The generalised distribution semantics and projective families of distributions
Felix Weitkämper
TL;DR
This work presents a generalised distribution semantics that separates a free probabilistic component from a deterministic layer, enabling a unified view across probabilistic logic programming, probabilistic databases, and related frameworks. It provides a complete characterisation of projective families of distributions representable within this generalised framework, showing that while many projective families remain out of reach, those that are representable are precisely reducts of relational stochastic block models not essentially asymmetric, and can be captured by acyclic determinate PLPs. The paper further connects projectivity to exchangeability and provides a category-theoretic perspective, establishing SIP (and semantic SIP) as central structural constraints that govern expressivity and learning across domain sizes. These results clarify fundamental limits of the distribution semantics, inform how learning can be performed on subdomains, and highlight the central role of determinacy in achieving a faithful yet tractable representation of projective distributions. Overall, the work delineates the boundary between expressivity and tractability for statistical relational formalisms based on a two-part, modular interpretation of probability and logic.
Abstract
We generalise the distribution semantics underpinning probabilistic logic programming by distilling its essential concept, the separation of a free random component and a deterministic part. This abstracts the core ideas beyond logic programming as such to encompass frameworks from probabilistic databases, probabilistic finite model theory and discrete lifted Bayesian networks. To demonstrate the usefulness of such a general approach, we completely characterise the projective families of distributions representable in the generalised distribution semantics and we demonstrate both that large classes of interesting projective families cannot be represented in a generalised distribution semantics and that already a very limited fragment of logic programming (acyclic determinate logic programs) in the determinsitic part suffices to represent all those projective families that are representable in the generalised distribution semantics at all.