# Lifted Relational Probabilistic Inference via Implicit Learning

arXiv: https://arxiv.org/abs/2602.14890 · HTML: https://mlsanity.com/paper/2602.14890/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 16 of its 37 references that are in the map.
- First posted on arXiv 16 Feb 2026.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 16 of 16)

- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [Faith and Fate: Limits of Transformers on Compositionality](https://mlsanity.com/paper/2305.18654/) (2023)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Embers of Autoregression: Understanding Large Language Models Through the Problem They are Trained to Solve](https://mlsanity.com/paper/2309.13638/) (2023)
- [End-to-End Differentiable Proving](https://mlsanity.com/paper/1705.11040/) (2017)
- [Logic Tensor Networks for Semantic Image Interpretation](https://mlsanity.com/paper/1705.08968/) (2017)
- [Inductive logic programming at 30: a new introduction](https://mlsanity.com/paper/2008.07912/) (2020)
- [Large Language Models Are Not Strong Abstract Reasoners](https://mlsanity.com/paper/2305.19555/) (2023)
- [Solving SDP Faster: A Robust IPM Framework and Efficient Implementation](https://mlsanity.com/paper/2101.08208/) (2021)
- [Neural Markov Logic Networks](https://mlsanity.com/paper/1905.13462/) (2019)
- [When a language model is optimized for reasoning, does it still show embers of autoregression? An analysis of OpenAI o1](https://mlsanity.com/paper/2410.01792/) (2024)
- [A Faster Interior-Point Method for Sum-of-Squares Optimization](https://mlsanity.com/paper/2202.08489/) (2022)
- [Implicitly Learning to Reason in First-Order Logic](https://mlsanity.com/paper/1906.10106/) (2019)
- [Polynomial-time probabilistic reasoning with partial observations via implicit learning in probability logics](https://mlsanity.com/paper/1806.11204/) (2018)
- [Learning Implicitly with Noisy Data in Linear Arithmetic](https://mlsanity.com/paper/2010.12619/) (2020)
- [Polynomial-Time Relational Probabilistic Inference in Open Universes](https://mlsanity.com/paper/2505.04115/) (2025)

## Abstract (arXiv)

Reconciling the tension between inductive learning and deductive reasoning in first-order relational domains is a longstanding challenge in AI. We study the problem of answering queries in a first-order relational probabilistic logic through a joint effort of learning and reasoning, without ever constructing an explicit model. Traditional lifted inference assumes access to a complete model and exploits symmetry to evaluate probabilistic queries; however, learning such models from partial, noisy observations is intractable in general. We reconcile these two challenges through implicit learning to reason and first-order relational probabilistic inference techniques. More specifically, we merge incomplete first-order axioms with independently sampled, partially observed examples into a bounded-degree fragment of the sum-of-squares (SOS) hierarchy in polynomial time. Our algorithm performs two lifts simultaneously: (i) grounding-lift, where renaming-equivalent ground moments share one variable, collapsing the domain of individuals; and (ii) world-lift, where all pseudo-models (partial world assignments) are enforced in parallel, producing a global bound that holds across all worlds consistent with the learned constraints. These innovations yield the first polynomial-time framework that implicitly learns a first-order probabilistic logic and performs lifted inference over both individuals and worlds.