{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T02:31:56Z","timestamp":1775097116863,"version":"3.50.1"},"reference-count":42,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T00:00:00Z","timestamp":1745280000000},"content-version":"unspecified","delay-in-days":111,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2025]]},"abstract":"<jats:title>Abstract<\/jats:title>\n\t  <jats:p>We investigate causal computations, which take sequences of inputs to sequences of outputs such that the <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000331_inline1.png\"\/>\n\t\t<jats:tex-math>\n$n$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula>th output depends on the first <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000331_inline2.png\"\/>\n\t\t<jats:tex-math>\n$n$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> inputs only. We model these in category theory via a construction taking a Cartesian category <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000331_inline3.png\"\/>\n\t\t<jats:tex-math>\n$\\mathbb{C}$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> to another category <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000331_inline4.png\"\/>\n\t\t<jats:tex-math>\n$\\mathrm{St}(\\mathbb{C})$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> with a novel trace-like operation called \u201cdelayed trace,\u201d which misses yanking and dinaturality axioms of the usual trace. The delayed trace operation provides a feedback mechanism in <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000331_inline5.png\"\/>\n\t\t<jats:tex-math>\n$\\mathrm{St}(\\mathbb{C})$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> with an implicit guardedness guarantee. When <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000331_inline6.png\"\/>\n\t\t<jats:tex-math>\n$\\mathbb{C}$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> is equipped with a Cartesian differential operator, we construct a differential operator for <jats:inline-formula>\n\t      <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000331_inline7.png\"\/>\n\t\t<jats:tex-math>\n$\\mathrm{St}(\\mathbb{C})$\n<\/jats:tex-math>\n\t      <\/jats:alternatives>\n\t    <\/jats:inline-formula> using an abstract version of backpropagation through time (BPTT), a technique from machine learning based on unrolling of functions. This obtains a swath of properties for BPTT, including a chain rule and Schwartz theorem. Our differential operator is also able to compute the derivative of a stateful network without requiring the network to be unrolled.<\/jats:p>","DOI":"10.1017\/s0960129524000331","type":"journal-article","created":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T10:34:34Z","timestamp":1745318074000},"update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Differentiable causal computations via delayed trace (extended version)"],"prefix":"10.1017","volume":"35","author":[{"given":"David","family":"Sprunger","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7529-5489","authenticated-orcid":false,"given":"Shin-ya","family":"Katsumata","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,4,22]]},"reference":[{"key":"S0960129524000331_ref8","doi-asserted-by":"publisher","DOI":"10.1145\/2676726.2676993"},{"key":"S0960129524000331_ref18","first-page":"35","article-title":"Differential interaction nets","volume":"123","author":"Ehrhard","year":"2005","journal-title":"Theoretical Computer Science"},{"key":"S0960129524000331_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-44584-6_30"},{"key":"S0960129524000331_ref29","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-4049(96)00012-6"},{"key":"S0960129524000331_ref34","doi-asserted-by":"publisher","DOI":"10.1109\/LICS.2010.11"},{"key":"S0960129524000331_ref37","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129504004517"},{"key":"S0960129524000331_ref1","doi-asserted-by":"crossref","unstructured":"Abel, A. and Pientka, B. 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