{"id":"https://openalex.org/W3047914308","doi":"https://doi.org/10.1145/3434333","title":"A pre-expectation calculus for probabilistic sensitivity","display_name":"A pre-expectation calculus for probabilistic sensitivity","publication_year":2021,"publication_date":"2021-01-04","ids":{"openalex":"https://openalex.org/W3047914308","doi":"https://doi.org/10.1145/3434333","mag":"3047914308"},"language":"en","primary_location":{"id":"doi:10.1145/3434333","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3434333","pdf_url":"https://dl.acm.org/doi/pdf/10.1145/3434333","source":{"id":"https://openalex.org/S4210216081","display_name":"Proceedings of the ACM on Programming Languages","issn_l":"2475-1421","issn":["2475-1421"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the ACM on Programming Languages","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://dl.acm.org/doi/pdf/10.1145/3434333","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5073092626","display_name":"Alejandro Aguirre","orcid":"https://orcid.org/0000-0001-6746-2734"},"institutions":[{"id":"https://openalex.org/I4210162154","display_name":"IMDEA Software Institute","ror":"https://ror.org/04xvfkh51","country_code":"ES","type":"facility","lineage":["https://openalex.org/I105140100","https://openalex.org/I4210162154"]},{"id":"https://openalex.org/I88060688","display_name":"Universidad Polit\u00e9cnica de Madrid","ror":"https://ror.org/03n6nwv02","country_code":"ES","type":"education","lineage":["https://openalex.org/I88060688"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Alejandro Aguirre","raw_affiliation_strings":["IMDEA Software Institute, Spain / Universidad Polit\u00e9cnica de Madrid, Spain"],"affiliations":[{"raw_affiliation_string":"IMDEA Software Institute, Spain / Universidad Polit\u00e9cnica de Madrid, Spain","institution_ids":["https://openalex.org/I4210162154","https://openalex.org/I88060688"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075577852","display_name":"Gilles Barthe","orcid":"https://orcid.org/0000-0002-3853-1777"},"institutions":[{"id":"https://openalex.org/I4210162154","display_name":"IMDEA Software Institute","ror":"https://ror.org/04xvfkh51","country_code":"ES","type":"facility","lineage":["https://openalex.org/I105140100","https://openalex.org/I4210162154"]}],"countries":["ES"],"is_corresponding":false,"raw_author_name":"Gilles Barthe","raw_affiliation_strings":["MPI-SP, Germany / IMDEA Software Institute, Spain"],"affiliations":[{"raw_affiliation_string":"MPI-SP, Germany / IMDEA Software Institute, Spain","institution_ids":["https://openalex.org/I4210162154"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033286190","display_name":"Justin Hsu","orcid":"https://orcid.org/0000-0002-8953-7060"},"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":"Justin Hsu","raw_affiliation_strings":["University of Wisconsin-Madison, USA","University of Wisconsin, Madison,#R#                USA"],"affiliations":[{"raw_affiliation_string":"University of Wisconsin-Madison, USA","institution_ids":["https://openalex.org/I135310074"]},{"raw_affiliation_string":"University of Wisconsin, Madison,#R#                USA","institution_ids":["https://openalex.org/I135310074"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5045667283","display_name":"Benjamin Lucien Kaminski","orcid":"https://orcid.org/0000-0001-5185-2324"},"institutions":[{"id":"https://openalex.org/I45129253","display_name":"University College London","ror":"https://ror.org/02jx3x895","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I45129253"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Benjamin Lucien Kaminski","raw_affiliation_strings":["University College London, UK","University College; London; UK"],"affiliations":[{"raw_affiliation_string":"University College London, UK","institution_ids":["https://openalex.org/I45129253"]},{"raw_affiliation_string":"University College; London; UK","institution_ids":["https://openalex.org/I45129253"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5090819329","display_name":"Joost-Pieter Katoen","orcid":"https://orcid.org/0000-0002-6143-1926"},"institutions":[{"id":"https://openalex.org/I887968799","display_name":"RWTH Aachen University","ror":"https://ror.org/04xfq0f34","country_code":"DE","type":"education","lineage":["https://openalex.org/I887968799"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Joost-Pieter Katoen","raw_affiliation_strings":["RWTH Aachen University, Germany","RWTH Aachen University (Germany)"],"affiliations":[{"raw_affiliation_string":"RWTH Aachen University, Germany","institution_ids":["https://openalex.org/I887968799"]},{"raw_affiliation_string":"RWTH Aachen University (Germany)","institution_ids":["https://openalex.org/I887968799"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5089189715","display_name":"Christoph Matheja","orcid":"https://orcid.org/0000-0001-9151-0441"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]},{"id":"https://openalex.org/I887968799","display_name":"RWTH Aachen University","ror":"https://ror.org/04xfq0f34","country_code":"DE","type":"education","lineage":["https://openalex.org/I887968799"]}],"countries":["CH","DE"],"is_corresponding":false,"raw_author_name":"Christoph Matheja","raw_affiliation_strings":["RWTH Aachen University, Germany / ETH Zurich, Switzerland"],"affiliations":[{"raw_affiliation_string":"RWTH Aachen University, Germany / ETH Zurich, Switzerland","institution_ids":["https://openalex.org/I887968799","https://openalex.org/I35440088"]}]}],"institutions":[],"countries_distinct_count":5,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5073092626"],"corresponding_institution_ids":["https://openalex.org/I4210162154","https://openalex.org/I88060688"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.0229717,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"5","issue":"POPL","first_page":"1","last_page":"28"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","score":0.9983000159263611,"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":0.9983000159263611,"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/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9970999956130981,"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/T10462","display_name":"Reinforcement Learning in Robotics","score":0.9966999888420105,"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/probabilistic-logic","display_name":"Probabilistic logic","score":0.70675128698349},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5261504054069519},{"id":"https://openalex.org/keywords/total-variation","display_name":"Total variation","score":0.4801030457019806},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.471809983253479},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.46999678015708923},{"id":"https://openalex.org/keywords/sensitivity","display_name":"Sensitivity (control systems)","score":0.4663715958595276},{"id":"https://openalex.org/keywords/bounding-overwatch","display_name":"Bounding overwatch","score":0.4596758186817169},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.44499412178993225},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4286201298236847},{"id":"https://openalex.org/keywords/monotonic-function","display_name":"Monotonic function","score":0.42301174998283386},{"id":"https://openalex.org/keywords/probability-distribution","display_name":"Probability distribution","score":0.42295682430267334},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3925860524177551},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.16400650143623352},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10470640659332275},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.09634679555892944}],"concepts":[{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.70675128698349},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5261504054069519},{"id":"https://openalex.org/C62100291","wikidata":"https://www.wikidata.org/wiki/Q1936288","display_name":"Total variation","level":2,"score":0.4801030457019806},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.471809983253479},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.46999678015708923},{"id":"https://openalex.org/C21200559","wikidata":"https://www.wikidata.org/wiki/Q7451068","display_name":"Sensitivity (control systems)","level":2,"score":0.4663715958595276},{"id":"https://openalex.org/C63584917","wikidata":"https://www.wikidata.org/wiki/Q333286","display_name":"Bounding overwatch","level":2,"score":0.4596758186817169},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.44499412178993225},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4286201298236847},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.42301174998283386},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.42295682430267334},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3925860524177551},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.16400650143623352},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10470640659332275},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.09634679555892944},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C24326235","wikidata":"https://www.wikidata.org/wiki/Q126095","display_name":"Electronic engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1145/3434333","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3434333","pdf_url":"https://dl.acm.org/doi/pdf/10.1145/3434333","source":{"id":"https://openalex.org/S4210216081","display_name":"Proceedings of the ACM on Programming Languages","issn_l":"2475-1421","issn":["2475-1421"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the ACM on Programming Languages","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1901.06540","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1901.06540","pdf_url":"https://arxiv.org/pdf/1901.06540","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:www.research-collection.ethz.ch:20.500.11850/462800","is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.11850/462800.1","pdf_url":"http://hdl.handle.net/20.500.11850/462800","source":{"id":"https://openalex.org/S4306402302","display_name":"Repository for Publications and Research Data (ETH Zurich)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I35440088","host_organization_name":"ETH Zurich","host_organization_lineage":["https://openalex.org/I35440088"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Proceedings of the ACM on Programming Languages, 5 (POPL)","raw_type":"info:eu-repo/semantics/conferenceObject"},{"id":"pmh:oai:eprints.ucl.ac.uk.OAI2:10115803","is_oa":false,"landing_page_url":"https://discovery.ucl.ac.uk/id/eprint/10115803/","pdf_url":null,"source":{"id":"https://openalex.org/S4306400024","display_name":"UCL Discovery (University College London)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I45129253","host_organization_name":"University College London","host_organization_lineage":["https://openalex.org/I45129253"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"   Proceedings of the ACM on Programming Languages , 5  (POPL)    , Article 52. (2021)      ","raw_type":"Article"},{"id":"doi:10.48550/arxiv.1901.06540","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1901.06540","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"},{"id":"doi:10.18154/rwth-2021-05003","is_oa":true,"landing_page_url":"https://doi.org/10.18154/rwth-2021-05003","pdf_url":null,"source":{"id":"https://openalex.org/S4306401033","display_name":"RWTH Publications (RWTH Aachen)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I887968799","host_organization_name":"RWTH Aachen University","host_organization_lineage":["https://openalex.org/I887968799"],"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-journal"},{"id":"doi:10.3929/ethz-b-000462800","is_oa":true,"landing_page_url":"https://doi.org/10.3929/ethz-b-000462800","pdf_url":null,"source":{"id":"https://openalex.org/S7407051236","display_name":"ETH Z\u00fcrich Research Collection","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":"article-journal"},{"id":"mag:3047914308","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"doi:10.1145/3434333","is_oa":true,"landing_page_url":"https://doi.org/10.1145/3434333","pdf_url":"https://dl.acm.org/doi/pdf/10.1145/3434333","source":{"id":"https://openalex.org/S4210216081","display_name":"Proceedings of the ACM on Programming Languages","issn_l":"2475-1421","issn":["2475-1421"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the ACM on Programming Languages","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320308737","display_name":"Facebook","ror":"https://ror.org/01zbnvs85"},{"id":"https://openalex.org/F4320309434","display_name":"University of Wisconsin-Madison","ror":"https://ror.org/01y2jtd41"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3047914308.pdf","grobid_xml":"https://content.openalex.org/works/W3047914308.grobid-xml"},"referenced_works_count":40,"referenced_works":["https://openalex.org/W385466589","https://openalex.org/W601863383","https://openalex.org/W1602844557","https://openalex.org/W1734364899","https://openalex.org/W2000108089","https://openalex.org/W2007273498","https://openalex.org/W2014354084","https://openalex.org/W2019411264","https://openalex.org/W2027193703","https://openalex.org/W2097061283","https://openalex.org/W2100677568","https://openalex.org/W2131275552","https://openalex.org/W2139338362","https://openalex.org/W2141670510","https://openalex.org/W2149996206","https://openalex.org/W2168256531","https://openalex.org/W2222812367","https://openalex.org/W2252661985","https://openalex.org/W2294463722","https://openalex.org/W2536060102","https://openalex.org/W2562060745","https://openalex.org/W2571169473","https://openalex.org/W2745179117","https://openalex.org/W2753900993","https://openalex.org/W2760914093","https://openalex.org/W2782235253","https://openalex.org/W2799266534","https://openalex.org/W2803683238","https://openalex.org/W2860675612","https://openalex.org/W2894048303","https://openalex.org/W2899000992","https://openalex.org/W2906125418","https://openalex.org/W2949292350","https://openalex.org/W2963794891","https://openalex.org/W2995734336","https://openalex.org/W2995951356","https://openalex.org/W3037319651","https://openalex.org/W3041202696","https://openalex.org/W3119972339","https://openalex.org/W3121327281"],"related_works":["https://openalex.org/W1559786788","https://openalex.org/W3013875875","https://openalex.org/W2135587265","https://openalex.org/W2900413721","https://openalex.org/W2255026220","https://openalex.org/W3045080532","https://openalex.org/W2548641835","https://openalex.org/W1914062059","https://openalex.org/W2951243569","https://openalex.org/W2293992861","https://openalex.org/W79686148","https://openalex.org/W3199797246","https://openalex.org/W2745049847","https://openalex.org/W2050681940","https://openalex.org/W300219822","https://openalex.org/W2343406609","https://openalex.org/W3132304818","https://openalex.org/W2117355432","https://openalex.org/W1795606407","https://openalex.org/W3024982789"],"abstract_inverted_index":{"Sensitivity":[0],"properties":[1],"describe":[2],"how":[3],"changes":[4],"to":[5,100,147,153,163],"the":[6,12,18,21,31,34,41,65,68,78,90,103,154],"input":[7],"of":[8,23,30,57,67,109,121,127,150,167],"a":[9,27,46,54,59,96,110,122,128],"program":[10,168],"affect":[11],"output,":[13],"typically":[14],"by":[15,26,63,72,117],"upper":[16,101],"bounding":[17],"distance":[19,32,42,47,60,66,76,105],"between":[20,33,43,48,61,106],"outputs":[22,44],"two":[24,107],"runs":[25],"monotone":[28],"function":[29],"corresponding":[35],"inputs.":[36],"When":[37],"programs":[38],"are":[39],"probabilistic,":[40],"is":[45],"distributions.":[49],"The":[50],"Kantorovich":[51,104],"lifting":[52,64],"provides":[53],"general":[55],"way":[56],"defining":[58],"distributions":[62],"underlying":[69],"sample":[70],"space;":[71],"choosing":[73],"an":[74,160],"appropriate":[75],"on":[77],"base":[79],"space,":[80],"one":[81],"can":[82],"recover":[83],"other":[84],"usual":[85],"probabilistic":[86,111],"distances,":[87],"such":[88],"as":[89],"Total":[91],"Variation":[92],"distance.":[93],"We":[94,113,139],"develop":[95],"relational":[97],"pre-expectation":[98],"calculus":[99,146],"bound":[102],"executions":[108,169],"program.":[112],"illustrate":[114],"our":[115,145],"methods":[116],"proving":[118],"algorithmic":[119],"stability":[120],"machine":[123],"learning":[124,130],"algorithm,":[125,131],"convergence":[126,149],"reinforcement":[129],"and":[132,159],"fast":[133],"mixing":[134],"for":[135],"card":[136],"shuffling":[137],"algorithms.":[138],"also":[140],"consider":[141],"some":[142],"extensions:":[143],"using":[144],"show":[148],"Markov":[151],"chains":[152],"uniform":[155],"distribution":[156],"over":[157],"states":[158],"asynchronous":[161],"extension":[162],"reason":[164],"about":[165],"pairs":[166],"with":[170],"different":[171],"control":[172],"flow.":[173]},"counts_by_year":[{"year":2019,"cited_by_count":1}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-10T00:00:00"}