An asymptotic analysis of probabilistic logic programming, with\n implications for expressing projective families of distributions

Probabilistic logic programming is a major part of statistical relational\nartificial intelligence, where approaches from logic and probability are\nbrought together to reason about and learn from relational domains in a setting\nof uncertainty. However, the behaviour of statistical relational\nrepresentations across variable domain sizes is complex, and scaling inference\nand learning to large domains remains a significant challenge. In recent years,\nconnections have emerged between domain size dependence, lifted inference and\nlearning from sampled subpopulations. The asymptotic behaviour of statistical\nrelational representations has come under scrutiny, and projectivity was\ninvestigated as the strongest form of domain-size dependence, in which query\nmarginals are completely independent of the domain size.\n In this contribution we show that every probabilistic logic program under the\ndistribution semantics is asymptotically equivalent to an acyclic probabilistic\nlogic program consisting only of determinate clauses over probabilistic facts.\nWe conclude that every probabilistic logic program inducing a projective family\nof distributions is in fact everywhere equivalent to a program from this\nfragment, and we investigate the consequences for the projective families of\ndistributions expressible by probabilistic logic programs.\n To facilitate the application of classical results from finite model theory,\nwe introduce the abstract distribution semantics, defined as an arbitrary\nlogical theory over probabilistic facts. This bridges the gap to the\ndistribution semantics underlying probabilistic logic programming. In this\nrepresentation, determinate logic programs correspond to quantifier-free\ntheories, making asymptotic quantifier elimination results available for the\nsetting of probabilistic logic programming.\n This paper is under consideration for acceptance in TPLP.\n