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Polynomial-Time Relational Probabilistic Inference in Open Universes

Luise Ge, Brendan Juba, Kris Nilsson

TL;DR

This work addresses efficient probabilistic inference over open, relational universes by extending sum-of-squares reasoning to a first-order probability-relational logic with expectation constraints. It introduces grounding and lifting techniques that yield a polynomial-time fragment when the quantifier rank and degree are fixed, even with a countably infinite object set. Key contributions include a formal open-universe grounding $OU(\Delta)$ via $k$ generic names, lifted SOS over equivalence classes, and a soundness-completeness theory that ties lifted, finite-degree SDPs to satisfiability. The approach offers certifiable bounds on all distributions within the fragment, providing robust, scalable reasoning for hybrid (discrete-continuous) relational problems and surpassing several existing open-universe models in theoretical guarantees and expressivity.

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.

Polynomial-Time Relational Probabilistic Inference in Open Universes

TL;DR

This work addresses efficient probabilistic inference over open, relational universes by extending sum-of-squares reasoning to a first-order probability-relational logic with expectation constraints. It introduces grounding and lifting techniques that yield a polynomial-time fragment when the quantifier rank and degree are fixed, even with a countably infinite object set. Key contributions include a formal open-universe grounding via generic names, lifted SOS over equivalence classes, and a soundness-completeness theory that ties lifted, finite-degree SDPs to satisfiability. The approach offers certifiable bounds on all distributions within the fragment, providing robust, scalable reasoning for hybrid (discrete-continuous) relational problems and surpassing several existing open-universe models in theoretical guarantees and expressivity.

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.
Paper Structure (21 sections, 7 theorems, 20 equations, 1 table)

This paper contains 21 sections, 7 theorems, 20 equations, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background and Notation
  4. (First-Order) Probability Relational Logic
  5. Propositional Sum-of-squares Refutations
  6. Tractability of Constant-Degree Sum-of-Squares (SOS)
  7. Efficient Reasoning in Open Universes
  8. Grounding in Probability Logic
  9. Satisfiability
  10. Equivalence Classes
  11. Lifted Sum-of-Squares
  12. Soundness and Completeness
  13. Discussion of Tractability
  14. Comparison to Other Models
  15. Tractable Markov Logic (TML)
  16. ...and 6 more sections

Key Result

Theorem 1

(Soundness shor87nesterov00parrilo00lasserre01). Let $\{g_i\geq 0\}_{i\in I}, \{h_j=0\}_{j\in J}, \{b_k\geq 0\}_{k\in K}$ be a system of constraints that is explicitly compact. Then either there is a degree-d sum-of-squares refutation or there is a solution to the degree-d sum-of-squares semidefinit

Theorems & Definitions (22)

  • Example 1
  • Example 2
  • Example 3
  • Theorem 1
  • Theorem 2
  • Definition 1
  • Example 4
  • Definition 2
  • Definition 3
  • Example 5
  • ...and 12 more