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Lifted Inference beyond First-Order Logic

Sagar Malhotra, Davide Bizzaro, Luciano Serafini

TL;DR

This work extends domain-liftability from the two-variable fragment to FO^2 with counting by introducing acyclicity, connectivity, and forest constraints via a general counting-by-splitting approach. It provides polynomial-time WFOMC algorithms for universally quantified FO^2/C^2 sentences under these graph-theoretic axioms by modular reductions to DAG/connected graph counting and by leveraging inclusion-exclusion and component-splitting techniques. The results unify probabilistic inference in SRL with enumerative combinatorics, enabling tractable counting of complex relational constraints and illustrating applications to DAGs, phylogenetic networks, and connected/forest structures, with empirical validation in MLNs and combinatorial benchmarks. The framework offers a versatile, modular toolkit for counting constrained relational structures and suggests avenues for more expressive lifted inference in practical SRL systems.

Abstract

Weighted First Order Model Counting (WFOMC) is fundamental to probabilistic inference in statistical relational learning models. As WFOMC is known to be intractable in general ($\#$P-complete), logical fragments that admit polynomial time WFOMC are of significant interest. Such fragments are called domain liftable. Recent works have shown that the two-variable fragment of first order logic extended with counting quantifiers ($\mathrm{C^2}$) is domain-liftable. However, many properties of real-world data, like acyclicity in citation networks and connectivity in social networks, cannot be modeled in $\mathrm{C^2}$, or first order logic in general. In this work, we expand the domain liftability of $\mathrm{C^2}$ with multiple such properties. We show that any $\mathrm{C^2}$ sentence remains domain liftable when one of its relations is restricted to represent a directed acyclic graph, a connected graph, a tree (resp. a directed tree) or a forest (resp. a directed forest). All our results rely on a novel and general methodology of "counting by splitting". Besides their application to probabilistic inference, our results provide a general framework for counting combinatorial structures. We expand a vast array of previous results in discrete mathematics literature on directed acyclic graphs, phylogenetic networks, etc.

Lifted Inference beyond First-Order Logic

TL;DR

This work extends domain-liftability from the two-variable fragment to FO^2 with counting by introducing acyclicity, connectivity, and forest constraints via a general counting-by-splitting approach. It provides polynomial-time WFOMC algorithms for universally quantified FO^2/C^2 sentences under these graph-theoretic axioms by modular reductions to DAG/connected graph counting and by leveraging inclusion-exclusion and component-splitting techniques. The results unify probabilistic inference in SRL with enumerative combinatorics, enabling tractable counting of complex relational constraints and illustrating applications to DAGs, phylogenetic networks, and connected/forest structures, with empirical validation in MLNs and combinatorial benchmarks. The framework offers a versatile, modular toolkit for counting constrained relational structures and suggests avenues for more expressive lifted inference in practical SRL systems.

Abstract

Weighted First Order Model Counting (WFOMC) is fundamental to probabilistic inference in statistical relational learning models. As WFOMC is known to be intractable in general (P-complete), logical fragments that admit polynomial time WFOMC are of significant interest. Such fragments are called domain liftable. Recent works have shown that the two-variable fragment of first order logic extended with counting quantifiers () is domain-liftable. However, many properties of real-world data, like acyclicity in citation networks and connectivity in social networks, cannot be modeled in , or first order logic in general. In this work, we expand the domain liftability of with multiple such properties. We show that any sentence remains domain liftable when one of its relations is restricted to represent a directed acyclic graph, a connected graph, a tree (resp. a directed tree) or a forest (resp. a directed forest). All our results rely on a novel and general methodology of "counting by splitting". Besides their application to probabilistic inference, our results provide a general framework for counting combinatorial structures. We expand a vast array of previous results in discrete mathematics literature on directed acyclic graphs, phylogenetic networks, etc.
Paper Structure (28 sections, 28 theorems, 70 equations, 8 figures, 2 tables, 4 algorithms)

This paper contains 28 sections, 28 theorems, 70 equations, 8 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Basic Notation
  5. First-Order Logic
  6. FO² and its Extensions
  7. Types and Tables
  8. Weighted First Order Model Counting
  9. WFOMC in FO²
  10. WFOMC in C²
  11. Main Approach: Counting by Splitting
  12. WFOMC with DAG Axiom
  13. Principle of Inclusion-Exclusion
  14. Counting Directed Acyclic Graphs
  15. WFOMC with DAG Axiom
  16. ...and 13 more sections

Key Result

Proposition 1

Let $\Psi$ be a first order logic sentence such that $\Psi$ is of the form $\forall \bm{x}. \Phi(\bm{x})$, where $\bm{x} = x_1, ..., x_k$ represents the free variables in $\Phi(\bm{x})$, and $\Phi(\bm{x})$ is quantifier-free. If $\omega$ is an interpretation over a domain $\Delta$, and $\omega \mode

Figures (8)

  • Figure 1: Run-times of all experiments as function of the domain cardinality $n$. For the WFOMC encoding see the referred examples. The number of active $1$-types for each formula, and if cardinality constraints on binary predicates were used, is reported in Table \ref{['tab:active']} of the appendix.
  • Figure 2: Expected values of the number of edges of the directed/undirected graphs produced by MLNs with only the predicate $R/2$. The legends report the hard constraints, while the soft constraint is always $-1: R(x,y)$.
  • Figure 3: Distributions of smokers for different weights. The legends report the hard constraints, while the titles report the weights of the soft constraints. The domain size is fixed to $n=20$.
  • Figure 4: Run-times of computing the partition function of the "smokers and friends" MLNs.
  • Figure 5: Probability distributions of the number of edges of the directed/undirected graphs produced by MLNs with only the predicate $R/2$. The legends report the hard constraints, while the soft constraint is always $-1: R(x,y)$. The domain size is fixed to $n=20$.
  • ...and 3 more figures

Theorems & Definitions (79)

  • Example 1
  • Proposition 1
  • proof
  • Example 2
  • Definition 1: 1-type Cardinality Vector
  • Definition 2: Symmetric Weight Function
  • Definition 3
  • Definition 4
  • Example 3
  • Proposition 2
  • ...and 69 more