bayesian framework, the optimal selection strategy can be defined based on the posterior probability of the observed passing states between solutions and tests. The problem of identifying the best solution is then framed as an integer programming problem. Second, we propose an efficient approach for approximating this optimal (yet uncomputable) strategy, where the approximation error is bounded by the correctness of prior knowledge. We then incorporate effective prior knowledge to tailor code generation tasks. Both theoretical and empirical studies confirm that existing heuristics are limited in selecting the best solutions with plausible test cases. Our proposed approximated optimal strategy B4 significantly surpasses existing heuristics in selecting code solutions generated by large language models (LLMs) with LLM-generated tests, achieving a relative performance improvement by up to 50% over the strongest heuristic and 246% over the random selection in the most challenging scenarios. Our code is publicly available at https://github.com/ZJU-CTAG/B4.

This paper proposes a novel column generation framework for combinatorial software testing. In particular, it combines Mathematical Programming and Constraint Programming in a hybrid decomposition to generate covering arrays. The approach allows generating parameterized test cases with coverage guarantees between parameter interactions of a given application. Compared to exhaustive testing, combinatorial test case generation reduces the number of tests to run significantly. Our column generation algorithm is generic and can accommodate mixed coverage arrays over heterogeneous alphabets. The algorithm is realized in practice as a cloud service and recognized as one of the five winners of the company-wide cloud application challenge at Oracle. The service is currently helping software developers from a range of different product teams in their testing efforts while exposing declarative constraint models and hybrid optimization techniques to a broader audience.

Large language models (LLMs) have recently demonstrated an impressive ability to perform arithmetic and symbolic reasoning tasks, when provided with a few examples at test time ("few-shot prompting"). Much of this success can be attributed to prompting methods such as "chain-of-thought'', which employ LLMs for both understanding the problem description by decomposing it into steps, as well as solving each step of the problem. While LLMs seem to be adept at this sort of step-by-step decomposition, LLMs often make logical and arithmetic mistakes in the solution part, even when the problem is decomposed correctly. In this paper, we present Program-Aided Language models (PAL): a novel approach that uses the LLM to read natural language problems and generate programs as the intermediate reasoning steps, but offloads the solution step to a runtime such as a Python interpreter. With PAL, decomposing the natural language problem into runnable steps remains the only learning task for the LLM, while solving is delegated to the interpreter. We demonstrate this synergy between a neural LLM and a symbolic interpreter across 13 mathematical, symbolic, and algorithmic reasoning tasks from BIG-Bench Hard and other benchmarks. In all these natural language reasoning tasks, generating code using an LLM and reasoning using a Python interpreter leads to more accurate results than much larger models. For example, PAL using Codex achieves state-of-the-art few-shot accuracy on the GSM8K benchmark of math word problems, surpassing PaLM-540B which uses chain-of-thought by absolute 15% top-1. Our code and data are publicly available at http://reasonwithpal.com/ .

Recently, deep learning models have made great progress in MWP solving on answer accuracy. However, they are uninterpretable since they mainly rely on shallow heuristics to achieve high performance without understanding and reasoning the grounded math logic. To address this issue and make a step towards interpretable MWP solving, we first construct a high-quality MWP dataset named InterMWP which consists of 11,495 MWPs and annotates interpretable logical formulas based on algebraic knowledge as the grounded linguistic logic of each solution equation. Different from existing MWP datasets, our InterMWP benchmark asks for a solver to not only output the solution expressions but also predict the corresponding logical formulas. We further propose a novel approach with logical prompt and interpretation generation, called LogicSolver. For each MWP, our LogicSolver first retrieves some highly-correlated algebraic knowledge and then passes them to the backbone model as prompts to improve the semantic representations of MWPs. With these improved semantic representations, our LogicSolver generates corresponding solution expressions and interpretable knowledge formulas in accord with the generated solution expressions, simultaneously. Experimental results show that our LogicSolver has stronger logical formula-based interpretability than baselines while achieving higher answer accuracy with the help of logical prompts, simultaneously. The source code and dataset is available at https://github.com/yangzhch6/InterMWP.

We introduce self-invoking code generation, a new task designed to evaluate the progressive reasoning and problem-solving capabilities of LLMs. In this task, models are presented with a base problem and a related, more complex problem. They must solve the base problem and then utilize its solution to address the more complex one. This work features three key contributions. First, we propose a general recipe for generating more challenging versions of existing benchmarks, resulting in three new benchmarks: HumanEval Pro, MBPP Pro, and BigCodeBench-Lite Pro, specifically designed to assess LLMs on self-invoking code generation. Second, from the analysis of experimental results over twenty LLMs on our benchmarks, we have two important observations: (i) Most LLMs excel in traditional code generation benchmarks like HumanEval and MBPP, but their performance declines on self-invoking tasks. For example, o1-mini achieves 96.2% pass@1 on HumanEval but only 76.2% on HumanEval Pro. (ii) On self-invoking code generation task, the instruction-tuned models demonstrate only marginal improvements compared to the base models. Third, we disclose the types of failure modes that exist in our evaluation results. All these results underscore the need for further advancements in self-invoking code generation tasks and provide a new direction for future research on enhancing LLMs' code reasoning capabilities.

Despite recent progress made by large language models in code generation, they still struggle with programs that meet complex requirements. Recent work utilizes plan-and-solve decomposition to decrease the complexity and leverage self-tests to refine the generated program. Yet, planning deep-inside requirements in advance can be challenging, and the tests need to be accurate to accomplish self-improvement. To this end, we propose FunCoder, a code generation framework incorporating the divide-and-conquer strategy with functional consensus. Specifically, FunCoder recursively branches off sub-functions as smaller goals during code generation, represented by a tree hierarchy. These sub-functions are then composited to attain more complex objectives. Additionally, we designate functions via a consensus formed by identifying similarities in program behavior, mitigating error propagation. FunCoder outperforms state-of-the-art methods by +9.8% on average in HumanEval, MBPP, xCodeEval and MATH with GPT-3.5 and GPT-4. Moreover, our method demonstrates superiority on smaller models: With FunCoder, StableCode-3b surpasses GPT-3.5 by +18.6% and achieves 97.7% of GPT-4's performance on HumanEval. Further analysis reveals that our proposed dynamic function decomposition is capable of handling complex requirements, and the functional consensus prevails over self-testing in correctness evaluation.

There is growing excitement about the potential of Language Models (LMs) to accelerate scientific discovery. Falsifying hypotheses is key to scientific progress, as it allows claims to be iteratively refined over time. This process requires significant researcher effort, reasoning, and ingenuity. Yet current benchmarks for LMs predominantly assess their ability to generate solutions rather than challenge them. We advocate for developing benchmarks that evaluate this inverse capability - creating counterexamples for subtly incorrect solutions. To demonstrate this approach, we start with the domain of algorithmic problem solving, where counterexamples can be evaluated automatically using code execution. Specifically, we introduce REFUTE, a dynamically updating benchmark that includes recent problems and incorrect submissions from programming competitions, where human experts successfully identified counterexamples. Our analysis finds that the best reasoning agents, even OpenAI o3-mini (high) with code execution feedback, can create counterexamples for only <9% of incorrect solutions in REFUTE, even though ratings indicate its ability to solve up to 48% of these problems from scratch. We hope our work spurs progress in evaluating and enhancing LMs' ability to falsify incorrect solutions - a capability that is crucial for both accelerating research and making models self-improve through reliable reflective reasoning.

Programming-by-example is the task of synthesizing a program that is consistent with a set of user-provided input-output examples. As examples are often an under-specification of one's intent, a good synthesizer must choose the intended program from the many that are consistent with the given set of examples. Prior work frames program synthesis as a cooperative game between a listener (that synthesizes programs) and a speaker (a user choosing examples), and shows that models of computational pragmatic inference are effective in choosing the user intended programs. However, these models require counterfactual reasoning over a large set of programs and examples, which is infeasible in realistic program spaces. In this paper, we propose a novel way to amortize this search with neural networks. We sample pairs of programs and examples via self-play between listener and speaker models, and use pragmatic inference to choose informative training examples from this sample.We then use the informative dataset to train models to improve the synthesizer's ability to disambiguate user-provided examples without human supervision. We validate our method on the challenging task of synthesizing regular expressions from example strings, and find that our method (1) outperforms models trained without choosing pragmatic examples by 23% (a 51% relative increase) (2) matches the performance of supervised learning on a dataset of pragmatic examples provided by humans, despite using no human data in training.

Large language models (LLMs) have exhibited their problem-solving abilities in mathematical reasoning. Solving realistic optimization (OPT) problems in application scenarios requires advanced and applied mathematics ability. However, current OPT benchmarks that merely solve linear programming are far from complex realistic situations. In this work, we propose OptiBench, a benchmark for End-to-end optimization problem-solving with human-readable inputs and outputs. OptiBench contains rich optimization problems, including linear and nonlinear programming with or without tabular data, which can comprehensively evaluate LLMs' solving ability. In our benchmark, LLMs are required to call a code solver to provide precise numerical answers. Furthermore, to alleviate the data scarcity for optimization problems, and to bridge the gap between open-source LLMs on a small scale (e.g., Llama-3-8b) and closed-source LLMs (e.g., GPT-4), we further propose a data synthesis method namely ReSocratic. Unlike general data synthesis methods that proceed from questions to answers, \ReSocratic first incrementally synthesizes formatted optimization demonstration with mathematical formulations step by step and then back-translates the generated demonstrations into questions. Based on this, we synthesize the ReSocratic-29k dataset. We further conduct supervised fine-tuning with ReSocratic-29k on multiple open-source models. Experimental results show that ReSocratic-29k significantly improves the performance of open-source models.

Symbolic execution is a powerful technique for software testing, but suffers from limitations when encountering external functions, such as native methods or third-party libraries. Existing solutions often require additional context, expensive SMT solvers, or manual intervention to approximate these functions through symbolic stubs. In this work, we propose a novel approach to automatically generate symbolic stubs for external functions during symbolic execution that leverages Genetic Programming. When the symbolic executor encounters an external function, AutoStub generates training data by executing the function on randomly generated inputs and collecting the outputs. Genetic Programming then derives expressions that approximate the behavior of the function, serving as symbolic stubs. These automatically generated stubs allow the symbolic executor to continue the analysis without manual intervention, enabling the exploration of program paths that were previously intractable. We demonstrate that AutoStub can automatically approximate external functions with over 90% accuracy for 55% of the functions evaluated, and can infer language-specific behaviors that reveal edge cases crucial for software testing.

Recent progress in reasoning models suggests that generating plausible attempts for research-level mathematics may be within reach, but verification remains a bottleneck, consuming scarce expert time. We hypothesize that a meaningful solution should contain enough method-level information that, when applied to a neighborhood of related questions, it should yield better downstream performance than incorrect solutions. Building on this idea, we propose Consequence-Based Utility, an oracle-free evaluator that scores each candidate by testing its value as an in-context exemplar in solving related yet verifiable questions. Our approach is evaluated on an original set of research-level math problems, each paired with one expert-written solution and nine LLM-generated solutions. Notably, Consequence-Based Utility consistently outperforms reward models, generative reward models, and LLM judges on ranking quality. Specifically, for GPT-OSS-120B, it improves Acc@1 from 67.2 to 76.3 and AUC from 71.4 to 79.6, with similarly large AUC gains on GPT-OSS-20B (69.0 to 79.2). Furthermore, compared to LLM-Judges, it also exhibits a larger solver-evaluator gap, maintaining a stronger correct-wrong separation even on instances where the underlying solver often fails to solve.

The task of generating code solutions for a given programming problem can benefit from the use of pre-trained language models such as Codex, which can produce multiple diverse samples. However, a major challenge for this task is to select the most appropriate solution from the multiple samples generated by the pre-trained language models. A natural way to evaluate the quality and correctness of a code solution is to run it against a set of test cases, but the manual creation of such test cases is often costly and time-consuming. In this paper, we propose a novel method, CodeT, that leverages the same pre-trained language models to automatically generate test cases for the code samples, thus reducing the human effort and increasing the coverage of the test scenarios. CodeT then executes the code samples using the generated test cases, and performs a dual execution agreement, which considers both the consistency of the outputs against the generated test cases and the agreement of the outputs with other code samples. We conduct comprehensive experiments on four benchmarks, HumanEval, MBPP, APPS and CodeContests, using five different pre-trained language models with varying sizes and capabilities. Our results show that CodeT can significantly improve the performance of code solution selection over previous methods, achieving remarkable and consistent gains across different models and benchmarks. For instance, CodeT improves the pass@1 metric on HumanEval to 65.8%, which represents an absolute improvement of 18.8% over the code-davinci-002 model, and an absolute improvement of more than 20% over the previous state-of-the-art results.

As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a O(1/k^2) and O(1/k) rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves O(1/k^2) and mathcal{O}(1/k^2) rates. We then provide a matching complexity lower bound to establish that Theta(1/k^2) is indeed the optimal accelerated rate.

Solving a linear system Ax=b is a fundamental scientific computing primitive for which numerous solvers and preconditioners have been developed. These come with parameters whose optimal values depend on the system being solved and are often impossible or too expensive to identify; thus in practice sub-optimal heuristics are used. We consider the common setting in which many related linear systems need to be solved, e.g. during a single numerical simulation. In this scenario, can we sequentially choose parameters that attain a near-optimal overall number of iterations, without extra matrix computations? We answer in the affirmative for Successive Over-Relaxation (SOR), a standard solver whose parameter omega has a strong impact on its runtime. For this method, we prove that a bandit online learning algorithm--using only the number of iterations as feedback--can select parameters for a sequence of instances such that the overall cost approaches that of the best fixed omega as the sequence length increases. Furthermore, when given additional structural information, we show that a contextual bandit method asymptotically achieves the performance of the instance-optimal policy, which selects the best omega for each instance. Our work provides the first learning-theoretic treatment of high-precision linear system solvers and the first end-to-end guarantees for data-driven scientific computing, demonstrating theoretically the potential to speed up numerical methods using well-understood learning algorithms.

The ability of large language models to solve complex mathematical problems has progressed significantly, particularly for tasks requiring advanced reasoning. However, the scarcity of sufficiently challenging problems, particularly at the Olympiad level, hinders further advancements. In this work, we introduce PromptCoT, a novel approach for automatically generating high-quality Olympiad-level math problems. The proposed method synthesizes complex problems based on mathematical concepts and the rationale behind problem construction, emulating the thought processes of experienced problem designers. We provide a theoretical analysis demonstrating that an optimal rationale should maximize both the likelihood of rationale generation given the associated concepts and the likelihood of problem generation conditioned on both the rationale and the concepts. Our method is evaluated on standard benchmarks including GSM8K, MATH-500, and AIME2024, where it consistently outperforms existing problem generation methods. Furthermore, we demonstrate that PromptCoT exhibits superior data scalability, consistently maintaining high performance as the dataset size increases, outperforming the baselines. The implementation is available at https://github.com/zhaoxlpku/PromptCoT.

Large language models (LLMs) have shown increasing competence in solving mathematical reasoning problems. However, many open-source LLMs still struggle with errors in calculation and semantic understanding during intermediate reasoning steps. In this work, we introduce Prove, a simple yet effective framework that leverages translated programs derived from natural language solutions as a verification mechanism to filter out potentially incorrect reasoning paths before aggregating final answers. Unlike vanilla majority voting, our approach filters out solutions whose corresponding program output is inconsistent with the generated solution, aggregating only those that pass verification. We conducted extensive experiments using 13 open-source LLMs from various model families and sizes, ranging from 0.5B to 13B parameters, across eight mathematical benchmarks. Our results show that Prove consistently outperforms vanilla majority voting as a heuristic for solving mathematical reasoning tasks across all model sizes and datasets, achieving improvements of up to 18% on GSM8K and 8% on MATH-500. Our codes are available at https://github.com/declare-lab/prove.

Large language models have demonstrated remarkable capabilities, but their performance is heavily reliant on effective prompt engineering. Automatic prompt optimization (APO) methods are designed to automate this and can be broadly categorized into those targeting instructions (instruction optimization, IO) vs. those targeting exemplars (exemplar selection, ES). Despite their shared objective, these have evolved rather independently, with IO recently receiving more research attention. This paper seeks to bridge this gap by comprehensively comparing the performance of representative IO and ES techniques, both isolation and combination, on a diverse set of challenging tasks. Our findings reveal that intelligently reusing model-generated input-output pairs obtained from evaluating prompts on the validation set as exemplars consistently improves performance over IO methods but is currently under-investigated. We also find that despite the recent focus on IO, how we select exemplars can outweigh how we optimize instructions, with ES strategies as simple as random search outperforming state-of-the-art IO methods with seed instructions without any optimization. Moreover, we observe synergy between ES and IO, with optimal combinations surpassing individual contributions. We conclude that studying exemplar selection as a standalone method and its optimal combination with instruction optimization remains a crucial aspect of APO and deserves greater consideration in future research, even in the era of highly capable instruction-following models.

Though LLMs are capable of generating plausible programs, it's challenging to interact with the LLMs further to revise the program, especially if the user's specific requirements are different from the initial proposal. In this paper, we introduce ANPL, an interactive programming system that ensures users can always refine the generated code towards their specific programmatic intents via structured decompositions. Borrowing the paradigm of sketching from program synthesis, an ANPL program consists of a set of input-outputs that it must satisfy, a ``sketch'' -- control/data flow expressed in precise code (e.g. Python), and ``holes'' -- sub-modules to be implemented by the LLM specified with natural language. The user revises an ANPL program by either modifying the sketch, changing the language used to describe the holes, or providing additional input-outputs to a particular hole, turning it into a sub-ANPL program that can be solved recursively. This workflow allows the users to offload programming burdens to the LLM as much as possible while retaining the ability to pinpoint and resolve bugs locally, without exposing the rest of the program to the LLM. We deploy ANPL on the Abstraction and Reasoning Corpus (ARC), a set of unique tasks that are challenging for state-of-the-art AI systems, showing it outperforms baseline programming systems that (a) without the ability to decompose tasks interactively and (b) without the guarantee that the modules can be correctly composed together. Additional evaluations on APPS, HumanEval, and real-world programming tasks have validated that the ANPL framework is applicable to multiple programming domains. We release the ANPL solutions to the ARC tasks as a dataset, providing insights into how humans decompose novel tasks programmatically. See our code at https://iprc-dip.github.io/ANPL/.

Instructing large language models (LLMs) to solve elementary school math problems has shown great success using Chain of Thought (CoT). However, the CoT approach relies on an LLM to generate a sequence of arithmetic calculations which can be prone to cascaded calculation errors. We hypothesize that an LLM should focus on extracting predicates and generating symbolic formulas from the math problem description so that the underlying calculation can be done via an external code interpreter. We investigate using LLM to generate Prolog programs to solve mathematical questions. Experimental results show that our Prolog-based arithmetic problem-solving outperforms CoT generation in the GSM8K benchmark across three distinct LLMs. In addition, given the insensitive ordering of predicates and symbolic formulas in Prolog, we propose to permute the ground truth predicates for more robust LLM training via data augmentation.

Recently, Large Language Models (LLMs) have been applied to scientific equation discovery, leveraging their embedded scientific knowledge for hypothesis generation. However, current methods typically confine LLMs to the role of an equation proposer within search algorithms like genetic programming. In this paper, we present SR-Scientist, a framework that elevates the LLM from a simple equation proposer to an autonomous AI scientist that writes code to analyze data, implements the equation as code, submits it for evaluation, and optimizes the equation based on experimental feedback. Specifically, we wrap the code interpreter into a set of tools for data analysis and equation evaluation. The agent is instructed to optimize the equation by utilizing these tools over a long horizon with minimal human-defined pipelines. Empirical results show that SR-Scientist outperforms baseline methods by an absolute margin of 6% to 35% on datasets covering four science disciplines. Additionally, we demonstrate our method's robustness to noise, the generalization of the discovered equations to out-of-domain data, and their symbolic accuracy. Furthermore, we develop an end-to-end reinforcement learning framework to enhance the agent's capabilities.

In this paper, we present a novel approach, called LogicPro, to enhance Large Language Models (LLMs) complex Logical reasoning through Program Examples. We do this effectively by simply utilizing widely available algorithmic problems and their code solutions. First, we constructed diverse test samples input based on algorithmic questions and code solutions. Then, we designed different complex reasoning questions based on algorithmic problems and test samples. Finally, combining the intermediate variable outputs of the code solutions and the complex reasoning questions, we derived the reasoning process and the final answer. With this approach, we can construct a dataset that is sufficiently difficult (all models are ineffective), diverse (synthesized from 2,360 different algorithmic questions), and scalable (building different test samples and collecting more algorithmic questions). In addition, we obtain a high-quality reasoning process guided by the values of intermediate variables. As a result, our approach achieves significant improvements in multiple models for the BBH^{27}, GSM8K, HellSwag, Logicqa, Reclor, and RTE datasets, outperforming a wide range of existing reasoning datasets.

Training models through self-play alone (without any human data) has been a longstanding goal in AI, but its effectiveness for training large language models remains unclear, particularly in code generation where rewards based on unit tests are brittle and prone to error propagation. We study self-play in the verified code generation setting, where formal verification provides reliable correctness signals. We introduce Propose, Solve, Verify (PSV) a simple self-play framework where formal verification signals are used to create a proposer capable of generating challenging synthetic problems and a solver trained via expert iteration. We use PSV to train PSV-Verus, which across three benchmarks improves pass@1 by up to 9.6x over inference-only and expert-iteration baselines. We show that performance scales with the number of generated questions and training iterations, and through ablations identify formal verification and difficulty-aware proposal as essential ingredients for successful self-play.

Code completion has become an essential tool for daily software development. Existing evaluation benchmarks often employ static methods that do not fully capture the dynamic nature of real-world coding environments and face significant challenges, including limited context length, reliance on superficial evaluation metrics, and potential overfitting to training datasets. In this work, we introduce a novel framework for enhancing code completion in software development through the creation of a repository-level benchmark ExecRepoBench and the instruction corpora Repo-Instruct, aim at improving the functionality of open-source large language models (LLMs) in real-world coding scenarios that involve complex interdependencies across multiple files. ExecRepoBench includes 1.2K samples from active Python repositories. Plus, we present a multi-level grammar-based completion methodology conditioned on the abstract syntax tree to mask code fragments at various logical units (e.g. statements, expressions, and functions). Then, we fine-tune the open-source LLM with 7B parameters on Repo-Instruct to produce a strong code completion baseline model Qwen2.5-Coder-Instruct-C based on the open-source model. Qwen2.5-Coder-Instruct-C is rigorously evaluated against existing benchmarks, including MultiPL-E and ExecRepoBench, which consistently outperforms prior baselines across all programming languages. The deployment of can be used as a high-performance, local service for programming development\url{https://execrepobench.github.io/}.

Large Language Models (LLMs) have demonstrated impressive progress in optimization modeling, fostering a rapid expansion of new methodologies and evaluation benchmarks. However, the boundaries of their capabilities in automated formulation and problem solving remain poorly understood, particularly when extending to complex, real-world tasks. To bridge this gap, we propose OPT-ENGINE, an extensible benchmark framework designed to evaluate LLMs on optimization modeling with controllable and scalable difficulty levels. OPT-ENGINE spans 10 canonical tasks across operations research, with five Linear Programming and five Mixed-Integer Programming. Utilizing OPT-ENGINE, we conduct an extensive study of LLMs' reasoning capabilities, addressing two critical questions: 1.) Do LLMs' performance remain robust when generalizing to out-of-distribution optimization tasks that scale in complexity beyond current benchmark levels? and 2.) At what stage, from problem interpretation to solution generation, do current LLMs encounter the most significant bottlenecks? Our empirical results yield two key insights: first, tool-integrated reasoning with external solvers exhibits significantly higher robustness as task complexity escalates, while pure-text reasoning reaches a ceiling; second, the automated formulation of constraints constitutes the primary performance bottleneck. These findings provide actionable guidance for developing next-generation LLMs for advanced optimization. Our code is publicly available at blue{https://github.com/Cardinal-Operations/OPTEngine}.

Restart strategies are an important factor in the performance of conflict-driven Davis Putnam style SAT solvers. Selecting a good restart strategy for a problem instance can enhance the performance of a solver. Inspired by recent success applying machine learning techniques to predict the runtime of SAT solvers, we present a method which uses machine learning to boost solver performance through a smart selection of the restart strategy. Based on easy to compute features, we train both a satisfiability classifier and runtime models. We use these models to choose between restart strategies. We present experimental results comparing this technique with the most commonly used restart strategies. Our results demonstrate that machine learning is effective in improving solver performance.

Model-driven engineering (MDE) provides abstraction and analytical rigour, but industrial adoption in many domains has been limited by the cost of developing and maintaining models. Large language models (LLMs) can help shift this cost balance by supporting direct generation of models from natural-language (NL) descriptions. For domain-specific languages (DSLs), however, LLM-generated models may be less accurate than LLM-generated code in mainstream languages such as Python, due to the latter's dominance in LLM training corpora. We investigate this issue in mathematical optimization, with AMPL, a DSL with established industrial use. We introduce EXEOS, an LLM-based approach that derives AMPL models and Python code from NL problem descriptions and iteratively refines them with solver feedback. Using a public optimization dataset and real-world supply-chain cases from our industrial partner Kinaxis, we evaluate generated AMPL models against Python code in terms of executability and correctness. An ablation study with two LLM families shows that AMPL is competitive with, and sometimes better than, Python, and that our design choices in EXEOS improve the quality of generated specifications.

Large Language Models (LLMs) have demonstrated remarkable performance in solving math problems, a hallmark of human intelligence. Despite high success rates on current benchmarks; however, these often feature simple problems with only one or two unknowns, which do not sufficiently challenge their reasoning capacities. This paper introduces a novel benchmark, BeyondX, designed to address these limitations by incorporating problems with multiple unknowns. Recognizing the challenges in proposing multi-unknown problems from scratch, we developed BeyondX using an innovative automated pipeline that progressively increases complexity by expanding the number of unknowns in simpler problems. Empirical study on BeyondX reveals that the performance of existing LLMs, even those fine-tuned specifically on math tasks, significantly decreases as the number of unknowns increases - with a performance drop of up to 70\% observed in GPT-4. To tackle these challenges, we propose the Formulate-and-Solve strategy, a generalized prompting approach that effectively handles problems with an arbitrary number of unknowns. Our findings reveal that this strategy not only enhances LLM performance on the BeyondX benchmark but also provides deeper insights into the computational limits of LLMs when faced with more complex mathematical challenges.

Advancing code reasoning in large language models (LLMs) is fundamentally limited by the scarcity of high-difficulty datasets, especially those with verifiable input-output test cases necessary for rigorous solution validation at scale. We introduce rStar-Coder, which significantly improves LLM code reasoning capabilities by constructing a large-scale, verified dataset of 418K competition-level code problems, 580K long-reasoning solutions along with rich test cases of varying difficulty. This is achieved through three core contributions: (1) we curate competitive programming code problems and oracle solutions to synthesize new, solvable problems; (2) we introduce a reliable input-output test case synthesis pipeline that decouples the generation into a three-step input generation method and a mutual verification mechanism for effective output labeling; (3) we augment problems with high-quality, test-case-verified long-reasoning solutions. Extensive experiments on Qwen models (1.5B-14B) across various code reasoning benchmarks demonstrate the superiority of rStar-Coder dataset, achieving leading performance comparable to frontier reasoning LLMs with much smaller model sizes. On LiveCodeBench, rStar-Coder improves Qwen2.5-7B from 17.4% to an impressive 57.3%, and Qwen2.5-14B from 23.3% to 62.5%, surpassing o3-mini (low) by3.1%. On the more challenging USA Computing Olympiad, our 7B model achieves an average pass@1 accuracy of 16.15%, outperforming the frontier-level QWQ-32B. Code and the dataset will be released at https://github.com/microsoft/rStar.

We explore the novel application of Large Language Models to code optimization. We present a 7B-parameter transformer model trained from scratch to optimize LLVM assembly for code size. The model takes as input unoptimized assembly and outputs a list of compiler options to best optimize the program. Crucially, during training, we ask the model to predict the instruction counts before and after optimization, and the optimized code itself. These auxiliary learning tasks significantly improve the optimization performance of the model and improve the model's depth of understanding. We evaluate on a large suite of test programs. Our approach achieves a 3.0% improvement in reducing instruction counts over the compiler, outperforming two state-of-the-art baselines that require thousands of compilations. Furthermore, the model shows surprisingly strong code reasoning abilities, generating compilable code 91% of the time and perfectly emulating the output of the compiler 70% of the time.

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

Code Executing Reasoning is becoming a new non-functional metric that assesses the ability of large language models (LLMs) in programming tasks. State-of-the-art frameworks (CodeMind or REval) and benchmarks (CruxEval) usually focus on LLM's prediction of a given code's input/output or intermediate variable states/values on limited programs. However, there is no tool for more in-depth analysis of the results. Without such a tool, the observations about LLM's code execution reasoning cannot be generalized to more datasets, preventing the research community and practitioners from devising the next generation of LLMs with better code execution reasoning abilities. This paper introduces ExeRScope, a series of tools and heuristics to analyze the result of code execution reasoning frameworks to understand better the impact of code properties in the studied benchmarks on the code execution reasoning. With such tooling, analysis can be generalized to code with similar properties without the urgent need to design more benchmarks, which is a cumbersome effort.

Code execution is a fundamental aspect of programming language semantics that reflects the exact behavior of the code. However, most pre-trained models for code intelligence ignore the execution trace and only rely on source code and syntactic structures. In this paper, we investigate how well pre-trained models can understand and perform code execution. We develop a mutation-based data augmentation technique to create a large-scale and realistic Python dataset and task for code execution, which challenges existing models such as Codex. We then present CodeExecutor, a Transformer model that leverages code execution pre-training and curriculum learning to enhance its semantic comprehension. We evaluate CodeExecutor on code execution and show its promising performance and limitations. We also demonstrate its potential benefits for code intelligence tasks such as zero-shot code-to-code search and text-to-code generation. Our analysis provides insights into the learning and generalization abilities of pre-trained models for code execution.

Decompilation aims to restore compiled code to human-readable source code, but struggles with details like names and structure. Large language models (LLMs) show promise for programming tasks, motivating their application to decompilation. However, there does not exist any open-source LLM for decompilation. Moreover, existing decompilation evaluation systems mainly consider token-level accuracy and largely ignore code executability, which is the most important feature of any program. Therefore, we release the first open-access decompilation LLMs ranging from 1B to 33B pre-trained on 4 billion tokens of C source code and the corresponding assembly code. The open-source LLMs can serve as baselines for further development in the field. To ensure practical program evaluation, we introduce Decompile-Eval, the first dataset that considers re-compilability and re-executability for decompilation. The benchmark emphasizes the importance of evaluating the decompilation model from the perspective of program semantics. Experiments indicate that our LLM4Decompile has demonstrated the capability to accurately decompile 21% of the assembly code, which achieves a 50% improvement over GPT-4. Our code, dataset, and models are released at https://github.com/albertan017/LLM4Decompile

Tool-integrated reasoning (TIR) has become a key approach for improving large reasoning models (LRMs) on complex problems. Prior work has mainly studied when to invoke tools, while overlooking how tools are applied. We identify two common patterns: a calculator pattern that uses code for direct computation, and an algorithmic pattern that encodes problems as programs. Misaligned choices often cause failures even when reasoning is sound. We propose a two-stage framework that first builds code competence from both patterns and then aligns pattern selection with teacher preferences. Across challenging math datasets, our pattern-aware method substantially improves both code usage and accuracy, for instance raising Code@1 on MATH500 from 64.0% to 70.5% and on AIME24 from 26.7% to 50.0%. These gains highlight the effectiveness of a pattern-aware approach for tool-integrated reasoning.

We study a class of optimization problems motivated by automating the design and update of AI systems like coding assistants, robots, and copilots. We propose an end-to-end optimization framework, Trace, which treats the computational workflow of an AI system as a graph akin to neural networks, based on a generalization of back-propagation. Optimization of computational workflows often involves rich feedback (e.g. console output or user's responses), heterogeneous parameters (e.g. prompts, hyper-parameters, codes), and intricate objectives (beyond maximizing a score). Moreover, its computation graph can change dynamically with the inputs and parameters. We frame a new mathematical setup of iterative optimization, Optimization with Trace Oracle (OPTO), to capture and abstract these properties so as to design optimizers that work across many domains. In OPTO, an optimizer receives an execution trace along with feedback on the computed output and updates parameters iteratively. Trace is the tool to implement OPTO in practice. Trace has a Python interface that efficiently converts a computational workflow into an OPTO instance using a PyTorch-like interface. Using Trace, we develop a general-purpose LLM-based optimizer called OptoPrime that can effectively solve OPTO problems. In empirical studies, we find that OptoPrime is capable of first-order numerical optimization, prompt optimization, hyper-parameter tuning, robot controller design, code debugging, etc., and is often competitive with specialized optimizers for each domain. We believe that Trace, OptoPrime and the OPTO framework will enable the next generation of interactive agents that automatically adapt using various kinds of feedback. Website: https://microsoft.github.io/Trace

Important tasks such as reasoning and planning are fundamentally algorithmic, meaning that solving them robustly requires acquiring true reasoning or planning algorithms, rather than shortcuts. Large Language Models lack true algorithmic ability primarily because of the limitations of neural network optimization algorithms, their optimization data and optimization objective, but also due to architectural inexpressivity. To solve this, our paper proposes augmenting LLMs with a library of fundamental operations and sophisticated differentiable programs, so that common algorithms do not need to be learned from scratch. We add memory, registers, basic operations, and adaptive recurrence to a transformer architecture built on LLaMA3. Then, we define a method for directly compiling algorithms into a differentiable starting library, which is used natively and propagates gradients for optimization. In this preliminary study, we explore the feasability of augmenting LLaMA3 with a differentiable computer, for instance by fine-tuning small transformers on simple algorithmic tasks with variable computational depth.

Mathematical equations have been unreasonably effective in describing complex natural phenomena across various scientific disciplines. However, discovering such insightful equations from data presents significant challenges due to the necessity of navigating extremely high-dimensional combinatorial and nonlinear hypothesis spaces. Traditional methods of equation discovery largely focus on extracting equations from data alone, often neglecting the rich domain-specific prior knowledge that scientists typically depend on. To bridge this gap, we introduce LLM-SR, a novel approach that leverages the extensive scientific knowledge and robust code generation capabilities of Large Language Models (LLMs) to discover scientific equations from data in an efficient manner. Specifically, LLM-SR treats equations as programs with mathematical operators and combines LLMs' scientific priors with evolutionary search over equation programs. The LLM iteratively proposes new equation skeletons, drawing from its physical understanding, which are then optimized against data to estimate skeleton parameters. We demonstrate LLM-SR's effectiveness across three diverse scientific domains, where it discovers physically accurate equations that provide significantly better fits to in-domain and out-of-domain data compared to the well-established equation discovery baselines

Large Language Models (LLMs) have already become quite proficient at solving simpler programming tasks like those in HumanEval or MBPP benchmarks. However, solving more complex and competitive programming tasks is still quite challenging for these models - possibly due to their tendency to generate solutions as monolithic code blocks instead of decomposing them into logical sub-tasks and sub-modules. On the other hand, experienced programmers instinctively write modularized code with abstraction for solving complex tasks, often reusing previously developed modules. To address this gap, we propose CodeChain, a novel framework for inference that elicits modularized code generation through a chain of self-revisions, each being guided by some representative sub-modules generated in previous iterations. Concretely, CodeChain first instructs the LLM to generate modularized codes through chain-of-thought prompting. Then it applies a chain of self-revisions by iterating the two steps: 1) extracting and clustering the generated sub-modules and selecting the cluster representatives as the more generic and re-usable implementations, and 2) augmenting the original chain-of-thought prompt with these selected module-implementations and instructing the LLM to re-generate new modularized solutions. We find that by naturally encouraging the LLM to reuse the previously developed and verified sub-modules, CodeChain can significantly boost both modularity as well as correctness of the generated solutions, achieving relative pass@1 improvements of 35% on APPS and 76% on CodeContests. It is shown to be effective on both OpenAI LLMs as well as open-sourced LLMs like WizardCoder. We also conduct comprehensive ablation studies with different methods of prompting, number of clusters, model sizes, program qualities, etc., to provide useful insights that underpin CodeChain's success.

Program synthesis aims to create accurate, executable code from natural language descriptions. This field has leveraged the power of reinforcement learning (RL) in conjunction with large language models (LLMs), significantly enhancing code generation capabilities. This integration focuses on directly optimizing functional correctness, transcending conventional supervised losses. While current literature predominantly favors policy-based algorithms, attributes of program synthesis suggest a natural compatibility with value-based methods. This stems from rich collection of off-policy programs developed by human programmers, and the straightforward verification of generated programs through automated unit testing (i.e. easily obtainable rewards in RL language). Diverging from the predominant use of policy-based algorithms, our work explores the applicability of value-based approaches, leading to the development of our B-Coder (pronounced Bellman coder). Yet, training value-based methods presents challenges due to the enormous search space inherent to program synthesis. To this end, we propose an initialization protocol for RL agents utilizing pre-trained LMs and a conservative Bellman operator to reduce training complexities. Moreover, we demonstrate how to leverage the learned value functions as a dual strategy to post-process generated programs. Our empirical evaluations demonstrated B-Coder's capability in achieving state-of-the-art performance compared with policy-based methods. Remarkably, this achievement is reached with minimal reward engineering effort, highlighting the effectiveness of value-based RL, independent of reward designs.

A fundamental challenge in formal theorem proving by LLMs is the lack of high-quality training data. Although reinforcement learning or expert iteration partially mitigates this issue by alternating between LLM generating proofs and finetuning them on correctly generated ones, performance quickly plateaus due to the scarcity of correct proofs (sparse rewards). To keep improving the models with limited data, we draw inspiration from mathematicians, who continuously develop new results, partly by proposing novel conjectures or exercises (which are often variants of known results) and attempting to solve them. We design the Self-play Theorem Prover (STP) that simultaneously takes on two roles, conjecturer and prover, each providing training signals to the other. The conjecturer is trained iteratively on previously generated conjectures that are barely provable by the current prover, which incentivizes it to generate increasingly challenging conjectures over time. The prover attempts to prove the conjectures with standard expert iteration. We evaluate STP with both Lean and Isabelle formal versifiers. With 19.8 billion tokens generated during the training in Lean, STP proves 26.3% of the statements in the LeanWorkbook dataset, doubling the previous best result of 13.2% achieved through expert iteration. The final model achieves state-of-the-art performance among whole-proof generation methods on miniF2F-test (61.7%, pass@3200), Proofnet-test (23.1%, pass@3200) and PutnamBench (8/644, pass@3200).

Solving problems through tool use under explicit constraints constitutes a highly challenging yet unavoidable scenario for large language models (LLMs), requiring capabilities such as function calling, instruction following, and self-refinement. However, progress has been hindered by the absence of dedicated evaluations. To address this, we introduce CCTU, a benchmark for evaluating LLM tool use under complex constraints. CCTU is grounded in a taxonomy of 12 constraint categories spanning four dimensions (i.e., resource, behavior, toolset, and response). The benchmark comprises 200 carefully curated and challenging test cases across diverse tool-use scenarios, each involving an average of seven constraint types and an average prompt length exceeding 4,700 tokens. To enable reliable evaluation, we develop an executable constraint validation module that performs step-level validation and enforces compliance during multi-turn interactions between models and their environments. We evaluate nine state-of-the-art LLMs in both thinking and non-thinking modes. Results indicate that when strict adherence to all constraints is required, no model achieves a task completion rate above 20%. Further analysis reveals that models violate constraints in over 50% of cases, particularly in the resource and response dimensions. Moreover, LLMs demonstrate limited capacity for self-refinement even after receiving detailed feedback on constraint violations, highlighting a critical bottleneck in the development of robust tool-use agents. To facilitate future research, we release the data and code.

Effective code optimization in compilers plays a central role in computer and software engineering. While compilers can be made to automatically search the optimization space without the need for user interventions, this is not a standard practice since the search is slow and cumbersome. Here we present CodeZero, an artificial intelligence agent trained extensively on large data to produce effective optimization strategies instantly for each program in a single trial of the agent. To overcome the huge range of possible test programs, we prepare a large dataset of training programs that emphasize quality, naturalness, and diversity. To tackle the vast space of possible optimizations, we adapt deep reinforcement learning to train the agent in a sample-efficient manner through interacting with a world model of the compiler environment. Evaluation on both benchmark suites and production-level code optimization problems demonstrates our agent's supercompiler performances and zero-shot generalization abilities, outperforming built-in optimization options designed by compiler experts. Our methodology kindles the great potential of artificial intelligence for engineering and paves the way for scaling machine learning techniques in the realm of code optimization.

Automatic software system optimization can improve software speed, reduce operating costs, and save energy. Traditional approaches to optimization rely on manual tuning and compiler heuristics, limiting their ability to generalize across diverse codebases and system contexts. Recent methods using Large Language Models (LLMs) offer automation to address these limitations, but often fail to scale to the complexity of real-world software systems and applications. We present SysLLMatic, a system that integrates LLMs with profiling-guided feedback and system performance insights to automatically optimize software code. We evaluate it on three benchmark suites: HumanEval_CPP (competitive programming in C++), SciMark2 (scientific kernels in Java), and DaCapoBench (large-scale software systems in Java). Results show that SysLLMatic can improve system performance, including latency, throughput, energy efficiency, memory usage, and CPU utilization. It consistently outperforms state-of-the-art LLM baselines on microbenchmarks. On large-scale application codes, it surpasses traditional compiler optimizations, achieving average relative improvements of 1.85x in latency and 2.24x in throughput. Our findings demonstrate that LLMs, guided by principled systems thinking and appropriate performance diagnostics, can serve as viable software system optimizers. We further identify limitations of our approach and the challenges involved in handling complex applications. This work provides a foundation for generating optimized code across various languages, benchmarks, and program sizes in a principled manner.

Automated math word problem solvers based on neural networks have successfully managed to obtain 70-80\% accuracy in solving arithmetic word problems. However, it has been shown that these solvers may rely on superficial patterns to obtain their equations. In order to determine what information math word problem solvers use to generate solutions, we remove parts of the input and measure the model's performance on the perturbed dataset. Our results show that the model is not sensitive to the removal of many words from the input and can still manage to find a correct answer when given a nonsense question. This indicates that automatic solvers do not follow the semantic logic of math word problems, and may be overfitting to the presence of specific words.

Automatic program repair seeks to generate correct code from buggy programs, with most approaches searching the correct program in a discrete, symbolic space of source code tokens. This symbolic search is fundamentally limited by its inability to directly reason about program behavior. We introduce Gradient-Based Program Repair (GBPR), a new paradigm that reframes program repair as continuous optimization in a differentiable numerical program space. Our core insight is to compile symbolic programs into differentiable numerical representations, enabling search in the numerical program space directly guided by program behavior. To evaluate GBPR, we present RaspBugs, a new benchmark of 1,466 buggy symbolic RASP programs and their respective numerical representations. Our experiments demonstrate that GBPR can effectively repair buggy symbolic programs by gradient-based optimization in the numerical program space, with convincing repair trajectories. To our knowledge, we are the first to state program repair as continuous optimization in a numerical program space. Our work establishes a new direction for program repair research, bridging two rich worlds: continuous optimization and program behavior.

Building precise simulations of the real world and invoking numerical solvers to answer quantitative problems is an essential requirement in engineering and science. We present FEABench, a benchmark to evaluate the ability of large language models (LLMs) and LLM agents to simulate and solve physics, mathematics and engineering problems using finite element analysis (FEA). We introduce a comprehensive evaluation scheme to investigate the ability of LLMs to solve these problems end-to-end by reasoning over natural language problem descriptions and operating COMSOL Multiphysics^circledR, an FEA software, to compute the answers. We additionally design a language model agent equipped with the ability to interact with the software through its Application Programming Interface (API), examine its outputs and use tools to improve its solutions over multiple iterations. Our best performing strategy generates executable API calls 88% of the time. LLMs that can successfully interact with and operate FEA software to solve problems such as those in our benchmark would push the frontiers of automation in engineering. Acquiring this capability would augment LLMs' reasoning skills with the precision of numerical solvers and advance the development of autonomous systems that can tackle complex problems in the real world. The code is available at https://github.com/google/feabench

Large language model (LLM) coding agents increasingly operate at the repository level, motivating benchmarks that evaluate their ability to optimize entire codebases under realistic constraints. Existing code benchmarks largely rely on synthetic tasks, binary correctness signals, or single-objective evaluation, limiting their ability to assess holistic optimization behavior. We introduce FormulaCode, a benchmark for evaluating agentic optimization on large, real-world codebases with fine-grained, multi-objective performance metrics. FormulaCode comprises 957 performance bottlenecks mined from scientific Python repositories on GitHub, each paired with expert-authored patches and, on average, 264.6 community-maintained performance workloads per task, enabling the holistic ability of LLM agents to optimize codebases under realistic correctness and performance constraints. Our evaluations reveal that repository-scale, multi-objective optimization remains a major challenge for frontier LLM agents. Project website at: https://formula-code.github.io

This paper explores the limits of the current generation of large language models for program synthesis in general purpose programming languages. We evaluate a collection of such models (with between 244M and 137B parameters) on two new benchmarks, MBPP and MathQA-Python, in both the few-shot and fine-tuning regimes. Our benchmarks are designed to measure the ability of these models to synthesize short Python programs from natural language descriptions. The Mostly Basic Programming Problems (MBPP) dataset contains 974 programming tasks, designed to be solvable by entry-level programmers. The MathQA-Python dataset, a Python version of the MathQA benchmark, contains 23914 problems that evaluate the ability of the models to synthesize code from more complex text. On both datasets, we find that synthesis performance scales log-linearly with model size. Our largest models, even without finetuning on a code dataset, can synthesize solutions to 59.6 percent of the problems from MBPP using few-shot learning with a well-designed prompt. Fine-tuning on a held-out portion of the dataset improves performance by about 10 percentage points across most model sizes. On the MathQA-Python dataset, the largest fine-tuned model achieves 83.8 percent accuracy. Going further, we study the model's ability to engage in dialog about code, incorporating human feedback to improve its solutions. We find that natural language feedback from a human halves the error rate compared to the model's initial prediction. Additionally, we conduct an error analysis to shed light on where these models fall short and what types of programs are most difficult to generate. Finally, we explore the semantic grounding of these models by fine-tuning them to predict the results of program execution. We find that even our best models are generally unable to predict the output of a program given a specific input.

Instance-specific algorithm configuration and algorithm portfolios have been shown to offer significant improvements over single algorithm approaches in a variety of application domains. In the SAT and CSP domains algorithm portfolios have consistently dominated the main competitions in these fields for the past five years. For a portfolio approach to be effective there are two crucial conditions that must be met. First, there needs to be a collection of complementary solvers with which to make a portfolio. Second, there must be a collection of problem features that can accurately identify structural differences between instances. This paper focuses on the latter issue: feature representation, because, unlike SAT, not every problem has well-studied features. We employ the well-known SATzilla feature set, but compute alternative sets on different SAT encodings of CSPs. We show that regardless of what encoding is used to convert the instances, adequate structural information is maintained to differentiate between problem instances, and that this can be exploited to make an effective portfolio-based CSP solver.

Recent reasoning-oriented LLMs have demonstrated strong performance on challenging tasks such as mathematics and science examinations. However, core cognitive faculties of human intelligence, such as abstract reasoning and generalization, remain underexplored. To address this, we evaluate recent reasoning-oriented LLMs on the Abstraction and Reasoning Corpus (ARC) benchmark, which explicitly demands both faculties. We formulate ARC as a program synthesis task and propose nine candidate solvers. Experimental results show that repeated-sampling planning-aided code generation (RSPC) achieves the highest test accuracy and demonstrates consistent generalization across most LLMs. To further improve performance, we introduce an ARC solver, Knowledge Augmentation for Abstract Reasoning (KAAR), which encodes core knowledge priors within an ontology that classifies priors into three hierarchical levels based on their dependencies. KAAR progressively expands LLM reasoning capacity by gradually augmenting priors at each level, and invokes RSPC to generate candidate solutions after each augmentation stage. This stage-wise reasoning reduces interference from irrelevant priors and improves LLM performance. Empirical results show that KAAR maintains strong generalization and consistently outperforms non-augmented RSPC across all evaluated LLMs, achieving around 5% absolute gains and up to 64.52% relative improvement. Despite these achievements, ARC remains a challenging benchmark for reasoning-oriented LLMs, highlighting future avenues of progress in LLMs.

Chain-of-Thought (CoT) plays a crucial role in reasoning for math problem solving. We conduct a comprehensive examination of methods for designing CoT, comparing conventional natural language CoT with various program CoTs, including the self-describing program, the comment-describing program, and the non-describing program. Furthermore, we investigate the impact of programming language on program CoTs, comparing Python and Wolfram Language. Through extensive experiments on GSM8K, MATHQA, and SVAMP, we find that program CoTs often have superior effectiveness in math problem solving. Notably, the best performing combination with 30B parameters beats GPT-3.5-turbo by a significant margin. The results show that self-describing program offers greater diversity and thus can generally achieve higher performance. We also find that Python is a better choice of language than Wolfram for program CoTs. The experimental results provide a valuable guideline for future CoT designs that take into account both programming language and coding style for further advancements. Our datasets and code are publicly available.

Language Model Programs, i.e. sophisticated pipelines of modular language model (LM) calls, are increasingly advancing NLP tasks, but they require crafting prompts that are jointly effective for all modules. We study prompt optimization for LM programs, i.e. how to update these prompts to maximize a downstream metric without access to module-level labels or gradients. To make this tractable, we factorize our problem into optimizing the free-form instructions and few-shot demonstrations of every module and introduce several strategies to craft task-grounded instructions and navigate credit assignment across modules. Our strategies include (i) program- and data-aware techniques for proposing effective instructions, (ii) a stochastic mini-batch evaluation function for learning a surrogate model of our objective, and (iii) a meta-optimization procedure in which we refine how LMs construct proposals over time. Using these insights we develop MIPRO, a novel algorithm for optimizing LM programs. MIPRO outperforms baseline optimizers on five of seven diverse multi-stage LM programs using a best-in-class open-source model (Llama-3-8B), by as high as 13% accuracy. We have released our new optimizers and benchmark in DSPy at http://dspy.ai

In recent times, a plethora of Large Code Generation Models (LCGMs) have been proposed, showcasing significant potential in assisting developers with complex programming tasks. Benchmarking LCGMs necessitates the creation of a set of diverse programming problems, and each problem comprises the prompt (including the task description), canonical solution, and test inputs. The existing methods for constructing such a problem set can be categorized into two main types: manual methods and perturbation-based methods. However, manual methods demand high effort and lack scalability, while also risking data integrity due to LCGMs' potentially contaminated data collection, and perturbation-based approaches mainly generate semantically homogeneous problems with the same canonical solutions and introduce typos that can be easily auto-corrected by IDE, making them ineffective and unrealistic. In this work, we propose the idea of programming problem merging (PPM) and provide two implementation of this idea, we utilize our tool on two widely-used datasets and compare it against nine baseline methods using eight code generation models. The results demonstrate the effectiveness of our tool in generating more challenging, diverse, and natural programming problems, comparing to the baselines.

We introduce KAPSO, a modular framework for autonomous program synthesis and optimization. Given a natural language goal and an evaluation method, KAPSO iteratively performs ideation, code synthesis and editing, execution, evaluation, and learning to improve a runnable artifact toward measurable objectives. Rather than treating synthesis as the endpoint, KAPSO uses synthesis as an operator within a long-horizon optimization loop, where progress is defined by evaluator outcomes. KAPSO targets long-horizon failures common in coding agents, including lost experimental state, brittle debugging, and weak reuse of domain expertise, by integrating three tightly coupled components. First, a git-native experimentation engine isolates each attempt as a branch, producing reproducible artifacts and preserving provenance across iterations. Second, a knowledge system ingests heterogeneous sources, including repositories, internal playbooks, and curated external resources such as documentation, scientific papers, and web search results, and organizes them into a structured representation that supports retrieval over workflows, implementations, and environment constraints. Third, a cognitive memory layer coordinates retrieval and maintains an episodic store of reusable lessons distilled from experiment traces (run logs, diffs, and evaluator feedback), reducing repeated error modes and accelerating convergence. We evaluated KAPSO on MLE-Bench (Kaggle-style ML competitions) and ALE-Bench (AtCoder heuristic optimization), and report end-to-end performance. Code Available at: https://github.com/Leeroo-AI/kapso

Semidefinite programming is a fundamental problem class in convex optimization, but despite recent advances in solvers, solving large-scale semidefinite programs remains challenging. Generally the matrix functions involved are spectral or unitarily invariant, i.e., they depend only on the eigenvalues or singular values of the matrix. This paper investigates how spectral matrix cones -- cones defined from epigraphs and perspectives of spectral or unitarily invariant functions -- can be used to enhance first-order conic solvers for semidefinite programs. Our main result shows that projecting a matrix can be reduced to projecting its eigenvalues or singular values, which we demonstrate can be done at a negligible cost compared to the eigenvalue or singular value decomposition itself. We have integrated support for spectral matrix cone projections into the Splitting Conic Solver (SCS). Numerical experiments show that SCS with this enhancement can achieve speedups of up to an order of magnitude for solving semidefinite programs arising in experimental design, robust principal component analysis, and graph partitioning.

Can AI make progress on important, unsolved mathematical problems? Large language models are now capable of sophisticated mathematical and scientific reasoning, but whether they can perform novel research is still widely debated and underexplored. We introduce HorizonMath, a benchmark of over 100 predominantly unsolved problems spanning 8 domains in computational and applied mathematics, paired with an open-source evaluation framework for automated verification. Our benchmark targets a class of problems where discovery is hard, requiring meaningful mathematical insight, but verification is computationally efficient and simple. Because these solutions are unknown, HorizonMath is immune to data contamination, and most state-of-the-art models score near 0%. Existing research-level benchmarks instead rely on formal proof verification or manual review, both of which are expensive to scale. Using this platform, we find two problems for which GPT 5.4 Pro proposes solutions that improve on the best-known published results, representing potential novel contributions (pending expert review). We release HorizonMath as an open challenge and a growing community resource, where correct solutions to problems in the unsolved problem classes could constitute novel results in the mathematical literature.

Agentic AI systems built on large language models (LLMs) offer significant potential for automating complex workflows, from software development to customer support. However, LLM agents often underperform due to suboptimal configurations; poorly tuned prompts, tool descriptions, and parameters that typically require weeks of manual refinement. Existing optimization methods either are too complex for general use or treat components in isolation, missing critical interdependencies. We present ARTEMIS, a no-code evolutionary optimization platform that jointly optimizes agent configurations through semantically-aware genetic operators. Given only a benchmark script and natural language goals, ARTEMIS automatically discovers configurable components, extracts performance signals from execution logs, and evolves configurations without requiring architectural modifications. We evaluate ARTEMIS on four representative agent systems: the ALE Agent for competitive programming on AtCoder Heuristic Contest, achieving a 13.6% improvement in acceptance rate; the Mini-SWE Agent for code optimization on SWE-Perf, with a statistically significant 10.1\% performance gain; and the CrewAI Agent for cost and mathematical reasoning on Math Odyssey, achieving a statistically significant 36.9% reduction in the number of tokens required for evaluation. We also evaluate the MathTales-Teacher Agent powered by a smaller open-source model (Qwen2.5-7B) on GSM8K primary-level mathematics problems, achieving a 22\% accuracy improvement and demonstrating that ARTEMIS can optimize agents based on both commercial and local models.

We evaluate AI-assisted generative capabilities on fundamental numerical kernels in high-performance computing (HPC), including AXPY, GEMV, GEMM, SpMV, Jacobi Stencil, and CG. We test the generated kernel codes for a variety of language-supported programming models, including (1) C++ (e.g., OpenMP [including offload], OpenACC, Kokkos, SyCL, CUDA, and HIP), (2) Fortran (e.g., OpenMP [including offload] and OpenACC), (3) Python (e.g., numba, Numba, cuPy, and pyCUDA), and (4) Julia (e.g., Threads, CUDA.jl, AMDGPU.jl, and KernelAbstractions.jl). We use the GitHub Copilot capabilities powered by OpenAI Codex available in Visual Studio Code as of April 2023 to generate a vast amount of implementations given simple <kernel> + <programming model> + <optional hints> prompt variants. To quantify and compare the results, we propose a proficiency metric around the initial 10 suggestions given for each prompt. Results suggest that the OpenAI Codex outputs for C++ correlate with the adoption and maturity of programming models. For example, OpenMP and CUDA score really high, whereas HIP is still lacking. We found that prompts from either a targeted language such as Fortran or the more general-purpose Python can benefit from adding code keywords, while Julia prompts perform acceptably well for its mature programming models (e.g., Threads and CUDA.jl). We expect for these benchmarks to provide a point of reference for each programming model's community. Overall, understanding the convergence of large language models, AI, and HPC is crucial due to its rapidly evolving nature and how it is redefining human-computer interactions.

The recently released GPT-4 Code Interpreter has demonstrated remarkable proficiency in solving challenging math problems, primarily attributed to its ability to seamlessly reason with natural language, generate code, execute code, and continue reasoning based on the execution output. In this paper, we present a method to fine-tune open-source language models, enabling them to use code for modeling and deriving math equations and, consequently, enhancing their mathematical reasoning abilities. We propose a method of generating novel and high-quality datasets with math problems and their code-based solutions, referred to as MathCodeInstruct. Each solution interleaves natural language, code, and execution results. We also introduce a customized supervised fine-tuning and inference approach. This approach yields the MathCoder models, a family of models capable of generating code-based solutions for solving challenging math problems. Impressively, the MathCoder models achieve state-of-the-art scores among open-source LLMs on the MATH (45.2%) and GSM8K (83.9%) datasets, substantially outperforming other open-source alternatives. Notably, the MathCoder model not only surpasses ChatGPT-3.5 and PaLM-2 on GSM8K and MATH but also outperforms GPT-4 on the competition-level MATH dataset. The dataset and models will be released at https://github.com/mathllm/MathCoder.

Decompilation is widely used in reverse engineering to recover high-level language code from binary executables. While recent approaches leveraging Large Language Models (LLMs) have shown promising progress, they typically treat assembly code as a linear sequence of instructions, overlooking arbitrary jump patterns and isolated data segments inherent to binary files. This limitation significantly hinders their ability to correctly infer source code semantics from assembly code. To address this limitation, we propose \saltm, a novel binary decompilation method that abstracts stable logical features shared between binary and source code. The core idea of \saltm is to abstract selected binary-level operations, such as specific jumps, into a high-level logic framework that better guides LLMs in semantic recovery. Given a binary function, \saltm constructs a Source-level Abstract Logic Tree (\salt) from assembly code to approximate the logic structure of high-level language. It then fine-tunes an LLM using the reconstructed \salt to generate decompiled code. Finally, the output is refined through error correction and symbol recovery to improve readability and correctness. We compare \saltm to three categories of baselines (general-purpose LLMs, commercial decompilers, and decompilation methods) using three well-known datasets (Decompile-Eval, MBPP, Exebench). Our experimental results demonstrate that \saltm is highly effective in recovering the logic of the source code, significantly outperforming state-of-the-art methods (e.g., 70.4\% TCP rate on Decompile-Eval with a 10.6\% improvement). The results further validate its robustness against four commonly used obfuscation techniques. Additionally, analyses of real-world software and a user study confirm that our decompiled output offers superior assistance to human analysts in comprehending binary functions.

Maintaining legacy software requires many software and systems engineering hours. Assembly code programs, which demand low-level control over the computer machine state and have no variable names, are particularly difficult for humans to analyze. Existing conventional program translators guarantee correctness, but are hand-engineered for the source and target programming languages in question. Learned transpilation, i.e. automatic translation of code, offers an alternative to manual re-writing and engineering efforts. Automated symbolic program translation approaches guarantee correctness but struggle to scale to longer programs due to the exponentially large search space. Their rigid rule-based systems also limit their expressivity, so they can only reason about a reduced space of programs. Probabilistic neural language models (LMs) produce plausible outputs for every input, but do so at the cost of guaranteed correctness. In this work, we leverage the strengths of LMs and symbolic solvers in a neurosymbolic approach to learned transpilation for assembly code. Assembly code is an appropriate setting for a neurosymbolic approach, since assembly code can be divided into shorter non-branching basic blocks amenable to the use of symbolic methods. Guess & Sketch extracts alignment and confidence information from features of the LM then passes it to a symbolic solver to resolve semantic equivalence of the transpilation input and output. We test Guess & Sketch on three different test sets of assembly transpilation tasks, varying in difficulty, and show that it successfully transpiles 57.6% more examples than GPT-4 and 39.6% more examples than an engineered transpiler. We also share a training and evaluation dataset for this task.