[Paper Review] Complexity of Weighted First-Order Model Counting in the Two-Variable Fragment with Counting Quantifiers: A Bound to Beat

Jan Tóth, Ondřej Kuželka|arXiv (Cornell University)|Apr 19, 2024

Scientific Computing and Data ManagementDecision Sciences3 citations

TL;DR

This paper improves the time complexity bound for weighted first-order model counting (WFOMC) in the two-variable fragment with counting quantifiers (C²) by reducing the exponential dependency on counting quantifier parameters to quadratic, achieving a super-exponential speedup. The new method restructures WFOMC computation using a novel encoding that significantly tightens the upper bound on polynomial degree, setting a new benchmark for efficiency in domain-liftable logical reasoning.

ABSTRACT

We study the time complexity of the weighted first-order model counting (WFOMC) over the logical language with two variables and counting quantifiers. The problem is known to be solvable in time polynomial in the domain size. However, the degree of the polynomial, which turns out to be relatively high for most practical applications, has never been properly addressed. First, we formulate a time complexity bound for the existing techniques for solving WFOMC with counting quantifiers. The bound is already known to be a polynomial with its degree depending on the number of cells of the input formula. We observe that the number of cells depends, in turn, exponentially on the parameter of the counting quantifiers appearing in the formula. Second, we propose a new approach to dealing with counting quantifiers, reducing the exponential dependency to a quadratic one, therefore obtaining a tighter upper bound. It remains an open question whether the dependency of the polynomial degree on the counting quantifiers can be reduced further, thus making our new bound a bound to beat.

Motivation & Objective

• To analyze and formalize the time complexity of existing WFOMC techniques in the C² fragment with counting quantifiers.
• To identify that the polynomial degree in domain size depends exponentially on the counting quantifier parameter k, limiting practical efficiency.
• To develop a new algorithmic approach that reduces the dependency on k from exponential to quadratic, thereby tightening the upper complexity bound.
• To establish a new, more efficient complexity bound that serves as a 'bound to beat' for future improvements in WFOMC over C².
• To empirically validate the theoretical improvement through runtime measurements on C² formulas and their extensions.

Proposed method

• Derive an upper bound on the time complexity of existing WFOMC algorithms in C², showing that the polynomial degree grows exponentially with the parameter k of counting quantifiers.
• Propose a new encoding technique that restructures the WFOMC computation by grouping and precomputing subformulas based on quantifier structure and variable sharing.
• Introduce a canonical form for C² formulas that isolates counting quantifier dependencies and enables efficient aggregation of model counts.
• Apply dynamic programming over a refined partitioning of the Herbrand base, exploiting symmetry and variable equivalence to reduce redundant computation.
• Integrate the new encoding into a WFOMC pipeline, replacing the standard decomposition method used in prior approaches.
• Use bit complexity analysis to confirm that the new method remains within polynomial bounds even when handling large weights, preserving domain-liftability.

(a) $3$ -regular

Experimental results

Research questions

• RQ1 What is the exact degree of the polynomial in domain size for existing WFOMC algorithms in C² with counting quantifiers?
• RQ2 Can the exponential dependency on the counting quantifier parameter k be reduced to a lower-order polynomial?
• RQ3 Does the proposed encoding method yield a measurable performance improvement in practical WFOMC computation?
• RQ4 Is the new complexity bound tight enough to serve as a meaningful benchmark for future research?
• RQ5 Can the new method be extended to other domain-liftable fragments of first-order logic?

Key findings

• The existing WFOMC algorithms for C² exhibit a time complexity with a polynomial degree that grows exponentially in the parameter k of the counting quantifiers.
• The proposed method reduces the dependency on k from exponential to quadratic, resulting in a super-exponential speedup with respect to k.
• Runtime measurements on C² formulas and their extensions confirm a significant performance gain, especially for formulas with high k values.
• The new bound is tighter than any previously known upper bound for WFOMC in C², establishing a new standard for efficiency.
• The method preserves domain-liftability, as bit complexity remains polynomial, ensuring scalability even with large weights.
• The new complexity bound is now the best-known upper bound, and the authors position it as a 'bound to beat' for future research.

(b) $4$ -regular