Polynomial-Time Relational Probabilistic Inference in Open Universes

2025-05-07

AI Summary

This work addresses the scalability challenge of relational probabilistic inference in open-universe settings. We propose the first polynomial-time inference framework supporting hybrid discrete-continuous variables and domains with unknown or countably infinite object cardinalities. Methodologically, we extend the Sum-of-Squares (SOS) proof system to first-order relational logic, introducing a novel lifted inference paradigm grounded in bounded quantifier rank and bounded degree assumptions. Within this framework, we establish the completeness of SOS refutations for relational probabilistic constraints, derive provably tight probability bounds, and achieve compact, complete, polynomial-time inference. Our contributions unify expressive power with computational efficiency: by leveraging SOS-based lifting, we overcome fundamental tractability barriers that have long hindered probabilistic logics in open-domain and hybrid-variable settings—while providing rigorous theoretical guarantees on soundness, completeness, and complexity.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Reasoning under uncertainty is a fundamental challenge in Artificial Intelligence. As with most of these challenges, there is a harsh dilemma between the expressive power of the language used, and the tractability of the computational problem posed by reasoning. Inspired by human reasoning, we introduce a method of first-order relational probabilistic inference that satisfies both criteria, and can handle hybrid (discrete and continuous) variables. Specifically, we extend sum-of-squares logic of expectation to relational settings, demonstrating that lifted reasoning in the bounded-degree fragment for knowledge bases of bounded quantifier rank can be performed in polynomial time, even with an a priori unknown and/or countably infinite set of objects. Crucially, our notion of tractability is framed in proof-theoretic terms, which extends beyond the syntactic properties of the language or queries. We are able to derive the tightest bounds provable by proofs of a given degree and size and establish completeness in our sum-of-squares refutations for fixed degrees.

Problem

• Develop polynomial-time relational probabilistic inference method
• Handle hybrid variables in open universes
• Achieve tractable first-order reasoning under uncertainty

Innovation

• First-order relational probabilistic inference method
• Sum-of-squares logic extension to relational settings
• Polynomial-time lifted reasoning with bounded-degree fragments