{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,25]],"date-time":"2026-08-25T01:38:13Z","timestamp":1787621893867,"version":"build-2736575974"},"reference-count":68,"publisher":"The Royal Society","issue":"2324","license":[{"start":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T00:00:00Z","timestamp":1784160000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,7,16]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Weak supervision enables machine learning models to learn from limited or noisy labels, but it introduces challenges in reliability and semantic clarity, particularly in multi-instance partial label learning (MI-PLL), where models must resolve both ambiguous supervision signals and uncertain instance-label mappings. This paper proposes a semantics for a neuro-symbolic framework that integrates inductive logic programming (ILP) to structure MI-PLL through relational constraints. In this formulation, ILP defines a hypothesis space over label transitions, formalizes the semantics of per-instance classifiers and provides a relational scaffold for reasoning about weak supervision. Two inductive tasks are studied in this framework: inferring the transition predicate (TP) from the observed and classifier predicates (CPs), and inferring instance-level classifier assignments from the observed and TPs. This formal semantics facilitates constraint specification, consistency checking and the diagnosis of semantic failure modes that bag-level accuracy alone may conceal. No claim is made, however, that it renders the underlying neural classifier more transparent.<\/jats:p>\n                  <jats:p>This article is part of the theme issue \u2018Safe, secure and robust AI for safety-critical systems\u2019.<\/jats:p>","DOI":"10.1098\/rsta.2025.0115","type":"journal-article","created":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T23:39:01Z","timestamp":1784245141000},"source":"Crossref","is-referenced-by-count":1,"title":["Neuro-symbolic weak supervision: theory and semantics"],"prefix":"10.1098","volume":"384","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-9376-6045","authenticated-orcid":false,"given":"Nijesh","family":"Upreti","sequence":"first","affiliation":[{"name":"The University of Edinburgh, School of Informatics , Edinburgh, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5573-8465","authenticated-orcid":false,"given":"Vaishak","family":"Belle","sequence":"additional","affiliation":[{"name":"The University of Edinburgh, School of Informatics , Edinburgh, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"175","published-online":{"date-parts":[[2026,7,16]]},"reference":[{"key":"2026082410124125000_bib1","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1093\/nsr\/nwx106","article-title":"A brief introduction to weakly supervised learning","volume":"5","author":"Zhou","year":"2018","journal-title":"Natl Sci. 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