[Paper Review] Understanding the Complexity of Lifted Inference and Asymmetric Weighted Model Counting

Eric Gribkoff, Guy Van den Broeck+1 more|arXiv (Cornell University)|May 13, 2014

Bayesian Modeling and Causal Inference12 references14 citations

TL;DR

This paper introduces Lift${}^{ extbf{R}}$, a lifted inference algorithm for weighted first-order model counting (WFOMC) that generalizes prior approaches in SRL and PDB. It establishes a novel dichotomy theorem for a non-trivial fragment of FO CNF sentences, proving that WFOMC is either in PTIME or #P-hard based on sentence structure, with Lift${}^{ extbf{R}}$ solving tractable cases in polynomial time and failing on intractable ones.

ABSTRACT

In this paper we study lifted inference for the Weighted First-Order Model Counting problem (WFOMC), which counts the assignments that satisfy a given sentence in first-order logic (FOL); it has applications in Statistical Relational Learning (SRL) and Probabilistic Databases (PDB). We present several results. First, we describe a lifted inference algorithm that generalizes prior approaches in SRL and PDB. Second, we provide a novel dichotomy result for a non-trivial fragment of FO CNF sentences, showing that for each sentence the WFOMC problem is either in PTIME or #P-hard in the size of the input domain; we prove that, in the first case our algorithm solves the WFOMC problem in PTIME, and in the second case it fails. Third, we present several properties of the algorithm. Finally, we discuss limitations of lifted inference for symmetric probabilistic databases (where the weights of ground literals depend only on the relation name, and not on the constants of the domain), and prove the impossibility of a dichotomy result for the complexity of probabilistic inference for the entire language FOL.

Motivation & Objective

• To develop a unified lifted inference algorithm that extends PDB-style asymmetric WFOMC to handle arbitrary CNF sentences, enabling lifted inference for SRL models with soft or hard evidence.
• To establish a theoretical dichotomy for a non-trivial fragment of first-order logic in conjunctive normal form (FO CNF), classifying WFOMC as either tractable (PTIME) or intractable (#P-hard).
• To demonstrate that Lift${}^\textbf{R}$ correctly identifies tractable cases via polynomial-time computation and fails on #P-hard cases, providing a precise complexity boundary.
• To clarify the limitations of lifted inference in symmetric probabilistic databases and show that a full dichotomy for general FOL is impossible.
• To bridge theoretical insights from probabilistic databases and statistical relational learning by unifying asymmetric WFOMC under a single algorithmic and complexity framework.

Proposed method

• Proposes Lift${}^\textbf{R}}$, a lifted inference algorithm that generalizes existing PDB and SRL techniques by supporting arbitrary CNF sentences, including those with negation.
• Employs a recursive decomposition strategy based on variable elimination and factorization over first-order atoms, leveraging symmetries in the logical structure.
• Applies algebraic geometry tools, including Hilbert’s Nullstellensatz and quotient rings, to analyze algebraic independence of polynomials and derive hardness results.
• Uses a zigzag construction to reduce known #P-hard problems to WFOMC instances with negation, proving hardness under specific structural conditions.
• Employs a quotient ring construction $ R[x]/\langle q \rangle $ to analyze polynomial equivalence classes and derive contradictions in the existence of annihilating polynomials.
• Introduces a framework to treat unwanted components of probabilistic queries as constant factors $ c_0 $, enabling reduction of complex queries to simpler, equivalent forms.

Experimental results

Research questions

• RQ1 Can a lifted inference algorithm be designed to handle arbitrary CNF sentences in asymmetric WFOMC, including those with negation, beyond the current MDNF restriction?
• RQ2 Is there a structural dichotomy in the complexity of WFOMC for a non-trivial fragment of FO CNF sentences, such that each sentence is either in PTIME or #P-hard?
• RQ3 Does the proposed algorithm Lift${}^\textbf{R}}$ correctly classify tractable WFOMC problems as solvable in polynomial time and intractable ones as failing?
• RQ4 Can a full dichotomy result be established for the entire first-order logic language, or are there inherent limitations due to symmetry assumptions?
• RQ5 What are the theoretical and practical limitations of lifted inference in symmetric probabilistic databases, and why is a general dichotomy impossible for full FOL?

Key findings

• Lift${}^\textbf{R}}$ generalizes prior lifted inference techniques by supporting arbitrary CNF sentences, including those with negation, enabling lifted inference for conditional queries in SRL and PDB.
• A novel dichotomy theorem is proven for a non-trivial fragment of FO CNF: for each sentence in this class, WFOMC is either in PTIME or #P-hard, depending solely on the sentence’s structure.
• When WFOMC is tractable (PTIME), Lift${}^\textbf{R}}$ solves it in polynomial time; when #P-hard, the algorithm fails, providing a precise classification boundary.
• The algorithm’s correctness and complexity classification are grounded in algebraic geometry, using tools like Hilbert’s Nullstellensatz and quotient rings to analyze polynomial independence.
• The paper proves that a full dichotomy for general FOL is impossible due to the lack of symmetries in asymmetric probabilistic databases, limiting the scope of such results.
• The framework enables the treatment of complex queries with negation by reducing them to equivalent forms via constant-factor scaling, allowing hardness proofs using the zigzag construction.