{"id":4796,"date":"2022-03-17T10:50:14","date_gmt":"2022-03-17T09:50:14","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4796"},"modified":"2022-04-07T11:33:07","modified_gmt":"2022-04-07T09:33:07","slug":"is-there-a-fubini-tonelli-type-theorem-for-continuous-valuations-on-dcpo","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4796","title":{"rendered":"Is there a Fubini-Tonelli-type theorem for continuous valuations on Dcpo?"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">This is a talk I gave at Hunan University, Changsha, Hunan, China, on April 14th, 2022, based on common work with <a href=\"https:\/\/scholar.google.co.uk\/citations?user=B4D_a-MAAAAJ&amp;hl=en\">Xiaodong Jia<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Abstract<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Continuous valuations are a close cousin of measures which have had some success in denotational semantics.  They enjoy a Fubini-Tonelli-type theorem on <strong>Top<\/strong>, the category of topological spaces, and this is fine: this allows us to swap the order of integration, and semantically, this means that we can draw two independent random variables in the order we wish.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">However, the denotational semantics of programs operates in the smaller (full) subcategory <strong>Dcpo<\/strong> of dcpos, and, perhaps surprisingly, such a Fubini-Tonelli theorem is not known in that context\u2014except on continuous dcpos, but that causes other problems.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There are solutions to this conundrum.  Instead considering all continuous valuations, we may consider <em>minimal valuations<\/em>, or Heckmann&#8217;s larger class of <em>point-continuous<\/em> valuations.  Rather to the point, there are Fubini-Tonelli theorems for those restricted classes of continuous valuations, which were discovered pretty recently, and they suffice to give semantics of expressive, higher-order probabilistic programming languages.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This raises the following natural question: are all those notions of valuations on dcpos distinct, or are they all the same?  Every minimal valuation is point-continuous, and every point-continuous valuation is continuous.  We show that the three notions are distinct, by showing that:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>There is a continuous valuation on the Johnstone dcpo (a famous counterexample in domain theory) which is not minimal, while all of them are point-continuous.<\/li><li>There is a continuous valuation on the Smyth powerdomain of the Sorgenfrey line (a famous counterexample in topology) which is not point-continuous.<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Reference<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">This is based on the following paper:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Jean Goubault-Larrecq and Xiadong Jia, <em><a href=\"https:\/\/www.cambridge.org\/core\/journals\/mathematical-structures-in-computer-science\/article\/abs\/separating-minimal-valuations-pointcontinuous-valuations-and-continuous-valuations\/1F71202B8D0EAE2012C269026B84F4E6\">Separating minimal valuations, point-continuous valuations, and continuous valuations<\/a><\/em>, <a href=\"https:\/\/www.cambridge.org\/core\/journals\/mathematical-structures-in-computer-science\">Mathematical Structures in Computer Science<\/a>, <a href=\"https:\/\/www.cambridge.org\/core\/journals\/mathematical-structures-in-computer-science\/issue\/12F5EC5456CA97397AF20847E007F9BF\">Volume 31 Issue 6<\/a>, published online 07 December 2021 (available from <a href=\"https:\/\/arxiv.org\/abs\/2109.00426\">https:\/\/arxiv.org\/abs\/2109.00426<\/a>).<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Slides and videos<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Changsha-apr-14-2022\/fubini-step-by-step_compressed.pdf\">full slides<\/a>, with all animation steps; the <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Changsha-apr-14-2022\/fubini_compressed.pdf\">shorter<\/a> presentation, without them.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The videos:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li><a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Changsha-apr-14-2022\/fubini-1-valuations.mp4\">Introduction<\/a>, continuous valuations (8:02)<\/li><li><a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Changsha-apr-14-2022\/fubini-2-fubini-tonelli.mp4\">Fubini-Tonelli<\/a> theorems, and the problem at hand (11:48)<\/li><li>A <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Changsha-apr-14-2022\/fubini-3-computer-science.mp4\">computer scientist&#8217;s<\/a> perspective (11:45)<\/li><li><a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Changsha-apr-14-2022\/fubini-4-minimal-and-point-continuous-valuations.mp4\">Minimal and point-continuous<\/a> valuations (11:35)<\/li><li><a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Changsha-apr-14-2022\/fubini-5-mu-is-not-minimal.mp4\">Separating minimal from point-continuous<\/a> valuations (14:51)<\/li><li><a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Changsha-apr-14-2022\/fubini-6-lambda-bar-is-not-point-continuous.mp4\">Separating point-continuous from continuous<\/a> valuations (13:54)<\/li><li><a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/Talks\/Changsha-apr-14-2022\/fubini-7-conclusion.mp4\">Conclusion<\/a> (3:32)<\/li><\/ol>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"alignright is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-content\/uploads\/2016\/08\/jgl-2011.png\" alt=\"jgl-2011\" class=\"wp-image-993\" width=\"60\" height=\"83\"\/><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">\u2014 <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/?l=en\">Jean Goubault-Larrecq<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is a talk I gave at Hunan University, Changsha, Hunan, China, on April 14th, 2022, based on common work with Xiaodong Jia. Abstract Continuous valuations are a close cousin of measures which have had some success in denotational semantics. &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=4796\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-4796","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/4796","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4796"}],"version-history":[{"count":21,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/4796\/revisions"}],"predecessor-version":[{"id":4925,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/4796\/revisions\/4925"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4796"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}