{"id":1145442,"date":"2025-07-22T03:41:26","date_gmt":"2025-07-22T10:41:26","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-blog-post&#038;p=1145442"},"modified":"2025-07-22T03:41:29","modified_gmt":"2025-07-22T10:41:29","slug":"cor","status":"publish","type":"msr-blog-post","link":"https:\/\/www.microsoft.com\/en-us\/research\/articles\/cor\/","title":{"rendered":"CoR\uff1a\u591a\u8303\u5f0f\u601d\u7ef4\u52a0\u6301\uff0c\u5927\u6a21\u578b\u4e5f\u80fd\u5b66\u4f1a\u201c\u591a\u601d\u8def\u89e3\u9898\u201d"},"content":{"rendered":"\n<p>\u7f16\u8005\u6309\uff1a\u5927\u8bed\u8a00\u6a21\u578b\uff08LLMs\uff09\u5728\u8bed\u8a00\u751f\u6210\u4e0e\u57fa\u7840\u63a8\u7406\u4e2d\u5df2\u5c55\u73b0\u51fa\u5f3a\u5927\u7684\u80fd\u529b\uff0c\u4f46\u5b83\u4eec\u5728\u6570\u5b66\u89e3\u9898\u4e0a\u7684\u80fd\u529b\u4ecd\u5b58\u5728\u660e\u663e\u77ed\u677f\uff0c\u5c24\u5176\u662f\u96be\u4ee5\u517c\u987e\u590d\u6742\u8ba1\u7b97\u4e0e\u5b9a\u7406\u8bc1\u660e\u3002\u8fd9\u80cc\u540e\u7684\u5173\u952e\u539f\u56e0\u5728\u4e8e\uff0c\u73b0\u6709\u6a21\u578b\u666e\u904d\u4f9d\u8d56\u4e8e\u5355\u4e00\u7684\u63a8\u7406\u8303\u5f0f\uff08\u5982\u81ea\u7136\u8bed\u8a00\u3001\u4ee3\u7801\u6216\u7b26\u53f7\u63a8\u7406\uff09\uff0c\u7f3a\u4e4f\u4eba\u7c7b\u601d\u8003\u95ee\u9898\u65f6\u90a3\u79cd\u7075\u6d3b\u7684\u63a8\u7406\u80fd\u529b\u3002<\/p>\n\n\n\n<p>\u4e3a\u6b64\uff0c\u5fae\u8f6f\u4e9a\u6d32\u7814\u7a76\u9662\u4e0e\u6e05\u534e\u5927\u5b66\u8054\u5408\u63d0\u51fa\u4e86\u201c\u63a8\u7406\u94fe\u201d\uff08Chain-of-Reasoning, CoR\uff09\u6846\u67b6\uff0c\u5f15\u5165\u4e86\u81ea\u7136\u8bed\u8a00\u3001\u4ee3\u7801\u4e0e\u7b26\u53f7\u4e09\u79cd\u8303\u5f0f\u534f\u540c\u63a8\u7406\u7684\u65b0\u89c6\u89d2\uff0c\u652f\u6301\u6a21\u578b\u8df3\u8dc3\u5f0f\u201c\u601d\u8003\u201d\uff0c\u663e\u8457\u63d0\u5347\u4e86\u6a21\u578b\u7684\u901a\u7528\u6570\u5b66\u80fd\u529b\u3002\u76f8\u5173\u8bba\u6587\u5df2\u88ab ACL 2025 \u6536\u5f55\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>\u6570\u5b66\u9898\u901a\u5e38\u53ef\u4ee5\u88ab\u5206\u4e3a\u8ba1\u7b97\u548c\u5b9a\u7406\u8bc1\u660e\u4e24\u5927\u7c7b\u4efb\u52a1\u3002\u5c3d\u7ba1\u5927\u8bed\u8a00\u6a21\u578b\uff08LLMs\uff09\u5728\u5bf9\u8bdd\u3001\u5199\u4f5c\u7b49\u4efb\u52a1\u4e2d\u8868\u73b0\u60ca\u8273\uff0c\u4f46\u76ee\u524d\u7684\u5f00\u6e90 LLMs \u5728\u89e3\u51b3\u6570\u5b66\u95ee\u9898\u65f6\u5f80\u5f80\u96be\u4ee5\u540c\u65f6\u80dc\u4efb\u8fd9\u4e24\u7c7b\u95ee\u9898\u3002\u5176\u539f\u56e0\u5728\u4e8e\u73b0\u6709\u5de5\u4f5c\u4f1a\u5206\u522b\u9488\u5bf9\u8fd9\u4e24\u4e2a\u4efb\u52a1\u8fdb\u884c\u8bad\u7ec3\uff0c\u501f\u52a9\u589e\u5f3a\u6a21\u578b\u57fa\u4e8e\u7279\u5b9a\u7ed3\u6784\u5316\u7684\u77e5\u8bc6\u8868\u793a\uff08\u5373\u63a8\u7406\u8303\u5f0f\uff09\u6765\u63a8\u5bfc\u7b54\u6848\u3002\u5e38\u89c1\u7684\u63a8\u7406\u8303\u5f0f\u4e3b\u8981\u6709\u81ea\u7136\u8bed\u8a00\u63a8\u7406\u3001\u4ee3\u7801\u63a8\u7406\u548c\u7b26\u53f7\u63a8\u7406\u3002<\/p>\n\n\n\n<p>\u5177\u4f53\u6765\u8bf4\uff0c\u81ea\u7136\u8bed\u8a00\u63a8\u7406\u662f\u5229\u7528\u81ea\u7136\u8bed\u8a00\u6587\u672c\uff0c\u6839\u636e\u4eba\u7c7b\u5e38\u8bc6\u548c\u8bed\u4e49\u4e0a\u4e0b\u6587\u8fdb\u884c\u63a8\u7406\uff0c\u5177\u6709\u660e\u786e\u7684\u9010\u6b65\u81ea\u7136\u8bed\u8a00\u89e3\u91ca\u3002\u4ee3\u7801\u63a8\u7406\u662f\u5c06\u4efb\u52a1\u8f6c\u5316\u4e3a\u4ee3\u7801\u6765\u6a21\u62df\u8ba1\u7b97\u673a\u7684\u64cd\u4f5c\u5e76\u6267\u884c\uff0c\u4ece\u800c\u5bf9\u6700\u7ec8\u76ee\u6807\u8fdb\u884c\u63a8\u7406\uff0c\u4f8b\u5982\u751f\u6210\u7528\u4e8e\u7f16\u8bd1\u7684 Python \u4ee3\u7801\u7b49\u3002\u7b26\u53f7\u63a8\u7406\u5219\u662f\u6307\u5229\u7528\u903b\u8f91\u7b26\u53f7\u548c\u516c\u7406\u7cfb\u7edf\u4f5c\u4e3a\u5f62\u5f0f\u5316\u63a8\u7406\u7684\u57fa\u672c\u5de5\u5177\uff0c\u6765\u6267\u884c\u63a8\u7406\u3002<\/p>\n\n\n\n<p>\u5f53\u524d\u7684\u4e3b\u6d41\u65b9\u6cd5\u5927\u81f4\u53ef\u5206\u4e3a\u4e24\u7c7b\uff1a\u7b2c\u4e00\u7c7b\u5982\u56fe1\u5de6\u56fe\u6240\u793a\uff0c\u901a\u8fc7\u94fe\u5f0f\u6216\u6811\u641c\u7d22\u5728\u5355\u4e00\u8303\u5f0f\u5185\u8fdb\u884c\u63a8\u7406\uff1b\u7b2c\u4e8c\u7c7b\u5982\u56fe1\u53f3\u56fe\uff0c\u662f\u5c06\u5176\u4ed6\u8303\u5f0f\u4f5c\u4e3a\u5de5\u5177\u96c6\u6210\u5230\u5355\u4e00\u8303\u5f0f\u7684\u63a8\u7406\u4e2d\uff0c\u6216\u4f7f\u7528\u6765\u81ea\u4e0d\u540c\u4efb\u52a1\u7684\u5355\u4e00\u8303\u5f0f\u63a8\u7406\u6570\u636e\u8fdb\u884c\u6df7\u5408\u8bad\u7ec3\u3002\u7136\u800c\uff0c\u8fd9\u4e24\u7c7b\u65b9\u6cd5\u8981\u4e48\u5ffd\u7565\u4e86\u4e0d\u540c\u63a8\u7406\u8303\u5f0f\u95f4\u5b58\u5728\u7684\u534f\u540c\u4e0e\u4e92\u8865\u6027\uff0c\u8981\u4e48\u672a\u610f\u8bc6\u5230\u4e0d\u540c\u8303\u5f0f\u672c\u8eab\u5c31\u53ef\u4ee5\u72ec\u7acb\u5b8c\u6210\u63a8\u7406\u4e14\u5177\u6709\u76f8\u540c\u7684\u9690\u5f0f\u903b\u8f91\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"614\" height=\"812\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-1.png\" alt=\"diagram\" class=\"wp-image-1145445\" style=\"width:477px;height:auto\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-1.png 614w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-1-227x300.png 227w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-1-136x180.png 136w\" sizes=\"auto, (max-width: 614px) 100vw, 614px\" \/><figcaption class=\"wp-element-caption\">\u56fe1\uff1a\u5f53\u524d\u7684\u4e24\u7c7b\u4e3b\u6d41\u65b9\u6cd5<\/figcaption><\/figure>\n\n\n\n<p>\u8fd9\u79cd\u5c40\u9650\u6027\u8ba9\u6a21\u578b\u5728\u5904\u7406\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u5b66\u4efb\u52a1\u65f6\u7f3a\u4e4f\u901a\u7528\u6027\u548c\u5f39\u6027\u3002\u800c\u5728\u4eba\u7c7b\u65e5\u5e38\u89e3\u9898\u4e2d\uff0c\u6211\u4eec\u5f80\u5f80\u4f1a\u7075\u6d3b\u5207\u6362\u591a\u79cd\u601d\u7ef4\u65b9\u5f0f\uff1a\u7528\u81ea\u7136\u8bed\u8a00\u7406\u89e3\u9898\u610f\uff0c\u7528\u4ee3\u7801\u6216\u7b26\u53f7\u6f14\u7b97\u63a8\u5bfc\u4e2d\u95f4\u7ed3\u679c\uff0c\u6700\u7ec8\u603b\u7ed3\u7b54\u6848\u3002\u90a3\u4e48\u5982\u4f55\u8ba9\u5927\u6a21\u578b\u4e5f\u5177\u5907\u8fd9\u6837\u7684\u201c\u591a\u8303\u5f0f\u534f\u540c\u63a8\u7406\u201d\u80fd\u529b\uff1f<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"cor-\u878d\u5408\u8bed\u8a00-\u4ee3\u7801\u4e0e\u7b26\u53f7\u7684\u591a\u8303\u5f0f\u6570\u5b66\u63a8\u7406\u6846\u67b6\">CoR\uff1a\u878d\u5408\u8bed\u8a00\u3001\u4ee3\u7801\u4e0e\u7b26\u53f7\u7684\u591a\u8303\u5f0f\u6570\u5b66\u63a8\u7406\u6846\u67b6<\/h2>\n\n\n\n<p>\u8fd1\u65e5\uff0c\u5fae\u8f6f\u4e9a\u6d32\u7814\u7a76\u9662\u4e0e\u6e05\u534e\u5927\u5b66\u5408\u4f5c\uff0c\u521b\u65b0\u6027\u5730\u63d0\u51fa\u4e86\u63a8\u7406\u94fe\uff08Chain-of-Reasoning, CoR\uff09\u6846\u67b6\uff0c\u5c06\u81ea\u7136\u8bed\u8a00\u3001\u4ee3\u7801\u548c\u7b26\u53f7\u4e09\u79cd\u63a8\u7406\u8303\u5f0f\u878d\u5165\u540c\u4e00\u6761\u63a8\u7406\u8f68\u8ff9\u4e2d\u3002\u8be5\u6846\u67b6\u5141\u8bb8\u6a21\u578b\u5148\u57fa\u4e8e\u67d0\u79cd\u8303\u5f0f\u8fdb\u884c\u63a8\u7406\uff0c\u7136\u540e\u6839\u636e\u9898\u76ee\u7684\u9700\u6c42\u5207\u6362\u8303\u5f0f\uff0c\u5728\u5148\u524d\u751f\u6210\u5185\u5bb9\u7684\u57fa\u7840\u4e0a\u7ee7\u7eed\u63a8\u7406\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u4ece\u800c\u5b9e\u73b0\u4e86\u5728\u6cdb\u5316\u7684\u901a\u7528\u6570\u5b66\u4efb\u52a1\u4e0a\u7684\u96f6\u6837\u672c\uff08zero-shot\uff09\u63a8\u7406\u80fd\u529b\u3002\u6b64\u5916\uff0c\u6a21\u578b\u4e5f\u53ef\u4ee5\u901a\u8fc7\u8c03\u6574\u63d0\u793a\u8bcd\u6765\u6539\u53d8\u63a8\u7406\u7684\u6df1\u5ea6\u548c\u4f7f\u7528\u7684\u8303\u5f0f\u6570\u91cf\uff0c\u8fdb\u4e00\u6b65\u63d0\u9ad8\u4e86\u5176\u5bf9\u4e0d\u540c\u4efb\u52a1\u7684\u9002\u5e94\u6027\u3002\u76f8\u5173\u8bba\u6587\u5df2\u88ab ACL 2025 \u6536\u5f55\u3002<\/p>\n\n\n\n<p><strong>Chain-of-Reasoning: Towards Unified Mathematical Reasoning in Large Language Models via a Multi-Paradigm Perspective<\/strong><\/p>\n\n\n\n<p>\u8bba\u6587\u94fe\u63a5\uff1a<a class=\"msr-external-link glyph-append glyph-append-open-in-new-tab glyph-append-xsmall\" rel=\"noopener noreferrer\" target=\"_blank\" href=\"https:\/\/arxiv.org\/abs\/2501.11110\">https:\/\/arxiv.org\/abs\/2501.11110<span class=\"sr-only\"> (opens in new tab)<\/span><\/a><\/p>\n\n\n\n<p>GitHub \u94fe\u63a5\uff1a<a class=\"msr-external-link glyph-append glyph-append-open-in-new-tab glyph-append-xsmall\" rel=\"noopener noreferrer\" target=\"_blank\" href=\"https:\/\/github.com\/microsoft\/CoR\">https:\/\/github.com\/microsoft\/CoR<span class=\"sr-only\"> (opens in new tab)<\/span><\/a><\/p>\n\n\n\n<p>\u7814\u7a76\u5458\u4eec\u4ece\u63a8\u7406\u7684\u672c\u8d28\u51fa\u53d1\uff0c\u63d0\u51fa LLMs \u751f\u6210\u7684\u63a8\u7406\u6587\u672c\u662f\u5177\u6709\u63a8\u7406\u5c42\u6b21\u7ed3\u6784\u7684\u3002\u8be5\u7ed3\u6784\u7531\u4e09\u4e2a\u5c42\u6b21\u7ec4\u6210\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u63a8\u7406\u6b65\u9aa4<\/strong>\u662f\u57fa\u672c\u5355\u4f4d\uff1b<\/li>\n\n\n\n<li><strong>\u63a8\u7406\u8def\u5f84<\/strong>\u7531\u591a\u4e2a\u63a8\u7406\u6b65\u9aa4\u7ec4\u6210\uff0c\u5f62\u6210\u4e00\u4e2a\u5b8c\u6574\u7684\u63a8\u7406\u7ebf\u8def\uff0c\u901a\u5e38\u5305\u62ec\u6700\u7ec8\u7b54\u6848\u548c\u89e3\u51b3\u65b9\u6848\u8fc7\u7a0b\uff1b<\/li>\n\n\n\n<li><strong>\u63a8\u7406\u8303\u5f0f<\/strong>\u5219\u4ee3\u8868\u5229\u7528\u5355\u4e00\u77e5\u8bc6\u5a92\u4f53\u7684\u4e00\u4e2a\u6216\u591a\u4e2a\u63a8\u7406\u8def\u5f84\uff0c\u5982\u81ea\u7136\u8bed\u8a00\u3001\u4ee3\u7801\u3001\u7b26\u53f7\u8bed\u8a00\u7b49\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u5f53\u524d\u7684\u7814\u7a76\u5de5\u4f5c\u57fa\u672c\u90fd\u96c6\u4e2d\u5728\u5355\u4e00\u8303\u5f0f\u5185\uff0c\u6cbf\u7740\u4e24\u4e2a\u7ef4\u5ea6\u8fdb\u884c\u4f18\u5316\u2014\u2014\u6df1\u5ea6\uff08\u63a8\u7406\u6b65\u9aa4\u7684\u6570\u91cf\uff09\u548c\u5bbd\u5ea6\uff08\u63a8\u7406\u8def\u5f84\u7684\u6570\u91cf\uff09\u3002\u800c CoR \u6846\u67b6\u63d0\u51fa\u4e86\u4e00\u4e2a\u4e0e\u5f53\u524d\u5de5\u4f5c\u6b63\u4ea4\u7684\u65b0\u9896\u89c6\u89d2\uff0c\u4ece\u6269\u5927\u63a8\u7406\u8303\u5f0f\u7684\u89d2\u5ea6\u8fdb\u884c\u6269\u5c55\uff0c\u4e14\u7406\u8bba\u4e0a\u53ef\u4ee5\u4e0e\u5f53\u524d\u57fa\u4e8e\u5355\u4e00\u8303\u5f0f\u63a8\u7406\u7684\u5404\u79cd\u4f18\u5316\u5de5\u4f5c\u53e0\u52a0\u4f7f\u7528\u3002<\/p>\n\n\n\n<p>CoR \u57285\u4e2a\u6570\u5b66\u63a8\u7406\u6570\u636e\u96c6\u4e0a\u5747\u53d6\u5f97\u4e86\u663e\u8457\u7684\u63d0\u5347\u6548\u679c\uff0c\u5c55\u73b0\u51fa\u4e86\u4ee4\u4eba\u60ca\u559c\u7684\u901a\u7528\u6570\u5b66\u89e3\u9898\u80fd\u529b\uff0c\u65e2\u80fd\u89e3\u51b3\u8ba1\u7b97\u9898\uff0c\u53c8\u80fd\u89e3\u51b3\u5f62\u5f0f\u5316\u8bc1\u660e\u9898\u3002\u4f8b\u5982\uff0c\u5728\u5b9a\u7406\u8bc1\u660e\u4efb\u52a1\u4e2d\uff0cCoR-Math-7B \u6bd4 GPT-4o \u53d6\u5f97\u7684\u6210\u7ee9\u9ad8\u51fa41.0%\uff1b\u5728\u8ba1\u7b97\u4efb\u52a1\u4e2d\uff0c\u6bd4 GPT-4 \u9ad8\u51fa24.2%\u3002\u5728\u6570\u5b66\u8ba1\u7b97\u4efb\u52a1\u4e2d\uff0cCoR-Math-7B \u5728\u8bad\u7ec3\u8d44\u6e90\u548c\u6027\u80fd\u4e4b\u95f4\u53d6\u5f97\u4e86\u6700\u4f73\u5e73\u8861\uff0c\u4e0e\u540c\u7c7b\u6a21\u578b\u76f8\u6bd4\uff0c\u4f4d\u4e8e\u5e15\u5229\u6258\u4f18\u5316\u66f2\u7ebf\u533a\u57df\u7684\u4e0a\u6cbf\u3002\u5728\u7f8e\u56fd\u6570\u5b66\u7ade\u8d5b\uff08AMC\uff09\u548c\u5965\u6797\u5339\u514b\u6570\u5b66\u7ade\u8d5b\uff08AIME\uff09\u7684\u8003\u8bd5\u4e2d\uff0cCoR-Math-7B \u7684\u6210\u7ee9\u5206\u522b\u7b54\u5bf9\u4e8634\u9053\uff08\u517140\u9898\uff09\u548c12\u9053\uff08\u517130\u9898\uff09\uff0c\u6027\u80fd\u63a5\u8fd1 o1-mini\u3002<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u7ed3\u679c\u8868\u660e\uff0cCoR \u6846\u67b6\u901a\u8fc7\u591a\u8303\u5f0f\u63a8\u7406\u5927\u5e45\u63d0\u5347\u4e86\u6a21\u578b\u89e3\u51b3\u6570\u5b66\u95ee\u9898\u7684\u80fd\u529b\uff0c\u540c\u65f6\u5bf9\u8bad\u7ec3\u6570\u636e\u7684\u4f9d\u8d56\u66f4\u5c11\uff0c\u5927\u5927\u964d\u4f4e\u4e86\u83b7\u53d6\u9ad8\u6548\u63a8\u7406\u80fd\u529b\u7684\u6210\u672c\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"491\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-2-1024x491.png\" alt=\"table\" class=\"wp-image-1145446\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-2-1024x491.png 1024w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-2-300x144.png 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-2-768x368.png 768w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-2-240x115.png 240w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-2.png 1121w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u88681\uff1aCoR \u57285\u4e2a\u6570\u5b66\u63a8\u7406\u6570\u636e\u96c6\u4e2d\u7684\u63d0\u5347\u6548\u679c<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"\u591a\u8303\u5f0f\u8bad\u7ec3-\u4e00\u6b65\u6b65\u6559\u4f1a\u6a21\u578b-\u5982\u4f55\u601d\u8003\">\u591a\u8303\u5f0f\u8bad\u7ec3\uff1a\u4e00\u6b65\u6b65\u6559\u4f1a\u6a21\u578b\u201c\u5982\u4f55\u601d\u8003\u201d<\/h2>\n\n\n\n<p>CoR \u80fd\u591f\u5b9e\u73b0\u8de8\u8303\u5f0f\u63a8\u7406\u7684\u5173\u952e\uff0c\u5728\u4e8e\u5176\u80cc\u540e\u7cbe\u5fc3\u8bbe\u8ba1\u7684\u6570\u636e\u6784\u5efa\u4e0e\u8bad\u7ec3\u7b56\u7565\u3002\u7814\u7a76\u5458\u4eec\u6784\u5efa\u4e86\u4e00\u4e2a\u591a\u8303\u5f0f\u6570\u5b66\uff08MPM\uff09\u7684\u8bad\u7ec3\u6570\u636e\u96c6\uff0c\u5305\u542b16.7\u4e07\u6761\u63a8\u7406\u8def\u5f84\uff0c\u540c\u65f6\u63d0\u51fa\u4e86\u6e10\u8fdb\u8303\u5f0f\u8bad\u7ec3\uff08PPT\uff09\u65b9\u6cd5\uff0c\u4f7f\u6a21\u578b\u80fd\u591f\u9010\u6b65\u638c\u63e1\u591a\u4e2a\u63a8\u7406\u8303\u5f0f\u3002<\/p>\n\n\n\n<p>\u9996\u5148\uff0c\u7814\u7a76\u5458\u4eec\u4ece Numina\u3001Lean-Workbook \u7b49\u6570\u636e\u96c6\u4e2d\u6536\u96c6\u4e8630\u4e07\u6761\u4e0d\u540c\u8303\u5f0f\u7684\u63a8\u7406\u8def\u5f84\uff0c\u6db5\u76d6\u81ea\u7136\u8bed\u8a00\u3001\u4ee3\u7801\uff08\u4ee5 Python \u4e3a\u4e3b\uff09\u548c\u7b26\u53f7\u8bed\u8a00\uff08\u4ee5 lean4 \u4e3a\u4e3b\uff09\u3002\u8fd9\u4e9b\u6837\u672c\u4efb\u52a1\u7c7b\u578b\u4e30\u5bcc\uff0c\u5305\u62ec\u51fd\u6570\u3001\u51e0\u4f55\u3001\u5b9a\u7406\u8bc1\u660e\u7b49\uff0c\u8574\u542b\u591a\u6837\u5316\u7684\u63a8\u7406\u6a21\u5f0f\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u652f\u6491 CoR \u7684\u8bad\u7ec3\uff0c\u7814\u7a76\u5458\u4eec\u5c06\u4f20\u7edf\u7684\u5355\u8303\u5f0f\u8bad\u7ec3\u6570\u636e\u6269\u5c55\u4e3a\u5305\u542b\u591a\u8303\u5f0f\u4fe1\u606f\u7684\u7ed3\u6784\u5316\u6837\u672c\uff0c\u8868\u793a\u4e3a< x, NLR, SR, AR, y >\u3002\u5982\u56fe2(a)\u6240\u793a\uff0c\u8bad\u7ec3\u6570\u636e\u7684\u6536\u96c6\u8fc7\u7a0b\u5206\u4e3a\u4e24\u4e2a\u9636\u6bb5\uff1a(a.1)\u7b2c\u4e00\u9636\u6bb5\uff1a\u91cd\u6784\u3001\u6269\u5c55\u63a8\u7406\u8def\u5f84\uff0c(a.2)\u7b2c\u4e8c\u9636\u6bb5\uff1a\u4fee\u8ba2\u8def\u5f84\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u7b2c\u4e00\u9636\u6bb5\uff1a\u5728\u8fdb\u884c\u521d\u6b65\u7684\u8fc7\u6ee4\u540e\uff0c\u7814\u7a76\u5458\u4eec\u6784\u5efa\u4e86\u4e00\u79cd\u9002\u7528\u4e8e\u591a\u8303\u5f0f\u63a8\u7406\u7684\u901a\u7528\u6a21\u677f\uff0c\u7528\u4e8e\u6269\u5c55\u7f3a\u5931\u7684\u63a8\u7406\u8303\u5f0f\u3002\u8be5\u6a21\u677f\u6807\u51c6\u5316\u4e86\u4e0d\u540c\u63a8\u7406\u8303\u5f0f\u7684\u4f4d\u7f6e\uff0c\u5e76\u5b9a\u4e49\u4e86\u5b83\u4eec\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u8fd8\u53ef\u4ee5\u9002\u5e94\u5404\u79cd\u63a8\u7406\u6df1\u5ea6\uff0c\u4e5f\u652f\u6301\u4e0d\u540c\u63a8\u7406\u8303\u5f0f\u7684\u7075\u6d3b\u7ec4\u5408\u3002\u8fd9\u4e00\u9636\u6bb5\u751f\u6210\u4e86\u7ea628.5\u4e07\u4e2a\u5408\u6210\u6837\u672c\uff0c\u6784\u6210\u4e86\u521d\u59cb\u7684 MPM-raw \u6570\u636e\u96c6\u3002<\/li>\n\n\n\n<li>\u7b2c\u4e8c\u9636\u6bb5\uff1a\u7814\u7a76\u5458\u4eec\u901a\u8fc7\u8ba9 MPM-raw \u6570\u636e\u96c6\u4e0e Lean Prover \u8fed\u4ee3\u4ea4\u4e92\uff0c\u4ee5\u9a8c\u8bc1\u7b26\u53f7\u8303\u5f0f\u5448\u73b0\u7684\u8bc1\u660e\u6b65\u9aa4\u662f\u5426\u6b63\u786e\u3002\u540c\u65f6\uff0c\u6839\u636e Prover \u7684\u53cd\u9988\u4fe1\u606f\u6765\u6307\u5bfc\u63a8\u7406\u8def\u5f84\u7684\u8fc7\u6ee4\u548c\u4fee\u6539\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u8be5\u8fc7\u7a0b\u5c06\u7b26\u53f7\u63a8\u7406\u8def\u5f84\u63d0\u4ea4\u7ed9 Prover\uff0c\u4e00\u65e6 Prover \u6210\u529f\u5b8c\u6210\u8bc1\u660e\u4e14\u672a\u8fd4\u56de\u4efb\u4f55\u9519\u8bef\uff0c\u6574\u4e2a\u591a\u8303\u5f0f\u63a8\u7406\u8def\u5f84\u5c06\u4f1a\u88ab\u6536\u96c6\u5230 MPM \u6570\u636e\u96c6\u4e2d\u3002\u5426\u5219\uff0cProver \u8fd4\u56de\u7684\u9519\u8bef\u4fe1\u606f\u5c06\u88ab\u5408\u5e76\u8fdb\u4e0b\u4e00\u8f6e\u8fed\u4ee3\u7684\u8f93\u5165\u4e2d\uff0c\u4f5c\u4e3a\u6307\u5bfc\u4e0b\u4e00\u8f6e\u8fed\u4ee3\u8f93\u51fa\u7684\u53cd\u601d\u6307\u5bfc\u4fe1\u606f\u3002\u5982\u6b64\u8fdb\u884c\u591a\u8f6e\uff0c\u6700\u7ec8\u7684 MPM \u6570\u636e\u96c6\u5305\u542b\u4e8682,770\u4e2a\u95ee\u9898\u548c167,412\u4e2a\u9ad8\u8d28\u91cf\u7684\u591a\u8303\u5f0f\u63a8\u7406\u89e3\u51b3\u65b9\u6848\u3002<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"943\" height=\"293\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-3.png\" alt=\"diagram\" class=\"wp-image-1145448\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-3.png 943w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-3-300x93.png 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-3-768x239.png 768w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-3-240x75.png 240w\" sizes=\"auto, (max-width: 943px) 100vw, 943px\" \/><figcaption class=\"wp-element-caption\">\u56fe2\uff1aCoR \u6574\u4f53\u8bad\u7ec3\u6846\u67b6<\/figcaption><\/figure>\n\n\n\n<p>\u6709\u4e86\u6570\u636e\uff0c\u8fd8\u9700\u8981\u65b9\u6cd5\u3002\u4e3a\u6b64\uff0c\u7814\u7a76\u5458\u4eec\u5f15\u5165\u4e86 PPT \u7684\u4e09\u9636\u6bb5\u8bad\u7ec3\u7b56\u7565\uff0c\u4f7f LLMs \u80fd\u591f\u9010\u6b65\u638c\u63e1\u4e0d\u540c\u7684\u63a8\u7406\u8303\u5f0f\u3002\u5982\u56fe2(b)\u6240\u793a\uff0c\u8bad\u7ec3\u8fc7\u7a0b\u5206\u4e3a\u4ee5\u4e0b\u4e09\u6b65\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u7b2c\u4e00\u9636\u6bb5\u662f\u4e00\u4e2a\u521d\u59cb\u5316\u7684\u6559\u5b66\u9636\u6bb5\uff0c\u9274\u4e8e\u81ea\u7136\u8bed\u8a00\u5728\u8bed\u8a00\u6a21\u578b\u9884\u8bad\u7ec3\u6570\u636e\u4e2d\u7684\u4e3b\u5bfc\u5730\u4f4d\uff0c\u8fd9\u4e00\u9636\u6bb5\u4e3b\u8981\u8bad\u7ec3\u6a21\u578b\u80fd\u591f\u5b66\u4e60\u4f7f\u7528\u81ea\u7136\u8bed\u8a00\u89e3\u51b3\u590d\u6742\u7684\u6570\u5b66\u95ee\u9898\u3002<\/li>\n\n\n\n<li>\u7b2c\u4e8c\u9636\u6bb5\uff0c\u8003\u8651\u5230\u9884\u8bad\u7ec3\u6570\u636e\u4e2d\u6709\u4e00\u5b9a\u6bd4\u4f8b\u7684\u4ee3\u7801\u8bed\u6599\u5e93\uff0c\u7814\u7a76\u5458\u4eec\u5c06\u63a8\u7406\u8def\u5f84\u6269\u5c55\u4e3a\u5305\u62ec\u81ea\u7136\u8bed\u8a00\u548c\u4ee3\u7801\u4e24\u79cd\u8303\u5f0f\u3002\u5728\u8fd9\u4e2a\u9636\u6bb5\u4e4b\u540e\uff0c\u6a21\u578b\u53ef\u4ee5\u5728\u4e00\u6b21\u8f93\u51fa\u4e2d\uff0c\u4ee5\u81ea\u7136\u8bed\u8a00\u63a8\u7406\u4e3a\u8d77\u70b9\uff0c\u540c\u65f6\u542f\u52a8\u4ee3\u7801\u8303\u5f0f\u5904\u7406\u9700\u8981\u7cbe\u786e\u7b54\u6848\u7684\u95ee\u9898\u3002<\/li>\n\n\n\n<li>\u7b2c\u4e09\u9636\u6bb5\uff0c\u901a\u8fc7\u5229\u7528 MPM \u6570\u636e\u96c6\uff0c\u7814\u7a76\u5458\u4eec\u5c06\u63a8\u7406\u8def\u5f84\u8fdb\u4e00\u6b65\u6269\u5c55\u81f3\u4e09\u4e2a\u63a8\u7406\u8303\u5f0f\u3002\u7ecf\u8fc7\u5b8c\u6574\u7684 PPT \u9636\u6bb5\u540e\uff0c\u8bad\u7ec3\u597d\u7684 CoR-Math-7B \u6a21\u578b\u4e0d\u4ec5\u638c\u63e1\u4e86\u81ea\u7136\u8bed\u8a00\u548c\u4ee3\u7801\u63a8\u7406\uff0c\u8fd8\u53ef\u4ee5\u6267\u884c\u4e25\u683c\u7684\u903b\u8f91\u7b26\u53f7\u63a8\u7406\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6b64\u65f6 CoR-Math-7B \u63a8\u7406\u7684\u6df1\u5ea6\uff08\u91c7\u7528\u51e0\u6b21\u4e0d\u540c\u8303\u5f0f\u95f4\u7684\u8f6c\u6362\u8fed\u4ee3\uff09\u548c\u91c7\u7528\u7684\u8303\u5f0f\u6570\u91cf\uff08\u4e00\u6761\u63a8\u7406\u8def\u5f84\u4e2d\u4f1a\u6d89\u53ca\u51e0\u79cd\u8303\u5f0f\uff09\u90fd\u662f\u53ef\u53d8\u7684\u3002\u5728 zero-shot \u7684\u60c5\u51b5\u4e0b\uff0c\u4e5f\u53ef\u4ee5\u6839\u636e\u7279\u5b9a\u4efb\u52a1\u7684\u7279\u5b9a\u9700\u6c42\uff0c\u7b80\u5355\u8c03\u6574\u63d0\u793a\u8bcd\u4ee5\u7075\u6d3b\u9002\u5e94\u5404\u79cd\u573a\u666f\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"\u5168\u9762\u9a8c\u8bc1\u591a\u8303\u5f0f\u63a8\u7406\u7684\u4f18\u8d8a\u6027\">\u5168\u9762\u9a8c\u8bc1\u591a\u8303\u5f0f\u63a8\u7406\u7684\u4f18\u8d8a\u6027<\/h2>\n\n\n\n<p>\u7814\u7a76\u5458\u4eec\u5728\u4e94\u4e2a\u4e0d\u540c\u7684\u5927\u8bed\u8a00\u6a21\u578b\u57fa\u7840\u4e0a\u8bc4\u4f30\u4e86 CoR \u6846\u67b6\u7684\u8868\u73b0\uff0c\u5305\u62ec DeepSeekMath-7B-base\u3001Qwen2.5-Math-7B-base\u3001Qwen2.5-Math-1.5B-base\u3001Llama-3.1-8B \u548c Llama-3.1-70B\u3002<\/p>\n\n\n\n<p>\u4e3a\u66f4\u7cfb\u7edf\u5730\u8bc4\u4f30\u63a8\u7406\u80fd\u529b\uff0c\u7814\u7a76\u5458\u4eec\u5c06\u57fa\u7ebf\u6a21\u578b\u5212\u5206\u4e3a\u4e24\u7c7b\uff1a\u901a\u7528\u6570\u5b66\u6a21\u578b\u548c\u9488\u5bf9\u7279\u5b9a\u4efb\u52a1\u4f18\u5316\u7684\u6570\u5b66\u4e13\u5bb6\u6a21\u578b\u3002\u5176\u4e2d\uff0c\u4e13\u5bb6\u6a21\u578b\u53c8\u53ef\u8fdb\u4e00\u6b65\u5206\u4e3a\u9762\u5411\u7b97\u672f\u8ba1\u7b97\u7684\u6a21\u578b\u548c\u9762\u5411\u5b9a\u7406\u8bc1\u660e\u7684\u6a21\u578b\u3002<\/p>\n\n\n\n<p>\u5728\u6db5\u76d6\u7b97\u672f\u4e0e\u5b9a\u7406\u8bc1\u660e\u4efb\u52a1\u7684\u4e94\u4e2a\u6570\u5b66\u6570\u636e\u96c6\u4e0a\uff0c\u7814\u7a76\u5458\u4eec\u5bf9 CoR \u8fdb\u884c\u4e86\u7cfb\u7edf\u6d4b\u8bd5\u3002\u7ed3\u679c\u663e\u793a\uff0c\u5728 zero-shot \u8bbe\u7f6e\u4e0b\uff0c\u7531 DeepSeekMath-7B-base \u5fae\u8c03\u5f97\u5230\u7684 CoR-Math-7B \u5728\u6240\u6709\u6570\u636e\u96c6\u4e0a\u5747\u53d6\u5f97\u4e86\u6700\u4f73\u6027\u80fd\uff0c\u5c55\u73b0\u51fa\u6781\u5f3a\u7684\u6cdb\u5316\u63a8\u7406\u80fd\u529b\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1123\" height=\"542\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-4.png\" alt=\"table\" class=\"wp-image-1145450\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-4.png 1123w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-4-300x145.png 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-4-1024x494.png 1024w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-4-768x371.png 768w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-4-240x116.png 240w\" sizes=\"auto, (max-width: 1123px) 100vw, 1123px\" \/><figcaption class=\"wp-element-caption\">\u88682\uff1aCoR-Math-7B \u57285\u4e2a\u5177\u6709\u4ee3\u8868\u6027\u7684\u6570\u5b66\u63a8\u7406\u6570\u636e\u96c6\u4e0a\u7684 zero-shot \u6548\u679c<\/figcaption><\/figure>\n\n\n\n<p>\u5728\u5b9a\u7406\u8bc1\u660e\u5b50\u4efb\u52a1\u4e2d\uff0c\u5982\u88683\u6240\u793a\uff0cCoR-Math-7B \u5728 MiniF2F \uff08\u8bc1\u660e\u9898\u578b\uff09\u6570\u636e\u96c6\u4e0a\u53d6\u5f97\u4e8666.0%\u7684\u51c6\u786e\u7387\uff0c\u5728\u65e0\u9700\u989d\u5916\u8bad\u7ec3\u7684\u524d\u63d0\u4e0b\u8fbe\u5230\u9886\u5148\u6c34\u5e73\uff0c\u517c\u987e\u51c6\u786e\u7387\u4e0e\u8ba1\u7b97\u6548\u7387\uff0c\u8868\u73b0\u5c24\u4e3a\u7a81\u51fa\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"690\" height=\"443\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-5.png\" alt=\"table\" class=\"wp-image-1145451\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-5.png 690w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-5-300x193.png 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-5-240x154.png 240w\" sizes=\"auto, (max-width: 690px) 100vw, 690px\" \/><figcaption class=\"wp-element-caption\">\u88683\uff1aCoR-Math-7B \u5728 MiniF2 \u4e0a\u7684\u6027\u80fd<\/figcaption><\/figure>\n\n\n\n<p>\u88684\u663e\u793a\u4e86 CoR \u4e0e\u7b97\u672f\u8ba1\u7b97\u4e13\u5bb6\u6a21\u578b\u7684\u6bd4\u8f83\u3002\u76f8\u8f83\u4e8e\u81ea\u7136\u8bed\u8a00\u548c\u4ee3\u7801\u788e\u7247\u5316\u4ea4\u7ec7\u7684\u5de5\u5177\u8c03\u7528\u7b97\u6cd5\uff0cCoR \u5c55\u73b0\u51fa\u4e86\u66f4\u5f3a\u7684\u7ade\u4e89\u529b\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"669\" height=\"574\" src=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-6.png\" alt=\"table\" class=\"wp-image-1145452\" srcset=\"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-6.png 669w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-6-300x257.png 300w, https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2025\/07\/cor-6-210x180.png 210w\" sizes=\"auto, (max-width: 669px) 100vw, 669px\" \/><figcaption class=\"wp-element-caption\">\u88684\uff1aCoR \u4e0e\u7b97\u672f\u8ba1\u7b97\u4e13\u5bb6\u6a21\u578b\u7684\u6bd4\u8f83<\/figcaption><\/figure>\n\n\n\n<p>\u603b\u4f53\u6765\u770b\uff0cCoR \u5728\u5b9a\u7406\u8bc1\u660e\u4efb\u52a1\u4e2d\u8d85\u8d8a\u4e86\u4f20\u7edf\u7684 SFT\u3001RL \u53ca\u5176\u53d8\u4f53\u65b9\u6cd5\uff0c\u751a\u81f3\u4f18\u4e8e GPT-4o \u548c o1-mini\uff1b\u5728\u7b97\u672f\u8ba1\u7b97\u4efb\u52a1\u4e2d\uff0cCoR \u5728\u6027\u80fd\u4e0e\u8d44\u6e90\u6548\u7387\u4e4b\u95f4\u4e5f\u5b9e\u73b0\u4e86\u4f18\u5f02\u7684\u5e73\u8861\uff0c\u663e\u793a\u51fa\u5728\u591a\u79cd\u6570\u5b66\u4efb\u52a1\u4e0a\u7684\u5e7f\u6cdb\u9002\u5e94\u6027\u4e0e\u9ad8\u6548\u6027\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"\u8fc8\u5411\u66f4\u5f3a\u63a8\u7406\u7cfb\u7edf\u7684\u672a\u6765\">\u8fc8\u5411\u66f4\u5f3a\u63a8\u7406\u7cfb\u7edf\u7684\u672a\u6765<\/h2>\n\n\n\n<p>CoR 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ACL 2025 \u6536\u5f55\u3002 Chain-of-Reasoning: Towards Unified Mathematical Reasoning in Large Language Models via a Multi-Paradigm Perspective \u8bba\u6587\u94fe\u63a5\uff1ahttps:\/\/arxiv.org\/abs\/2501.11110 (opens in new tab) GitHub \u94fe\u63a5\uff1ahttps:\/\/github.com\/microsoft\/CoR (opens in new tab) \u7814\u7a76\u5458\u4eec\u4ece\u63a8\u7406\u7684\u672c\u8d28\u51fa\u53d1\uff0c\u63d0\u51fa LLMs \u751f\u6210\u7684\u63a8\u7406\u6587\u672c\u662f\u5177\u6709\u63a8\u7406\u5c42\u6b21\u7ed3\u6784\u7684\u3002\u8be5\u7ed3\u6784\u7531\u4e09\u4e2a\u5c42\u6b21\u7ec4\u6210\uff1a \u5f53\u524d\u7684\u7814\u7a76\u5de5\u4f5c\u57fa\u672c\u90fd\u96c6\u4e2d\u5728\u5355\u4e00\u8303\u5f0f\u5185\uff0c\u6cbf\u7740\u4e24\u4e2a\u7ef4\u5ea6\u8fdb\u884c\u4f18\u5316\u2014\u2014\u6df1\u5ea6\uff08\u63a8\u7406\u6b65\u9aa4\u7684\u6570\u91cf\uff09\u548c\u5bbd\u5ea6\uff08\u63a8\u7406\u8def\u5f84\u7684\u6570\u91cf\uff09\u3002\u800c CoR 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