{"id":"https://openalex.org/W7161712223","doi":"https://doi.org/10.48550/arxiv.2605.18079","title":"The Expressive Power of Low Precision Softmax Transformers with (Summarized) Chain-of-Thought","display_name":"The Expressive Power of Low Precision Softmax Transformers with (Summarized) Chain-of-Thought","publication_year":2026,"publication_date":"2026-05-18","ids":{"openalex":"https://openalex.org/W7161712223","doi":"https://doi.org/10.48550/arxiv.2605.18079"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.18079","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.18079","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.18079","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5136465011","display_name":"Moritz Br\u00f6samle","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Br\u00f6samle, Moritz","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5047214859","display_name":"Stephan Eckstein","orcid":"https://orcid.org/0000-0002-6407-2498"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Eckstein, Stephan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T11975","display_name":"Evolutionary Algorithms and Applications","score":0.06129999831318855,"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/T11975","display_name":"Evolutionary Algorithms and Applications","score":0.06129999831318855,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.060100000351667404,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T12808","display_name":"Ferroelectric and Negative Capacitance Devices","score":0.053700000047683716,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/softmax-function","display_name":"Softmax function","score":0.8741000294685364},{"id":"https://openalex.org/keywords/transformer","display_name":"Transformer","score":0.677299976348877},{"id":"https://openalex.org/keywords/operand","display_name":"Operand","score":0.4699999988079071},{"id":"https://openalex.org/keywords/learnability","display_name":"Learnability","score":0.421999990940094},{"id":"https://openalex.org/keywords/expressive-power","display_name":"Expressive power","score":0.38420000672340393},{"id":"https://openalex.org/keywords/interfacing","display_name":"Interfacing","score":0.38269999623298645},{"id":"https://openalex.org/keywords/turing-machine","display_name":"Turing machine","score":0.3659999966621399},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.31380000710487366}],"concepts":[{"id":"https://openalex.org/C188441871","wikidata":"https://www.wikidata.org/wiki/Q7554146","display_name":"Softmax function","level":3,"score":0.8741000294685364},{"id":"https://openalex.org/C66322947","wikidata":"https://www.wikidata.org/wiki/Q11658","display_name":"Transformer","level":3,"score":0.677299976348877},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6769999861717224},{"id":"https://openalex.org/C55526617","wikidata":"https://www.wikidata.org/wiki/Q719375","display_name":"Operand","level":2,"score":0.4699999988079071},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.45489999651908875},{"id":"https://openalex.org/C2777723229","wikidata":"https://www.wikidata.org/wiki/Q4367921","display_name":"Learnability","level":2,"score":0.421999990940094},{"id":"https://openalex.org/C195818886","wikidata":"https://www.wikidata.org/wiki/Q5421724","display_name":"Expressive power","level":2,"score":0.38420000672340393},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3831999897956848},{"id":"https://openalex.org/C2776303644","wikidata":"https://www.wikidata.org/wiki/Q1020499","display_name":"Interfacing","level":2,"score":0.38269999623298645},{"id":"https://openalex.org/C29248071","wikidata":"https://www.wikidata.org/wiki/Q163310","display_name":"Turing machine","level":3,"score":0.3659999966621399},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.31380000710487366},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3001999855041504},{"id":"https://openalex.org/C152124472","wikidata":"https://www.wikidata.org/wiki/Q1204361","display_name":"Redundancy (engineering)","level":2,"score":0.2971000075340271},{"id":"https://openalex.org/C577917","wikidata":"https://www.wikidata.org/wiki/Q189223","display_name":"Turing test","level":2,"score":0.2824999988079071},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.2759000062942505},{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.27160000801086426},{"id":"https://openalex.org/C34736171","wikidata":"https://www.wikidata.org/wiki/Q918333","display_name":"Preprocessor","level":2,"score":0.27079999446868896},{"id":"https://openalex.org/C9870796","wikidata":"https://www.wikidata.org/wiki/Q490481","display_name":"Turing","level":2,"score":0.2678000032901764},{"id":"https://openalex.org/C127705205","wikidata":"https://www.wikidata.org/wiki/Q5748245","display_name":"Heuristics","level":2,"score":0.2603999972343445},{"id":"https://openalex.org/C60048801","wikidata":"https://www.wikidata.org/wiki/Q1433889","display_name":"Intelligibility (philosophy)","level":2,"score":0.2596000134944916},{"id":"https://openalex.org/C55439883","wikidata":"https://www.wikidata.org/wiki/Q360812","display_name":"Correctness","level":2,"score":0.2563000023365021},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.2500999867916107}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.18079","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.18079","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.18079","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.18079","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Existing":[0],"expressivity":[1],"results":[2,143],"for":[3,156],"transformers":[4,63,87],"typically":[5],"rely":[6],"on":[7,144],"hardmax":[8,62],"attention,":[9],"high":[10],"precision,":[11],"and":[12,37,41,47,67,113,149],"other":[13],"architectural":[14],"modifications":[15],"that":[16,71,96,115],"disconnect":[17],"them":[18],"from":[19],"the":[20,53,82,89,102],"models":[21],"used":[22],"in":[23,127],"practice.":[24],"We":[25,136],"bridge":[26],"this":[27],"gap":[28],"by":[29,141],"analyzing":[30],"standard":[31],"transformer":[32],"decoders":[33],"with":[34,52,64,122,153],"softmax":[35,86],"attention":[36,42,69],"rounding":[38],"of":[39],"activations":[40,66],"weights,":[43],"while":[44],"allowing":[45],"depth":[46],"width":[48],"to":[49,84],"grow":[50],"logarithmically":[51,126],"context":[54],"length.":[55],"As":[56],"an":[57],"intermediate":[58],"step,":[59],"we":[60,105],"construct":[61],"ternary":[65],"well-separated":[68],"scores":[70],"simulate":[72],"Turing":[73,118],"machines":[74,119],"using":[75],"Chain-of-Thought":[76],"(CoT).":[77],"This":[78],"lets":[79],"us":[80],"convert":[81],"constructions":[83],"equivalent":[85],"without":[88],"unrealistic":[90],"parameter":[91],"magnitudes":[92],"or":[93],"activation":[94],"precision":[95],"prior":[97,157],"approaches":[98],"would":[99],"require.":[100],"Using":[101],"same":[103],"technique,":[104],"analyze":[106],"a":[107,128,133,145],"recently":[108],"proposed":[109],"summarized":[110],"CoT":[111],"paradigm":[112],"show":[114],"it":[116],"simulates":[117],"more":[120],"efficiently,":[121],"model":[123],"size":[124],"scaling":[125],"space":[129],"bound":[130],"rather":[131],"than":[132,155],"time":[134],"bound.":[135],"empirically":[137],"test":[138],"predictions":[139],"made":[140],"our":[142],"Sudoku":[146],"reasoning":[147],"task":[148],"find":[150],"better":[151],"alignment":[152],"learnability":[154],"high-precision":[158],"results.":[159],"Our":[160],"code":[161],"is":[162],"available":[163],"at":[164],"https://github.com/moritzbroe/transformer-expressivity.":[165]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-05-20T00:00:00"}