Towards Counting Markov Equivalence Classes with Logical Constraints
Davide Bizzaro, Luciano Serafini, Sagar Malhotra
TL;DR
This work tackles counting Markov Equivalence Classes (MECs) of DAGs under logical constraints, focusing on MECs of size one and their essential DAG representatives. It combines tractable First-Order Model Counting (WFOMC) techniques for FO$^2$ and C$^2$ with a graph-theoretic treatment of essential DAGs, introducing the EssentialDAG axiom to count models efficiently in polynomial time with respect to the domain size. The main contributions include a polynomial-time algorithm for enumerating essential DAGs under arbitrary C$^2$ constraints, a modular WFOMC framework for FO$^2$ and C$^2$ with essential-DAG constraints, and a specialization to indegree-bounded essential DAGs enabling fine-grained MEC counting. This advances causal inference understanding by providing a scalable, logic-aware method to enumerate MECs and, consequently, bound the complexity of structure learning under knowledge constraints. The approach promises applicability to statistical-relational AI and causal discovery where background logical constraints govern permissible graph topologies.
Abstract
We initiate the study of counting Markov Equivalence Classes (MEC) under logical constraints. MECs are equivalence classes of Directed Acyclic Graphs (DAGs) that encode the same conditional independence structure among the random variables of a DAG model. Observational data can only allow to infer a DAG model up to Markov Equivalence. However, Markov equivalent DAGs can represent different causal structures, potentially super-exponentially many. Hence, understanding MECs combinatorially is critical to understanding the complexity of causal inference. In this paper, we focus on analysing MECs of size one, with logical constraints on the graph topology. We provide a polynomial-time algorithm (w.r.t. the number of nodes) for enumerating essential DAGs (the only members of an MEC of size one) with arbitrary logical constraints expressed in first-order logic with two variables and counting quantifiers (C^2). Our work brings together recent developments in tractable first-order model counting and combinatorics of MECs.