[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-289631-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-289631-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","divide-and-abstract-autoformalization-via-decomposition-and-abstraction-learning","DIVIDE AND ABSTRACT - AUTOFORMALIZATION VIA DECOMPOSITION AND ABSTRACTION LEARNING","","Autoformalization translates informal mathematics into formal, machine-verifiable languages for theorem provers. Existing methods rely on predefined libraries and direct LLM generation, but are bottlenecked by limited abstractions, struggle with realistic statement complexity, and generalize poorly across formal languages. Divide and Abstract (DNA) is a zero-training framework using a two-phase pipeline: extracting reusable mathematical abstractions and then decomposing, translating, and composing clauses into complete formalizations.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/divide-and-abstract-autoformalization-via-decomposition-and-abstraction-learning/289631/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/divide-and-abstract-autoformalization-via-decomposition-and-abstraction-learning/289631.png","ImageObject",300,407,{"name":42,"@type":43},"Theodora","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-11","2026-09-17",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",16,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What is the goal of autoformalization in this paper?","Question",{"text":63,"@type":64},"Autoformalization automatically translates informal mathematics into formal languages for theorem provers so that theorems and proofs can be mechanically verified.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How does Divide and Abstract (DNA) work in two phases?",{"text":68,"@type":64},"DNA first extracts common mathematical concepts from a corpus and formalizes them as reusable abstractions to extend the target DSL. Then it decomposes new statements into structured clauses, translates each clause using the learned abstractions, composes full formalizations, and refines with symbolic validator feedback.",{"name":70,"@type":61,"acceptedAnswer":71},"What limitations of previous approaches does DNA address?",{"text":72,"@type":64},"The paper identifies three limitations: dependence on existing abstractions, difficulty scaling to complex nested statements, and weak transfer across formal languages.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},289631,1789636062,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":112,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":113,"slug":114},8,30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":97,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":112,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":140,"language":141,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":12,"update_tm":80,"read_time":145,"preview_page_count":4},687197207919,"https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552","DIVIDE AND ABSTRACT: AUTOFORMALIZATION VIA DECOMPOSITION AND ABSTRACTION LEARNING  \nMarcus J. Min 1 Yeqi Gao2 Wilson Sy3 Zhaoyu Li3 Xujie Si3 Osbert Bastani 1 1University of Pennsylvania 2Tsinghua University 3University of Toronto  \n{marcmin,[obastani}@seas.upenn.edu](obastani}@seas.upenn.edu)[ ](obastani}@seas.upenn.edu)[six@cs.toronto.edu](six@cs.toronto.edu)  \nABSTRACT  \nExisting approaches to autoformalization—the task of translating informal mathematics into formal machine-verifiable languages—rely heavily on pre-defined libraries and expect LLMs to directly generate complete formalizations. These approaches face three fundamental limitations: they are bottlenecked by existing abstractions, they have difficulty handling the complexity of realistic statements, and they do not transfer well across formal languages. We propose Divide and Abstract (DNA), a zero-training framework that addresses these challenges through a two-phase approach. First, DNA extracts common mathematical concepts from the entire corpus and formalizes them as reusable abstractions, extending the target language’s capability. Second, DNA hierarchically decomposes new statements into structured informal clauses, translates each clause using the learned abstractions, and composes them into complete formalizations.  \nOur evaluation on the LeanEuclidPlus and ProofNet-Hard benchmarks demon  \nstrates consistent improvements across multiple model families, achieving up to  \n8.60× performance gains over baselines. Notably, DNA enables smaller models to match baselines using much larger models, and shows particularly strong performance on complex mathematical statements requiring nested reasoning.  \nFurthermore, our framework requires no training on target languages, making it effective for low-resource domain-specific languages. Our code is available at [https://github.com/marcusm117/DNA](https://github.com/marcusm117/DNA).  \n1 INTRODUCTION  \nAutoformalization is the task of automatically translating informal mathematics into formal languages designed for theorem provers such as Lean (Moura & Ullrich, 2021) and Z3 (De Moura & Bjørner, 2008), so that theorems and proofs written in natural languages can be mechanically verified. The success of autoformalization will significantly reduce the amount of labor from human experts, and thus empower three transformative applications:  \n1. Synthesizing informal-formal parallel data for training neural theorem provers (Xin et al., 2024) .  \n2. Grounding and guiding reasoning processes in natural languages (Yang et al., 2022) .  \n3. Accelerating verification of important mathematical theorems (Gonthier et al., 2013; Hales et al.,  \n2015) or engineering systems (Zhao et al., 2012; Reid et al., 2016) not only to identify potential gaps  \nand mistakes, but also to facilitate certified extensions and future mathematical discovery.  \nInspired by the breakthrough of Large Language Models (LLMs) in machine translation between natural languages, researchers have increasingly applied LLMs to autoformalization (Weng et al., 2025) . Even though the ultimate vision is to automatically formalize entire theories that include axioms, definitions, notations, theorems, and proofs, current LLMs struggle to even formalize an individual statement (Murphy et al., 2024; Liu et al., 2025a) .  \nPrevious work in LLM-based statement autoformalization has explored in-context learning (Wu et al., 2022), fine-tuning specialized autoformalizer models (Jiang et al., 2023a), invoking retrieval before LLM inference (Liu et al., 2025a), or employing post-inference techniques such as majority  \nFigure 1: Overview of the Divide and Abstract (DNA) framework. (a) Phase I (Language Extension): Given an informal corpus, we extract common mathematical concepts, filter out those already present in the target language, construct a concept dependency graph (CDG), and systematically formalize concepts following topological ordering to extend the DSL. (b) Phase II (S","cbCaihttUWdmZov6","https://ap.wps.com/l/cbCaihttUWdmZov6","pdf",14056242,51,"English","# Abstract\n# Introduction\n## Autoformalization task\n## Applications enabled by autoformalization\n## Limits of existing LLM-based approaches\n## DNA framework overview\n### Phase I: Language extension\n### Phase II: Statement formalization\n## Challenges with existing methods","[{\"question\":\"What is the goal of autoformalization in this paper?\",\"answer\":\"Autoformalization automatically translates informal mathematics into formal languages for theorem provers so that theorems and proofs can be mechanically verified.\"},{\"question\":\"How does Divide and Abstract (DNA) work in two phases?\",\"answer\":\"DNA first extracts common mathematical concepts from a corpus and formalizes them as reusable abstractions to extend the target DSL. Then it decomposes new statements into structured clauses, translates each clause using the learned abstractions, composes full formalizations, and refines with symbolic validator feedback.\"},{\"question\":\"What limitations of previous approaches does DNA address?\",\"answer\":\"The paper identifies three limitations: dependence on existing abstractions, difficulty scaling to complex nested statements, and weak transfer across formal languages.\"}]","DIVIDE AND ABSTRACT - AUTOFORMALIZATION VIA DECOMPOSITION AND ABSTRACTION LEARNING | PDF",129]