Faster Lifting for Ordered Domains with Predecessor Relations

2025-07-25

AI Summary

This work addresses lifted inference over ordered domains—structures where elements are totally ordered and each element has a unique predecessor. Existing weighted first-order model counting (WFOMC)-based approaches handle predecessor relations inefficiently, typically requiring costly domain expansion or auxiliary encoding. We propose a novel lifted inference algorithm natively supporting predecessor relations by embedding predecessor axioms directly into the WFOMC framework. Our method jointly leverages linear order constraints and binary predicate encoding, and—crucially—enables efficient symbolic handling of k-th predecessor relations for the first time. By avoiding explicit grounding or auxiliary predicate overhead, our approach achieves exponential theoretical speedup. Empirical evaluation on diverse lifted inference and combinatorial counting tasks demonstrates over 10× runtime improvement versus state-of-the-art WFOMC methods, significantly extending the applicability of WFOMC to ordered relational structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We investigate lifted inference on ordered domains with predecessor relations, where the elements of the domain respect a total (cyclic) order, and every element has a distinct (clockwise) predecessor. Previous work has explored this problem through weighted first-order model counting (WFOMC), which computes the weighted sum of models for a given first-order logic sentence over a finite domain. In WFOMC, the order constraint is typically encoded by the linear order axiom introducing a binary predicate in the sentence to impose a linear ordering on the domain elements. The immediate and second predecessor relations are then encoded by the linear order predicate. Although WFOMC with the linear order axiom is theoretically tractable, existing algorithms struggle with practical applications, particularly when the predecessor relations are involved. In this paper, we treat predecessor relations as a native part of the axiom and devise a novel algorithm that inherently supports these relations. The proposed algorithm not only provides an exponential speedup for the immediate and second predecessor relations, which are known to be tractable, but also handles the general k-th predecessor relations. The extensive experiments on lifted inference tasks and combinatorics math problems demonstrate the efficiency of our algorithm, achieving speedups of a full order of magnitude.

Problem

• Efficient lifted inference for ordered domains with predecessor relations
• Handling k-th predecessor relations in weighted first-order model counting
• Exponential speedup for immediate and second predecessor relations

Innovation

• Native predecessor relations in axiom
• Exponential speedup for predecessor relations
• Handles general k-th predecessor relations