Weighted First Order Model Counting with Directed Acyclic Graph Axioms

Abstract: Statistical Relational Learning (SRL) integrates First-Order Logic (FOL) and probability theory for learning and inference over relational data. Probabilistic inference and learning in many SRL models can be reduced to Weighted First Order Model Counting (WFOMC). However, WFOMC is known to be intractable ($\mathrm{#P_1-}$ complete). Hence, logical fragments that admit polynomial time WFOMC are of significant interest. Such fragments are called domain liftable. Recent line of works have shown the two-variable fragment of FOL, extended with counting quantifiers ($\mathrm{C^2}$) to be domain-liftable. However, many properties of real-world data can not be modelled in $\mathrm{C^2}$. In fact many ubiquitous properties of real-world data are inexressible in FOL. Acyclicity is one such property, found in citation networks, genealogy data, temporal data e.t.c. In this paper we aim to address this problem by investigating the domain liftability of directed acyclicity constraints. We show that the fragment $\mathrm{C^2}$ with a Directed Acyclic Graph (DAG) axiom, i.e., a predicate in the language is axiomatized to represent a DAG, is domain-liftable. We present a method based on principle of inclusion-exclusion for WFOMC of $\mathrm{C^2}$ formulas extended with DAG axioms.