Polynomial-Time Relational Probabilistic Inference in Open Universes
This work addresses a fundamental problem in AI for researchers and practitioners needing efficient uncertainty reasoning in complex, open-ended domains, representing a novel method for a known bottleneck rather than an incremental advance.
The paper tackles the challenge of balancing expressive power and computational tractability in first-order relational probabilistic inference, introducing a method that achieves polynomial-time reasoning for hybrid variables in open universes with bounded-degree and bounded quantifier rank knowledge bases.
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.