{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T10:10:43Z","timestamp":1784196643976,"version":"3.55.0"},"reference-count":88,"publisher":"Association for Computing Machinery (ACM)","issue":"PLDI","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["Proc. ACM Program. Lang."],"published-print":{"date-parts":[[2025,6,10]]},"abstract":"<jats:p>\n                    A\n                    <jats:italic toggle=\"yes\">singular function<\/jats:italic>\n                    is a partial function such that at one or more points, the left and\/or right limit diverge (e.g., the function 1\/x). Since programming languages typically support division, programs may denote singular functions. Although on its own, a singularity may be considered a bug, introducing a division-by-zero error,\n                    <jats:italic toggle=\"yes\">singular integrals\u2014<\/jats:italic>\n                    a version of the integral that is well-defined when the integrand is a singular function and the domain of integration contains a singularity\u2014arise in science and engineering, including in physics, aerodynamics, mechanical engineering, and computer graphics.\n                  <\/jats:p>\n                  <jats:p>\n                    In this paper, we present the first semantics of a programming language for singular integration. Our differentiable programming language,\n                    <jats:sc>SingularFlow,<\/jats:sc>\n                    supports the evaluation and differentiation of singular integrals. We formally define the denotational semantics of\n                    <jats:sc>SingularFlow,<\/jats:sc>\n                    deriving all the necessary mathematical machinery so that this work is rigorous and self-contained. We then define an operational semantics for\n                    <jats:sc>SingularFlow<\/jats:sc>\n                    that estimates integrals and their derivatives using Monte Carlo samples, and show that the operational semantics is a well-behaved estimator for the denotational semantics.\n                  <\/jats:p>\n                  <jats:p>\n                    We implement\n                    <jats:sc>SingularFlow<\/jats:sc>\n                    in JAX and evaluate the implementation on a suite of benchmarks that perform the\n                    <jats:italic toggle=\"yes\">finite Hilbert transform,<\/jats:italic>\n                    an integral transform related to the Fourier transform, which arises in domains such as physics and electrical engineering. We then use\n                    <jats:sc>SingularFlow<\/jats:sc>\n                    to approximate the solutions to four\n                    <jats:italic toggle=\"yes\">singular integral equations\u2014<\/jats:italic>\n                    equations where the unknown function is in the integrand of a singular integral\u2014arising in aerodynamics and mechanical engineering.\n                  <\/jats:p>","DOI":"10.1145\/3729263","type":"journal-article","created":{"date-parts":[[2025,6,13]],"date-time":"2025-06-13T16:02:27Z","timestamp":1749830547000},"page":"421-446","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Semantics of Integrating and Differentiating Singularities"],"prefix":"10.1145","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-8735-001X","authenticated-orcid":false,"given":"Jesse","family":"Michel","sequence":"first","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0301-0872","authenticated-orcid":false,"given":"Wonyeol","family":"Lee","sequence":"additional","affiliation":[{"name":"POSTECH, Pohang, Republic of Korea"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1502-2942","authenticated-orcid":false,"given":"Hongseok","family":"Yang","sequence":"additional","affiliation":[{"name":"KAIST, Daejeon, Republic of Korea"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2025,6,13]]},"reference":[{"key":"e_1_3_2_2_2","unstructured":"Ira H Abbott Albert E Von Doenhoff and Louis Stivers Jr.. 1945. 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