{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/autoformalizing-euclidean-geometry","title":"Autoformalizing Euclidean Geometry","arxiv_id":"2405.17216","date":"2024-05-27","proceeding":null,"authors":["Logan Murphy","Kaiyu Yang","Jialiang Sun","Zhaoyu Li","Anima Anandkumar","Xujie Si"],"abstract":"Autoformalization involves automatically translating informal math into formal theorems and proofs that are machine-verifiable. Euclidean geometry provides an interesting and controllable domain for studying autoformalization. In this paper, we introduce a neuro-symbolic framework for autoformalizing Euclidean geometry, which combines domain knowledge, SMT solvers, and large language models (LLMs). One challenge in Euclidean geometry is that informal proofs rely on diagrams, leaving gaps in texts that are hard to formalize. To address this issue, we use theorem provers to fill in such diagrammatic information automatically, so that the LLM only needs to autoformalize the explicit textual steps, making it easier for the model. We also provide automatic semantic evaluation for autoformalized theorem statements. We construct LeanEuclid, an autoformalization benchmark consisting of problems from Euclid's Elements and the UniGeo dataset formalized in the Lean proof assistant. Experiments with GPT-4 and GPT-4V show the capability and limitations of state-of-the-art LLMs on autoformalizing geometry problems. The data and code are available at https://github.com/loganrjmurphy/LeanEuclid.","url_abs":"https://arxiv.org/abs/2405.17216v1","url_pdf":"https://arxiv.org/pdf/2405.17216v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"autoformalizing-euclidean-geometry","repo_url":"https://github.com/loganrjmurphy/leaneuclid","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"math","task_name":"Math"}],"methods":[{"method_slug":"absolute-position-encodings","method_name":"Absolute Position Encodings"},{"method_slug":"adam","method_name":"Adam"},{"method_slug":"attention","method_name":"Attention"},{"method_slug":"bpe","method_name":"BPE"},{"method_slug":"dense-connections","method_name":"Dense Connections"},{"method_slug":"dropout","method_name":"Dropout"},{"method_slug":"gpt-4","method_name":"GPT-4"},{"method_slug":"label-smoothing","method_name":"Label Smoothing"},{"method_slug":"layer-normalization","method_name":"Layer Normalization"},{"method_slug":"linear-layer","method_name":"Linear Layer"},{"method_slug":"multi-head-attention","method_name":"Multi-Head Attention"},{"method_slug":"position-wise-feed-forward-layer","method_name":"Position-Wise Feed-Forward Layer"},{"method_slug":"residual-connection","method_name":"Residual Connection"},{"method_slug":"softmax","method_name":"Softmax"},{"method_slug":"transformer","method_name":"Transformer"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2405.17216","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2405.17216"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/loganrjmurphy/leaneuclid","reach":null}],"summary":{"ran":3,"ran_draft_wrong":4,"ran_violates":2,"unverified":4},"by_repo_kind":{"official":{"samples":13,"ran":9,"repositories":1}},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":0,"samples":[{"code_sha256_prefix":"b17a3745345f3d22","entry":"PermutationStruct","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":"deterministic","behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"b17a3745345f3d22"}},{"code_sha256_prefix":"f93e2527f3540da6","entry":"checkPerm","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"f93e2527f3540da6"}},{"code_sha256_prefix":"9ccf4b01d9d2129a","entry":"choosePermutations","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"OUTPUT_MISDECLARED","metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"9ccf4b01d9d2129a"}},{"code_sha256_prefix":"e9a46f8957968943","entry":"has_match","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"ran_violates","verification_level":1,"contract_check":"VIOLATES","metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"e9a46f8957968943"}},{"code_sha256_prefix":"4f18707c98469c94","entry":"has_permutation","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"ran_violates","verification_level":1,"contract_check":"VIOLATES","metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"4f18707c98469c94"}},{"code_sha256_prefix":"fdfee91915c790f0","entry":"is_permutation","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"fdfee91915c790f0"}},{"code_sha256_prefix":"cb6fe1699a17fcb6","entry":"removeGuards","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"OUTPUT_MISDECLARED","metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"cb6fe1699a17fcb6"}},{"code_sha256_prefix":"76c240a8d6747abf","entry":"remove_all_guards","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"OUTPUT_MISDECLARED","metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"76c240a8d6747abf"}},{"code_sha256_prefix":"78112d3724540dc8","entry":"replaceGuards","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"OUTPUT_MISDECLARED","metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"78112d3724540dc8"}},{"code_sha256_prefix":"bb00132b7cffb7fb","entry":"NameMap","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"bb00132b7cffb7fb"}},{"code_sha256_prefix":"1160a348cb5575f9","entry":"PropData","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"1160a348cb5575f9"}},{"code_sha256_prefix":"3e48503267b317e0","entry":"QuantifierData","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"3e48503267b317e0"}},{"code_sha256_prefix":"63f2931afa6efafd","entry":"checkAndInsert","repo":"loganrjmurphy/leaneuclid","repo_kind":"official","path":"E3/Engine/choosePerms.py","file_url":"https://github.com/loganrjmurphy/leaneuclid/blob/HEAD/E3/Engine/choosePerms.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"63f2931afa6efafd"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}