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Statistical Relational Learning (SRL) integrates First-Order Logic (FOL) andprobability theory for learning and inference over relational data.Probabilistic inference and learning in many SRL models can be reduced toWeighted First Order Model Counting (WFOMC). However, WFOMC is known to beintractable ($\mathrm{\#P_1-}$ complete). Hence, logical fragments that admitpolynomial time WFOMC are of significant interest. Such fragments are calleddomain liftable. Recent line of works have shown the two-variable fragment ofFOL, 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 areinexressible in FOL. Acyclicity is one such property, found in citationnetworks, genealogy data, temporal data e.t.c. In this paper we aim to addressthis problem by investigating the domain liftability of directed acyclicityconstraints. We show that the fragment $\mathrm{C^2}$ with a Directed AcyclicGraph (DAG) axiom, i.e., a predicate in the language is axiomatized torepresent a DAG, is domain-liftable. We present a method based on principle ofinclusion-exclusion for WFOMC of $\mathrm{C^2}$ formulas extended with DAGaxioms.

Weighted First Order Model Counting with Directed Acyclic Graph Axioms