{"id":"https://openalex.org/W7138962675","doi":"https://doi.org/10.1007/s10703-026-00494-9","title":"Symbolic computer algebra for multipliers revisited - demonstrating the significance of order and phase optimization","display_name":"Symbolic computer algebra for multipliers revisited - demonstrating the significance of order and phase optimization","publication_year":2026,"publication_date":"2026-03-19","ids":{"openalex":"https://openalex.org/W7138962675","doi":"https://doi.org/10.1007/s10703-026-00494-9"},"language":"en","primary_location":{"id":"doi:10.1007/s10703-026-00494-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10703-026-00494-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10703-026-00494-9.pdf","source":{"id":"https://openalex.org/S3845260","display_name":"Formal Methods in System Design","issn_l":"0925-9856","issn":["0925-9856","1572-8102"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formal Methods in System Design","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10703-026-00494-9.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057999267","display_name":"Alexander Konrad","orcid":null},"institutions":[{"id":"https://openalex.org/I161046081","display_name":"University of Freiburg","ror":"https://ror.org/0245cg223","country_code":"DE","type":"education","lineage":["https://openalex.org/I161046081"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Alexander Konrad","raw_affiliation_strings":["Department of Computer Science, University of Freiburg, Georges-K\u00f6hler-Allee 51, 79110, Freiburg i.Br., Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Freiburg, Georges-K\u00f6hler-Allee 51, 79110, Freiburg i.Br., Germany","institution_ids":["https://openalex.org/I161046081"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5014271471","display_name":"C. Scholl","orcid":null},"institutions":[{"id":"https://openalex.org/I161046081","display_name":"University of Freiburg","ror":"https://ror.org/0245cg223","country_code":"DE","type":"education","lineage":["https://openalex.org/I161046081"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Christoph Scholl","raw_affiliation_strings":["Department of Computer Science, University of Freiburg, Georges-K\u00f6hler-Allee 51, 79110, Freiburg i.Br., Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Freiburg, Georges-K\u00f6hler-Allee 51, 79110, Freiburg i.Br., Germany","institution_ids":["https://openalex.org/I161046081"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5057999267"],"corresponding_institution_ids":["https://openalex.org/I161046081"],"apc_list":{"value":2390,"currency":"USD","value_usd":2390},"apc_paid":{"value":2390,"currency":"USD","value_usd":2390},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.28821293,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"68","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","score":0.4207000136375427,"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"}},"topics":[{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","score":0.4207000136375427,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.2727000117301941,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.18529999256134033,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/rewriting","display_name":"Rewriting","score":0.6851999759674072},{"id":"https://openalex.org/keywords/symbolic-computation","display_name":"Symbolic computation","score":0.5246999859809875},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.450300008058548},{"id":"https://openalex.org/keywords/process","display_name":"Process (computing)","score":0.44359999895095825},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.42399999499320984},{"id":"https://openalex.org/keywords/formal-verification","display_name":"Formal verification","score":0.39649999141693115},{"id":"https://openalex.org/keywords/kleene-algebra","display_name":"Kleene algebra","score":0.37929999828338623},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.3659999966621399},{"id":"https://openalex.org/keywords/disjunctive-normal-form","display_name":"Disjunctive normal form","score":0.35659998655319214}],"concepts":[{"id":"https://openalex.org/C154690210","wikidata":"https://www.wikidata.org/wiki/Q1668499","display_name":"Rewriting","level":2,"score":0.6851999759674072},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5486999750137329},{"id":"https://openalex.org/C110812573","wikidata":"https://www.wikidata.org/wiki/Q175515","display_name":"Symbolic computation","level":2,"score":0.5246999859809875},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.459199994802475},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.450300008058548},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.44359999895095825},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4296000003814697},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.42399999499320984},{"id":"https://openalex.org/C111498074","wikidata":"https://www.wikidata.org/wiki/Q173326","display_name":"Formal verification","level":2,"score":0.39649999141693115},{"id":"https://openalex.org/C206470798","wikidata":"https://www.wikidata.org/wiki/Q2634506","display_name":"Kleene algebra","level":2,"score":0.37929999828338623},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.3659999966621399},{"id":"https://openalex.org/C136726353","wikidata":"https://www.wikidata.org/wiki/Q903789","display_name":"Disjunctive normal form","level":2,"score":0.35659998655319214},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3531999886035919},{"id":"https://openalex.org/C23123167","wikidata":"https://www.wikidata.org/wiki/Q7661193","display_name":"Symbolic trajectory evaluation","level":3,"score":0.3522000014781952},{"id":"https://openalex.org/C2778220771","wikidata":"https://www.wikidata.org/wiki/Q1522579","display_name":"Substitution (logic)","level":2,"score":0.3264999985694885},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.3165999948978424},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.3084000051021576},{"id":"https://openalex.org/C51823790","wikidata":"https://www.wikidata.org/wiki/Q504353","display_name":"Greedy algorithm","level":2,"score":0.30630001425743103},{"id":"https://openalex.org/C157922185","wikidata":"https://www.wikidata.org/wiki/Q173198","display_name":"Logic synthesis","level":3,"score":0.3003000020980835},{"id":"https://openalex.org/C162319229","wikidata":"https://www.wikidata.org/wiki/Q175263","display_name":"Data structure","level":2,"score":0.2953000068664551},{"id":"https://openalex.org/C161771561","wikidata":"https://www.wikidata.org/wiki/Q1970286","display_name":"Process calculus","level":2,"score":0.2948000133037567},{"id":"https://openalex.org/C96654402","wikidata":"https://www.wikidata.org/wiki/Q5469962","display_name":"Formal equivalence checking","level":3,"score":0.2924000024795532},{"id":"https://openalex.org/C75606506","wikidata":"https://www.wikidata.org/wiki/Q1049183","display_name":"Formal methods","level":2,"score":0.28450000286102295},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.2653999924659729},{"id":"https://openalex.org/C206880738","wikidata":"https://www.wikidata.org/wiki/Q431667","display_name":"Automated theorem proving","level":2,"score":0.26440000534057617},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.2513999938964844},{"id":"https://openalex.org/C81409106","wikidata":"https://www.wikidata.org/wiki/Q76505","display_name":"Combinational logic","level":3,"score":0.25029999017715454}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s10703-026-00494-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10703-026-00494-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10703-026-00494-9.pdf","source":{"id":"https://openalex.org/S3845260","display_name":"Formal Methods in System Design","issn_l":"0925-9856","issn":["0925-9856","1572-8102"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formal Methods in System Design","raw_type":"journal-article"},{"id":"pmh:oai:freidok.uni-freiburg.de:281752","is_oa":false,"landing_page_url":"https://nbn-resolving.org/urn:nbn:de:bsz:25-freidok-2817526","pdf_url":null,"source":{"id":"https://openalex.org/S4306401057","display_name":"FreiDok plus (Universit\u00e4tsbibliothek Freiburg)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I161046081","host_organization_name":"University of Freiburg","host_organization_lineage":["https://openalex.org/I161046081"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Formal methods in system design. - 68, 2 (2026) , 6, ISSN: 1572-8102","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1007/s10703-026-00494-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10703-026-00494-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10703-026-00494-9.pdf","source":{"id":"https://openalex.org/S3845260","display_name":"Formal Methods in System Design","issn_l":"0925-9856","issn":["0925-9856","1572-8102"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formal Methods in System Design","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7138962675.pdf","grobid_xml":"https://content.openalex.org/works/W7138962675.grobid-xml"},"referenced_works_count":42,"referenced_works":["https://openalex.org/W1528837436","https://openalex.org/W1844774176","https://openalex.org/W1963965124","https://openalex.org/W1976325818","https://openalex.org/W1983849809","https://openalex.org/W2000519336","https://openalex.org/W2005126221","https://openalex.org/W2026445983","https://openalex.org/W2075743607","https://openalex.org/W2080267935","https://openalex.org/W2127271657","https://openalex.org/W2127420616","https://openalex.org/W2143462372","https://openalex.org/W2168543008","https://openalex.org/W2282171944","https://openalex.org/W2329849566","https://openalex.org/W2752387439","https://openalex.org/W2767589661","https://openalex.org/W2768948804","https://openalex.org/W2798446820","https://openalex.org/W2899780216","https://openalex.org/W2946097251","https://openalex.org/W2974269117","https://openalex.org/W2983688775","https://openalex.org/W3036041789","https://openalex.org/W3043296681","https://openalex.org/W3092596670","https://openalex.org/W3113354335","https://openalex.org/W3137859721","https://openalex.org/W3158190002","https://openalex.org/W3165204519","https://openalex.org/W3186608118","https://openalex.org/W4223620618","https://openalex.org/W4236423066","https://openalex.org/W4242633448","https://openalex.org/W4250470207","https://openalex.org/W4253715966","https://openalex.org/W4280615637","https://openalex.org/W4285261767","https://openalex.org/W4293261720","https://openalex.org/W4393904817","https://openalex.org/W6946279046"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"Using":[1],"Symbolic":[2],"Computer":[3],"Algebra":[4],"(SCA)":[5],"enabled":[6],"a":[7,75,93,124],"huge":[8],"progress":[9],"in":[10,16,110],"formal":[11],"verification":[12,31,116],"of":[13,32,35,55,67,82,107,131,158],"arithmetic":[14],"circuits":[15,161],"recent":[17],"years.":[18],"Several":[19],"different":[20],"approaches":[21],"have":[22],"been":[23],"proposed":[24],"showing":[25],"great":[26],"success":[27],"especially":[28],"for":[29,44,97],"the":[30,68,80,98,105,111,115,129,159],"multipliers.":[33],"Some":[34],"them":[36],"are":[37,120],"based":[38,122],"on":[39,123],"precomputing":[40],"and":[41,63,85,102,150],"simplifying":[42],"polynomials":[43,113,133],"specific":[45],"circuit":[46],"structures":[47],"like":[48],"converging":[49],"cones":[50],"while":[51],"others":[52],"take":[53],"advantage":[54],"known":[56],"or":[57],"detected":[58],"hierarchy":[59],"information":[60],"to":[61],"replace":[62],"simplify":[64],"particular":[65],"subcircuits":[66],"design.":[69],"In":[70],"this":[71,141],"paper":[72],"we":[73],"propose":[74],"new":[76],"method":[77,142],"that":[78,140],"avoids":[79],"use":[81],"such":[83],"methods":[84,119],"applies":[86],"only":[87],"two":[88],"dynamic":[89],"approaches:":[90],"(1)":[91],"choosing":[92],"good":[94],"substitution":[95],"order":[96],"backward":[99],"rewriting":[100],"process":[101],"(2)":[103],"adjusting":[104],"phases":[106],"signals":[108],"occurring":[109],"intermediate":[112,132],"during":[114],"process.":[117],"Both":[118],"simply":[121],"greedy":[125],"local":[126],"search":[127],"taking":[128],"sizes":[130],"into":[134],"account.":[135],"Our":[136],"experimental":[137],"results":[138],"show":[139],"is":[143],"very":[144],"competitive":[145],"with":[146],"already":[147],"existing":[148],"tools":[149],"it":[151],"improves":[152],"their":[153],"robustness,":[154],"e.g.":[155],"against":[156],"optimizations":[157],"verified":[160],"using":[162],"logic":[163],"synthesis.":[164]},"counts_by_year":[],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2026-03-20T00:00:00"}
