Neuro-symbolic weak supervision: theory and semantics.

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. This article is part of the theme issue 'Safe, secure and robust AI for safety-critical systems'.