View source for Reasoning language model

{{unreliable sources|date=January 2025}}
{{Short description|Language models designed for reasoning tasks}}

'''Reasoning language models''' are [[artificial intelligence]] systems that combine [[natural language processing]] with structured reasoning capabilities. These models are usually constructed by [[Prompt engineering|prompting]], [[Fine-tuning (deep learning)|supervised finetuning]] (SFT), and [[reinforcement learning]] (RL) initialized with [[Pretrained language model|pretrained language models]].

== Prompting ==
{{Main|Prompt engineering}}
A language model is a generative model of a training dataset of texts. Prompting means constructing a text prompt, such that, conditional on the text prompt, the language model generates a solution to the task. Prompting can be applied to a pretrained model ("base model"), a base model that has undergone SFT, or RL, or both.<ref>{{Citation |last1=Qiao |first1=Shuofei |title=Reasoning with Language Model Prompting: A Survey |date=2023-09-18 |arxiv=2212.09597 |last2=Ou |first2=Yixin |last3=Zhang |first3=Ningyu |last4=Chen |first4=Xiang |last5=Yao |first5=Yunzhi |last6=Deng |first6=Shumin |last7=Tan |first7=Chuanqi |last8=Huang |first8=Fei |last9=Chen |first9=Huajun}}</ref>

=== Chain of thought ===
Chain of Thought prompting (CoT) prompts the model to answer a question by first generating a "chain of thought", i.e. steps of reasoning that mimic a [[train of thought]].<ref name="weipaper2">{{cite conference |last1=Wei |first1=Jason |last2=Wang |first2=Xuezhi |last3=Schuurmans |first3=Dale |last4=Bosma |first4=Maarten |last5=Ichter |first5=Brian |last6=Xia |first6=Fei |last7=Chi |first7=Ed H. |last8=Le |first8=Quoc V. |last9=Zhou |first9=Denny |date=31 October 2022 |title=Chain-of-Thought Prompting Elicits Reasoning in Large Language Models |conference=Advances in Neural Information Processing Systems (NeurIPS 2022) |language=en |volume=35 |arxiv=2201.11903}}</ref> It was published in 2022 by the [[Google Brain|Brain team]] of Google on the [[PaLM|PaLM-540B]] model.<ref>{{Cite web |author=Sharan Narang and Aakanksha Chowdhery |date=2022-04-04 |title=Pathways Language Model (PaLM): Scaling to 540 Billion Parameters for Breakthrough Performance |url=https://ai.googleblog.com/2022/04/pathways-language-model-palm-scaling-to.html}}</ref> In CoT prompting, the prompt is of the form "<Input> Let's think step by step", and the model would respond with a chain of reasoning steps, ended with an answer:<math display="block">
\text{Input} \rightarrow \underbrace{\text{Step}_1 \rightarrow \text{Step}_2 \rightarrow \cdots \rightarrow \text{Step}_n}_{\text{Reasoning chain}} \rightarrow \text{Answer}
</math>Similarly, Tree of Thought prompting generalizes CoT by prompting the model to generate one or more "possible next steps", and then running the model on each of the possible next steps by [[Breadth-first search|breadth-first]], [[Beam search|beam]], or some other method of tree search.<ref>{{Cite arXiv |eprint=2305.10601 |class=cs.CL |first1=Shunyu |last1=Yao |first2=Dian |last2=Yu |title=Tree of Thoughts: Deliberate Problem Solving with Large Language Models |date=2023-05-17 |last3=Zhao |first3=Jeffrey |last4=Shafran |first4=Izhak |last5=Griffiths |first5=Thomas L. |last6=Cao |first6=Yuan |last7=Narasimhan |first7=Karthik}}</ref> Similarly, Graph of Thought generalizes CoT so that the reasoning steps form a [[directed acyclic graph]].<ref>{{Cite journal |last1=Besta |first1=Maciej |last2=Blach |first2=Nils |last3=Kubicek |first3=Ales |last4=Gerstenberger |first4=Robert |last5=Podstawski |first5=Michal |last6=Gianinazzi |first6=Lukas |last7=Gajda |first7=Joanna |last8=Lehmann |first8=Tomasz |last9=Niewiadomski |first9=Hubert |last10=Nyczyk |first10=Piotr |last11=Hoefler |first11=Torsten |date=2024-03-24 |title=Graph of Thoughts: Solving Elaborate Problems with Large Language Models |url=https://ojs.aaai.org/index.php/AAAI/article/view/29720 |journal=Proceedings of the AAAI Conference on Artificial Intelligence |language=en |volume=38 |issue=16 |pages=17682–17690 |doi=10.1609/aaai.v38i16.29720 |issn=2374-3468|arxiv=2308.09687 }}</ref>

Self-consistency decoding performs several chain-of-thought rollouts, then selects the most commonly reached conclusion out of all the rollouts.<ref>{{cite arXiv |eprint=2203.11171 |class=cs.CL |first1=Xuezhi |last1=Wang |first2=Jason |last2=Wei |title=Self-Consistency Improves Chain of Thought Reasoning in Language Models |date=2022-03-01 |last3=Schuurmans |first3=Dale |last4=Le |first4=Quoc |last5=Chi |first5=Ed |last6=Narang |first6=Sharan |last7=Chowdhery |first7=Aakanksha |last8=Zhou |first8=Denny}}</ref> If the rollouts disagree by a lot, a human can be queried for the correct chain of thought.<ref>{{cite arXiv |eprint=2302.12246 |class=cs.CL |first1=Shizhe |last1=Diao |first2=Pengcheng |last2=Wang |title=Active Prompting with Chain-of-Thought for Large Language Models |date=2023-02-01 |last3=Lin |first3=Yong |last4=Zhang |first4=Tong}}</ref>

=== Retrieval-augmented generation ===
{{Main article|Retrieval-augmented generation}}
A language model may answer a query by first querying a database of documents using the query. The document retrieval can be via a [[vector database]], summary index, tree index, or keyword table index.<ref>{{Cite web |title=How Each Index Works - LlamaIndex 🦙 v0.10.17 |url=https://docs.llamaindex.ai/en/v0.10.17/module_guides/indexing/index_guide.html |access-date=2024-04-08 |website=docs.llamaindex.ai}}</ref> Following document retrieval, the LLM generates an output that incorporates information from both the query and the retrieved documents.<ref>{{Cite journal |last1=Lewis |first1=Patrick |last2=Perez |first2=Ethan |last3=Piktus |first3=Aleksandra |last4=Petroni |first4=Fabio |last5=Karpukhin |first5=Vladimir |last6=Goyal |first6=Naman |last7=Küttler |first7=Heinrich |last8=Lewis |first8=Mike |last9=Yih |first9=Wen-tau |last10=Rocktäschel |first10=Tim |last11=Riedel |first11=Sebastian |last12=Kiela |first12=Douwe |date=2020 |title=Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks |url=https://proceedings.neurips.cc/paper/2020/hash/6b493230205f780e1bc26945df7481e5-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=Curran Associates, Inc. |volume=33 |pages=9459–9474 |arxiv=2005.11401}}</ref>

=== Tool use ===
Language models can perform long reasoning steps by calling external methods, such as numerical recipes, program interpreters, API calls, and so on. This can be prompt-engineered by describing the external methods in-context (an example of in-context learning) or finetuned into the model.<ref>{{Cite journal |last1=Schick |first1=Timo |last2=Dwivedi-Yu |first2=Jane |last3=Dessi |first3=Roberto |last4=Raileanu |first4=Roberta |last5=Lomeli |first5=Maria |last6=Hambro |first6=Eric |last7=Zettlemoyer |first7=Luke |last8=Cancedda |first8=Nicola |last9=Scialom |first9=Thomas |date=2023-12-15 |title=Toolformer: Language Models Can Teach Themselves to Use Tools |url=https://proceedings.neurips.cc/paper_files/paper/2023/hash/d842425e4bf79ba039352da0f658a906-Abstract-Conference.html |journal=Advances in Neural Information Processing Systems |language=en |volume=36 |pages=68539–68551|arxiv=2302.04761 }}</ref>

== Supervised finetuning ==
A base model can be finetuned on a dataset of reasoning tasks with example solutions and reasoning traces. The finetuned model would then be able to generate reasoning traces for a given problem.<ref name=":0">{{Citation |last1=Uesato |first1=Jonathan |title=Solving math word problems with process- and outcome-based feedback |date=2022-11-25 |arxiv=2211.14275 |last2=Kushman |first2=Nate |last3=Kumar |first3=Ramana |last4=Song |first4=Francis |last5=Siegel |first5=Noah |last6=Wang |first6=Lisa |last7=Creswell |first7=Antonia |last8=Irving |first8=Geoffrey |last9=Higgins |first9=Irina}}</ref><ref name=":2" />

As it is expensive to get humans to write reasoning traces for a SFT dataset, researchers have proposed ways to automatically construct SFT datasets. In rejection sampling finetuning (RFT), new reasoning traces are collected via a loop:<ref>{{Citation |last1=Yuan |first1=Zheng |title=Scaling Relationship on Learning Mathematical Reasoning with Large Language Models |date=2023-09-13 |arxiv=2308.01825 |last2=Yuan |first2=Hongyi |last3=Li |first3=Chengpeng |last4=Dong |first4=Guanting |last5=Lu |first5=Keming |last6=Tan |first6=Chuanqi |last7=Zhou |first7=Chang |last8=Zhou |first8=Jingren}}</ref>

# Sample a task prompt
# Generate many reasoning traces for the prompt.
# Use a verifier to remove reasoning traces with the wrong final answer.
# For each remaining trace, extract the set of equations appearing in it. Deduplicate the traces so that each one has a different set of equations. Add those to the dataset.

== Reinforcement learning ==
A pretrained language model can be further trained by RL. In the RL formalism, a generative language model is a '''policy''' <math>\pi</math>. A prompt specifying a task to solve is an environmental '''state''' <math>x</math>, and the response of the language model to the prompt is an '''action''' <math>y</math>. The probability that the language model responds <math>x</math> with <math>y</math> is <math>\pi(y|x)</math>.

Training a reasoning language model by RL then consists of constructing a '''reward model''' <math>r(x, y)</math> to guide the RL process. Intuitively, a reward model describes how desirable/appropriate/good the response is for the prompt. For reasoning language model, the prompt describes a reasoning task, and the reward would be high if the response solves the task, and low if the response fails to solve the task.

For reasoning language models, the model's response <math>y</math> may be broken down into multiple steps, in which case it is written as <math>y_1, y_2, \dots, y_n</math>.

=== Outcome Reward Model ===
{{Anchor|Outcome Reward Model|ORM}}

Outcome reward model, or outcome-supervised RM (ORM),<ref name=":0" /> is a reward model that computes the reward of a step <math>r(x, y_1, \dots, y_i)</math> determined by the final answer: <math>r(x, y_1, \dots, y_i) = r(x, y_n)</math>. They are also called "verifiers". 

For tasks with an answer that is easy to verify, such as [[Word problem (mathematics education)|word problems in math]], the outcome reward can simply be binary: 1 if the final answer is correct, and 0 otherwise.<ref name=":0" /> If the answer is not easy to verify programmatically, humans can manually label the answers as correct or not, then the labels can be used to finetune a base model that predicts the human label.<ref name=":2">{{Citation |last1=Cobbe |first1=Karl |title=Training Verifiers to Solve Math Word Problems |date=2021-11-18 |arxiv=2110.14168 |last2=Kosaraju |first2=Vineet |last3=Bavarian |first3=Mohammad |last4=Chen |first4=Mark |last5=Jun |first5=Heewoo |last6=Kaiser |first6=Lukasz |last7=Plappert |first7=Matthias |last8=Tworek |first8=Jerry |last9=Hilton |first9=Jacob}}</ref> For other kinds of tasks, such as creative writing, where task performance is not binary true/false, one can train a reward model by finetuning a base model on human [[Ranking (statistics)|ranked preference]] data, such as used in [[reinforcement learning from human feedback]].<ref name=":1">{{Citation |last1=Lightman |first1=Hunter |title=Let's Verify Step by Step |date=2023-05-31 |arxiv=2305.20050 |last2=Kosaraju |first2=Vineet |last3=Burda |first3=Yura |last4=Edwards |first4=Harri |last5=Baker |first5=Bowen |last6=Lee |first6=Teddy |last7=Leike |first7=Jan |last8=Schulman |first8=John |last9=Sutskever |first9=Ilya}}</ref> A base model can also be finetuned to predict, given a partial thinking trace <math>x, y_1, \dots, y_m</math>, whether the final answer would be correct or not. This can then be used as a binary reward signal.<ref name=":0" />

The ORM is usually trained via [[logistic regression]], i.e. minimizing [[Cross-entropy|cross-entropy loss]].<ref name=":3" />

Given a PRM, an ORM can be constructed by multiplying the total process reward during the reasoning trace,<ref name=":1" /> or by taking the minimum,<ref name=":3" /> or some other method to aggregate the process rewards.

=== Process Reward Model ===
{{Anchor|Process Reward Model|PRM}}

Process reward model, or process-supervised RM (PRM),<ref name=":0" /> is a reward model that computes the reward of a step <math>r(x, y_1, \dots, y_i)</math> determined by the steps so far: <math>(x, y_1, \dots, y_i)</math>.

Given a partial thinking trace <math>x, y_1, \dots, y_m</math>, a human can be queried as to whether the steps ''so far'' are correct, regardless of whether the ultimate answer would be correct. This can then be used as a binary reward signal. As human labels are expensive, a base model can be finetuned to predict the human labels.<ref name=":0" /> The PRM is usually trained via [[logistic regression]], i.e. minimizing [[Cross-entropy|cross-entropy loss]].<ref name=":3" />

As an example, in a 2023 OpenAI paper, 800K process labels were collected for 75K solution traces. A labeler would be presented with a solution trace, and keep labelling "positive" if the step progresses towards the solution, "neutral" if it is not wrong, but does not progress towards solution, and "negative" if it is a mistake. As soon as a "negative" label is entered, the labeler stops labeling that thinking trace, and begins labeling another one. The idea was that, while labelling subsequent reasoning steps can provide even richer supervision signals, simply labeling up to the first error was sufficient for training a competent PRM.<ref name=":1" /><ref>{{Citation |title=openai/prm800k |date=2025-01-27 |url=https://github.com/openai/prm800k |access-date=2025-01-27 |publisher=OpenAI}}</ref>

As human labels are expensive, researchers have proposed methods to create PRM without human labels on the processes. Inspired by [[Monte Carlo tree search]] (MCTS), the Math-Shepherd method samples multiple continuations until the end, starting at each reasoning step <math>y_i</math>, and set the reward at that step to be either <math>\frac{\#\text{(correct answers)}}{\#\text{(total answers)}}</math> in the case of "soft estimation", or <math>\begin{cases}
1 & \text{if one of the answers is correct}\\
0 & \text{else}
\end{cases}</math> in the case of "hard estimation". This creates process reward using only an ORM, which is usually easier or cheaper to construct. After creating these process reward labels, a PRM can be trained on them.<ref name=":3">{{Cite journal |last1=Wang |first1=Peiyi |last2=Li |first2=Lei |last3=Shao |first3=Zhihong |last4=Xu |first4=Runxin |last5=Dai |first5=Damai |last6=Li |first6=Yifei |last7=Chen |first7=Deli |last8=Wu |first8=Yu |last9=Sui |first9=Zhifang |date=August 2024 |editor-last=Ku |editor-first=Lun-Wei |editor2-last=Martins |editor2-first=Andre |editor3-last=Srikumar |editor3-first=Vivek |title=Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations |url=https://aclanthology.org/2024.acl-long.510/ |journal=Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) |location=Bangkok, Thailand |publisher=Association for Computational Linguistics |pages=9426–9439 |doi=10.18653/v1/2024.acl-long.510|arxiv=2312.08935 }}</ref> Some have tried a fully MCTS approach.<ref>{{Citation |last1=Chen |first1=Guoxin |title=AlphaMath Almost Zero: Process Supervision without Process |date=2024-09-27 |arxiv=2405.03553 |last2=Liao |first2=Minpeng |last3=Li |first3=Chengxi |last4=Fan |first4=Kai}}</ref>

One can also use an ORM to implicitly construct a PRM, similar to [[direct preference optimization]].<ref>{{Citation |last1=Yuan |first1=Lifan |title=Free Process Rewards without Process Labels |date=2024-12-02 |arxiv=2412.01981 |last2=Li |first2=Wendi |last3=Chen |first3=Huayu |last4=Cui |first4=Ganqu |last5=Ding |first5=Ning |last6=Zhang |first6=Kaiyan |last7=Zhou |first7=Bowen |last8=Liu |first8=Zhiyuan |last9=Peng |first9=Hao}}</ref>

<ref>{{Citation |last1=DeepSeek-AI |title=DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning |date=2025-01-22 |arxiv=2501.12948 |last2=Guo |first2=Daya |last3=Yang |first3=Dejian |last4=Zhang |first4=Haowei |last5=Song |first5=Junxiao |last6=Zhang |first6=Ruoyu |last7=Xu |first7=Runxin |last8=Zhu |first8=Qihao |last9=Ma |first9=Shirong}}</ref><ref>{{Citation |last1=Shao |first1=Zhihong |title=DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models |date=2024-04-27 |arxiv=2402.03300 |last2=Wang |first2=Peiyi |last3=Zhu |first3=Qihao |last4=Xu |first4=Runxin |last5=Song |first5=Junxiao |last6=Bi |first6=Xiao |last7=Zhang |first7=Haowei |last8=Zhang |first8=Mingchuan |last9=Li |first9=Y. K.}}</ref>

=== Guided sampling ===
A trained ORM can be used to select the best response. The policy would rollout multiple responses, and a trained ORM would select the best response. This allows a simple form of [[Neural scaling law|test time compute scaling]] ("best-of-N").<ref name=":2" /> <ref>{{Citation |last1=Zhang |first1=Di |title=LLaMA-Berry: Pairwise Optimization for O1-like Olympiad-Level Mathematical Reasoning |date=2024-11-21 |arxiv=2410.02884 |last2=Wu |first2=Jianbo |last3=Lei |first3=Jingdi |last4=Che |first4=Tong |last5=Li |first5=Jiatong |last6=Xie |first6=Tong |last7=Huang |first7=Xiaoshui |last8=Zhang |first8=Shufei |last9=Pavone |first9=Marco}}</ref>

A trained PRM can also be used to guide reasoning by greedy [[Tree traversal|tree search]]. That is, the policy model generates several possible next reasoning steps, and the PRM selects the best one, and the process repeats. This is similar to how a trained ORM can be used to select the best response.<ref>{{Citation |last1=Ma |first1=Qianli |title=Let's reward step by step: Step-Level reward model as the Navigators for Reasoning |date=2023-10-16 |arxiv=2310.10080 |last2=Zhou |first2=Haotian |last3=Liu |first3=Tingkai |last4=Yuan |first4=Jianbo |last5=Liu |first5=Pengfei |last6=You |first6=Yang |last7=Yang |first7=Hongxia}}</ref> [[Beam search]] perform better than greedy search.

Lookahead search is another tree search method, where the policy model generates several possible next reasoning steps, then make a (partial) rollout for each. If a solution endpoint is reached during the forward simulation, the process halts early. Otherwise, the PRM is used to calculate the total reward for each rollout. The step with the highest rollout is selected.<ref>{{Citation |last1=Snell |first1=Charlie |title=Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters |date=2024-08-06 |arxiv=2408.03314 |last2=Lee |first2=Jaehoon |last3=Xu |first3=Kelvin |last4=Kumar |first4=Aviral}}</ref>

Self-consistency can be combined with an ORM. The model would be used to generate multiple answers, and the answers would be clustered, so that each cluster has the same answer. The ORM is used to compute the reward for each answer, and the rewards within each cluster is summed. The answer corresponding to the cluster with the highest summed reward is output.<ref name=":3" />

== Applications ==
Prompt engineering was discovered in [[GPT-3]] as "few-shot learning",<ref>{{Cite journal |last1=Brown |first1=Tom B. |last2=Mann |first2=Benjamin |last3=Ryder |first3=Nick |last4=Subbiah |first4=Melanie |last5=Kaplan |first5=Jared |last6=Dhariwal |first6=Prafulla |last7=Neelakantan |first7=Arvind |last8=Shyam |first8=Pranav |last9=Sastry |first9=Girish |last10=Askell |first10=Amanda |last11=Agarwal |first11=Sandhini |last12=Herbert-Voss |first12=Ariel |last13=Krueger |first13=Gretchen |last14=Henighan |first14=Tom |last15=Child |first15=Rewon |date=2020-12-06 |title=Language models are few-shot learners |url=https://dl.acm.org/doi/abs/10.5555/3495724.3495883 |journal=Proceedings of the 34th International Conference on Neural Information Processing Systems |series=NIPS '20 |location=Red Hook, NY, USA |publisher=Curran Associates Inc. |pages=1877–1901 |isbn=978-1-7138-2954-6}}</ref> which began a period of research into "eliciting" capacities of pretrained language models. It was then found that a model could be prompted to perform CoT reasoning, which improves its performance on reasoning tasks.

== Benchmark ==
{{Main|Benchmark (computing)|List of language model benchmarks}}

The reasoning ability of language models are usually tested on problems with unambiguous solutions that can be cheaply checked, and requires reasoning when solved by a human. Such problems are usually in mathematics and [[competitive programming]]. The answer is usually an array of integers, a multiple choice letter, or a program that passes [[Unit testing|unit tests]] within a limited runtime. Some common ones include: 

* GSM8K (Grade School Math): 8.5K linguistically diverse [[Primary school|elementary school]] [[Word problem (mathematics education)|math word problems]] that require 2 to 8 basic arithmetic operations to solve.<ref name=":2" />
* [[MMLU]] (Measuring Massive Multitask Language Understanding): 16,000 multiple-choice questions spanning 57 academic subjects including mathematics, philosophy, law, and medicine.<ref>{{Citation |last1=Hendrycks |first1=Dan |title=Measuring Massive Multitask Language Understanding |date=2021-01-12 |arxiv=2009.03300 |last2=Burns |first2=Collin |last3=Basart |first3=Steven |last4=Zou |first4=Andy |last5=Mazeika |first5=Mantas |last6=Song |first6=Dawn |last7=Steinhardt |first7=Jacob}}</ref>
* GPQA (Google-Proof Q&A): 448 multiple-choice questions written by domain experts in biology, physics, and chemistry, and requires PhD-level experts to solve.<ref>{{Citation |last1=Rein |first1=David |title=GPQA: A Graduate-Level Google-Proof Q&A Benchmark |date=2023-11-20 |arxiv=2311.12022 |last2=Hou |first2=Betty Li |last3=Stickland |first3=Asa Cooper |last4=Petty |first4=Jackson |last5=Pang |first5=Richard Yuanzhe |last6=Dirani |first6=Julien |last7=Michael |first7=Julian |last8=Bowman |first8=Samuel R.}}</ref>
* HumanEval: Programming problems where the solution is always a python function, often just a few lines long.<ref name=":4">{{Citation |last1=Chen |first1=Mark |title=Evaluating Large Language Models Trained on Code |date=2021-07-14 |arxiv=2107.03374 |last2=Tworek |first2=Jerry |last3=Jun |first3=Heewoo |last4=Yuan |first4=Qiming |last5=Pinto |first5=Henrique Ponde de Oliveira |last6=Kaplan |first6=Jared |last7=Edwards |first7=Harri |last8=Burda |first8=Yuri |last9=Joseph |first9=Nicholas}}</ref>

The benchmark scores are of the following kinds:

* pass@n: The model is given <math>n</math> attempts to solve each problem. If any attempt is correct, the model earns a point. The pass@n score is the model's average score over all problems.
* cons@n: The model is given <math>n</math> attempts to solve each problem. If the most common answer is correct, the model earns a point. The cons@n score is the model's average score over all problems. Here "cons" stands for "consensus" or "majority voting".<ref>{{Citation |last1=DeepSeek-AI |title=DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning |date=2025-01-22 |arxiv=2501.12948 |last2=Guo |first2=Daya |last3=Yang |first3=Dejian |last4=Zhang |first4=Haowei |last5=Song |first5=Junxiao |last6=Zhang |first6=Ruoyu |last7=Xu |first7=Runxin |last8=Zhu |first8=Qihao |last9=Ma |first9=Shirong}}</ref>
The pass@n score can be estimated more accurately by making <math>N > n</math> attempts, and use the unbiased estimator <math>1- \frac{\binom{N-c}{n}}{\binom{N}{n}}</math>, where <math>c</math> is the number of correct attempts.<ref name=":4" />

== See also ==
* [[Generative pre-trained transformer]]
* [[Neuro-symbolic AI]]
* [[Automated theorem proving]]
* [[Automated reasoning]]
* [[Reflection (artificial intelligence)]]
* [[Large language model]]

== References ==
{{Reflist|30em}}

== External links ==
* {{Citation |last=Fortes |first=Armando |title=atfortes/Awesome-LLM-Reasoning |date=2025-01-27 |url=https://github.com/atfortes/Awesome-LLM-Reasoning?tab=readme-ov-file |access-date=2025-01-27}}
* {{Citation |last1=Huang |first1=Jie |title=Towards Reasoning in Large Language Models: A Survey |date=2023-05-26 |arxiv=2212.10403 |last2=Chang |first2=Kevin Chen-Chuan}}
* {{Citation |last1=Besta |first1=Maciej |title=Reasoning Language Models: A Blueprint |date=2025-01-23 |arxiv=2501.11223 |last2=Barth |first2=Julia |last3=Schreiber |first3=Eric |last4=Kubicek |first4=Ales |last5=Catarino |first5=Afonso |last6=Gerstenberger |first6=Robert |last7=Nyczyk |first7=Piotr |last8=Iff |first8=Patrick |last9=Li |first9=Yueling}}
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