{"id":"https://openalex.org/W7127068361","doi":"https://doi.org/10.48550/arxiv.2601.22510","title":"Shattered Compositionality: Counterintuitive Learning Dynamics of Transformers for Arithmetic","display_name":"Shattered Compositionality: Counterintuitive Learning Dynamics of Transformers for Arithmetic","publication_year":2026,"publication_date":"2026-01-30","ids":{"openalex":"https://openalex.org/W7127068361","doi":"https://doi.org/10.48550/arxiv.2601.22510"},"language":null,"primary_location":{"id":"pmh:oai:arXiv.org:2601.22510","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2601.22510","pdf_url":"https://arxiv.org/pdf/2601.22510","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2601.22510","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5037909738","display_name":"Xingyu Zhao","orcid":"https://orcid.org/0000-0001-6541-3445","observed_orcids":["https://orcid.org/0000-0001-6541-3445","https://orcid.org/0000-0002-3474-349X","https://orcid.org/0009-0007-1357-2189"]},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhao, Xingyu","raw_affiliation_strings":["University of Wisconsin-Madison"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Wisconsin-Madison","institution_ids":["https://openalex.org/I135310074"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026824220","display_name":"Darsh Sharma","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Sharma, Darsh","raw_affiliation_strings":["University of Wisconsin-Madison"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Wisconsin-Madison","institution_ids":["https://openalex.org/I135310074"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065688288","display_name":"Rheeya Uppaal","orcid":"https://orcid.org/0000-0001-6798-8621","observed_orcids":["https://orcid.org/0000-0001-6798-8621"]},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Uppaal, Rheeya","raw_affiliation_strings":["University of Wisconsin-Madison"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Wisconsin-Madison","institution_ids":["https://openalex.org/I135310074"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100968975","display_name":"Yiqiao Zhong","orcid":"https://orcid.org/0000-0003-0642-662X","observed_orcids":["https://orcid.org/0000-0003-0642-662X"]},"institutions":[{"id":"https://openalex.org/I135310074","display_name":"University of Wisconsin\u2013Madison","ror":"https://ror.org/01y2jtd41","country_code":"US","type":"education","lineage":["https://openalex.org/I135310074"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhong, Yiqiao","raw_affiliation_strings":["University of Wisconsin-Madison"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Wisconsin-Madison","institution_ids":["https://openalex.org/I135310074"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I135310074"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10028","display_name":"Topic Modeling","score":0.35249999165534973,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10028","display_name":"Topic Modeling","score":0.35249999165534973,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12026","display_name":"Explainable Artificial Intelligence (XAI)","score":0.31349998712539673,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.07259999960660934,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/skill-composition","display_name":"skill composition","score":0.6259999871253967},{"id":"https://openalex.org/keywords/arithmetic-reasoning","display_name":"arithmetic reasoning","score":0.5210000276565552},{"id":"https://openalex.org/keywords/transformer","display_name":"transformer models","score":0.6629999876022339},{"id":"https://openalex.org/keywords/large-language-models","display_name":"large language models","score":0.8240000009536743},{"id":"https://openalex.org/keywords/learning-dynamics","display_name":"learning dynamics","score":0.8859999775886536},{"id":"https://openalex.org/keywords/distribution-shift-robustness","display_name":"distribution shift robustness","score":0.7310000061988831},{"id":"https://openalex.org/keywords/skill-competitions","display_name":"skill competitions","score":0.44999998807907104},{"id":"https://openalex.org/keywords/hierarchical-reasoning","display_name":"hierarchical reasoning","score":0.4950000047683716},{"id":"https://openalex.org/keywords/mechanistic-interpretability","display_name":"mechanistic interpretability","score":0.6159999966621399}],"concepts":[{"id":"https://openalex.org/C66322947","wikidata":"https://www.wikidata.org/wiki/Q11658","display_name":"Transformer","level":3,"score":0.5489000082015991},{"id":"https://openalex.org/C46686674","wikidata":"https://www.wikidata.org/wiki/Q466303","display_name":"Boosting (machine learning)","level":2,"score":0.5396000146865845},{"id":"https://openalex.org/C180747234","wikidata":"https://www.wikidata.org/wiki/Q23373","display_name":"Cognitive psychology","level":1,"score":0.5303000211715698},{"id":"https://openalex.org/C101097943","wikidata":"https://www.wikidata.org/wiki/Q5176983","display_name":"Counterintuitive","level":2,"score":0.5270000100135803},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.49480000138282776},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.45989999175071716},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.4521999955177307},{"id":"https://openalex.org/C188147891","wikidata":"https://www.wikidata.org/wiki/Q147638","display_name":"Cognitive science","level":1,"score":0.4032000005245209},{"id":"https://openalex.org/C121375916","wikidata":"https://www.wikidata.org/wiki/Q936559","display_name":"Principle of compositionality","level":2,"score":0.38940000534057617},{"id":"https://openalex.org/C40506919","wikidata":"https://www.wikidata.org/wiki/Q7452469","display_name":"Sequence learning","level":2,"score":0.3804999887943268},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.36419999599456787},{"id":"https://openalex.org/C50335755","wikidata":"https://www.wikidata.org/wiki/Q483247","display_name":"Phenomenon","level":2,"score":0.3555000126361847},{"id":"https://openalex.org/C2777379011","wikidata":"https://www.wikidata.org/wiki/Q938545","display_name":"Implicit learning","level":3,"score":0.337799996137619},{"id":"https://openalex.org/C2777152325","wikidata":"https://www.wikidata.org/wiki/Q108163","display_name":"Proposition","level":2,"score":0.325300008058548},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.2976999878883362},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.2930000126361847},{"id":"https://openalex.org/C145912823","wikidata":"https://www.wikidata.org/wiki/Q113558","display_name":"Dynamics (music)","level":2,"score":0.2874999940395355},{"id":"https://openalex.org/C2992503517","wikidata":"https://www.wikidata.org/wiki/Q620584","display_name":"Mental arithmetic","level":4,"score":0.28600001335144043},{"id":"https://openalex.org/C2992503517","wikidata":"https://www.wikidata.org/wiki/Q620584","display_name":"Mental arithmetic","level":4,"score":0.2858999967575073},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.27880001068115234},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.2777999937534332},{"id":"https://openalex.org/C100609095","wikidata":"https://www.wikidata.org/wiki/Q1335050","display_name":"Embodied cognition","level":2,"score":0.27390000224113464},{"id":"https://openalex.org/C511192102","wikidata":"https://www.wikidata.org/wiki/Q5156948","display_name":"Comprehension","level":2,"score":0.2703000009059906},{"id":"https://openalex.org/C2776889888","wikidata":"https://www.wikidata.org/wiki/Q1135789","display_name":"Unintended consequences","level":2,"score":0.2689000070095062},{"id":"https://openalex.org/C21963081","wikidata":"https://www.wikidata.org/wiki/Q11337567","display_name":"Working memory","level":3,"score":0.2671999931335449},{"id":"https://openalex.org/C204321447","wikidata":"https://www.wikidata.org/wiki/Q30642","display_name":"Natural language processing","level":1,"score":0.2612000107765198},{"id":"https://openalex.org/C132758656","wikidata":"https://www.wikidata.org/wiki/Q5307365","display_name":"Dreyfus model of skill acquisition","level":2,"score":0.26089999079704285},{"id":"https://openalex.org/C92393732","wikidata":"https://www.wikidata.org/wiki/Q1790374","display_name":"Learning theory","level":2,"score":0.26030001044273376},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.25780001282691956}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2601.22510","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2601.22510","pdf_url":"https://arxiv.org/pdf/2601.22510","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2601.22510","doi":"https://doi.org/10.48550/arxiv.2601.22510","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.22510","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2601.22510","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2601.22510","pdf_url":"https://arxiv.org/pdf/2601.22510","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"sustainable_development_goals_aurora":[{"display_name":"Quality education","id":"https://metadata.un.org/sdg/4","score":0.8252322673797607}],"study_designs":[],"awards":[{"id":"https://openalex.org/G5783629928","display_name":"Tackling High Dimensionality for Modern Machine Learning: Theory and Visualization","funder_award_id":"2412052","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320315389","display_name":"Open Philanthropy Project","ror":"https://ror.org/004d1k391"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7127068361.pdf","grobid_xml":"https://content.openalex.org/works/W7127068361.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Large":[0],"language":[1],"models":[2],"(LLMs)":[3],"often":[4,67],"achieve":[5],"strong":[6],"benchmark":[7],"accuracy":[8],"yet":[9],"remain":[10,42],"brittle":[11],"under":[12,132],"small":[13],"distribution":[14,134],"shifts.":[15,135],"While":[16],"recent":[17],"mechanistic":[18],"studies":[19],"reveal":[20],"the":[21,30,37,101,140,169],"discrepancy":[22],"between":[23,121,164],"LLMs":[24,147],"and":[25,36,129,148,168,178],"humans":[26],"in":[27,72,76,145],"skill":[28,34,61,166],"compositions,":[29,172],"learning":[31,111],"dynamics":[32],"of":[33,39,79],"acquisition":[35,118,167],"role":[38],"data":[40],"distributions":[41],"elusive.":[43],"In":[44],"this":[45,116],"study,":[46],"we":[47,84,94],"train":[48],"transformers":[49,66],"on":[50],"synthetic":[51],"arithmetic":[52,71],"tasks":[53],"with":[54,173],"black-box":[55],"model-agnostic":[56],"metrics":[57],"for":[58,70,175],"analyzing":[59],"non-human":[60,117],"compositions.":[62],"We":[63,136],"discover":[64],"that":[65,97,139],"acquire":[68],"skills":[69],"reverse":[73],"order":[74],"or":[75,107,156],"parallel":[77],"instead":[78],"human-like":[80,170],"sequential":[81],"rules--a":[82],"phenomenon":[83],"refer":[85],"to":[86,100],"as":[87],"shattered":[88],"compositionality.":[89],"To":[90],"explain":[91],"these":[92],"behaviors,":[93],"provide":[95],"evidence":[96],"correlational":[98],"matching":[99],"training":[102],"data,":[103],"rather":[104],"than":[105],"causal":[106],"procedural":[108],"composition,":[109],"shapes":[110],"dynamics.":[112],"As":[113],"a":[114,162],"consequence,":[115],"creates":[119],"competition":[120],"partially":[122],"learned":[123],"skills,":[124],"producing":[125],"characteristic":[126],"mixing":[127],"errors":[128],"weaker":[130],"robustness":[131],"controlled":[133],"further":[137],"show":[138],"same":[141],"qualitative":[142],"behavior":[143],"persists":[144],"modern":[146],"is":[149],"not":[150],"mitigated":[151],"by":[152],"pure":[153],"model":[154],"scaling":[155],"scratchpad":[157],"supervision.":[158],"Our":[159],"results":[160],"highlight":[161],"mismatch":[163],"training-time":[165],"hierarchical":[171],"implications":[174],"reasoning":[176],"reliability":[177],"out-of-distribution":[179],"robustness.":[180]},"counts_by_year":[],"updated_date":"2026-10-15T10:00:46.086208","created_date":"2026-02-03T00:00:00"}