Polynomial-Time Relational Probabilistic Inference in Open Universes

TL;DR

This paper introduces a polynomial-time relational probabilistic inference method for open universes that extends sum-of-squares logic, enabling efficient reasoning with hybrid variables and infinite domains.

Contribution

It extends sum-of-squares logic to relational, first-order probabilistic inference, achieving polynomial-time reasoning in open, infinite, and hybrid domains.

Findings

Lifted reasoning in bounded-degree fragments is polynomial-time.

The method handles hybrid (discrete and continuous) variables.

It provides tight bounds and completeness in sum-of-squares refutations.

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.