# Abstract Visual Reasoning: An Algebraic Approach for Solving Raven's Progressive Matrices

arXiv: https://arxiv.org/abs/2303.11730 · HTML: https://mlsanity.com/paper/2303.11730/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 32 of its 99 references that are in the map.
- First posted on arXiv 21 Mar 2023.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 30 of 32)

- [MMDetection: Open MMLab Detection Toolbox and Benchmark](https://mlsanity.com/paper/1906.07155/) (2019)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [RAVEN: A Dataset for Relational and Analogical Visual rEasoNing](https://mlsanity.com/paper/1903.02741/) (2019)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [Measuring abstract reasoning in neural networks](https://mlsanity.com/paper/1807.04225/) (2018)
- [RNNLogic: Learning Logic Rules for Reasoning on Knowledge Graphs](https://mlsanity.com/paper/2010.04029/) (2020)
- [Abstraction and Analogy-Making in Artificial Intelligence](https://mlsanity.com/paper/2102.10717/) (2021)
- [Are Disentangled Representations Helpful for Abstract Visual Reasoning?](https://mlsanity.com/paper/1905.12506/) (2019)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [DeepMath - Deep Sequence Models for Premise Selection](https://mlsanity.com/paper/1606.04442/) (2016)
- [Stratified Rule-Aware Network for Abstract Visual Reasoning](https://mlsanity.com/paper/2002.06838/) (2020)
- [On the Expressive Power of Deep Polynomial Neural Networks](https://mlsanity.com/paper/1905.12207/) (2019)
- [Learning Perceptual Inference by Contrasting](https://mlsanity.com/paper/1912.00086/) (2019)
- [Graph Representations for Higher-Order Logic and Theorem Proving](https://mlsanity.com/paper/1905.10006/) (2019)
- [Scale-Localized Abstract Reasoning](https://mlsanity.com/paper/2009.09405/) (2020)
- [INT: An Inequality Benchmark for Evaluating Generalization in Theorem Proving](https://mlsanity.com/paper/2007.02924/) (2020)
- [Learning to Prove Theorems by Learning to Generate Theorems](https://mlsanity.com/paper/2002.07019/) (2020)
- [Abstract Spatial-Temporal Reasoning via Probabilistic Abduction and Execution](https://mlsanity.com/paper/2103.14230/) (2021)
- [Why AI is Harder Than We Think](https://mlsanity.com/paper/2104.12871/) (2021)
- [Abstract Reasoning with Distracting Features](https://mlsanity.com/paper/1912.00569/) (2019)
- [Learning Algebraic Representation for Systematic Generalization in Abstract Reasoning](https://mlsanity.com/paper/2111.12990/) (2021)
- [Effective Abstract Reasoning with Dual-Contrast Network](https://mlsanity.com/paper/2205.13720/) (2022)
- [Few-shot Visual Reasoning with Meta-analogical Contrastive Learning](https://mlsanity.com/paper/2007.12020/) (2020)
- [Solving Raven's Progressive Matrices with Multi-Layer Relation Networks](https://mlsanity.com/paper/2003.11608/) (2020)
- [Learning to Guide a Saturation-Based Theorem Prover](https://mlsanity.com/paper/2106.03906/) (2021)
- [Solving Raven's Progressive Matrices with Neural Networks](https://mlsanity.com/paper/2002.01646/) (2020)
- [Generating Correct Answers for Progressive Matrices Intelligence Tests](https://mlsanity.com/paper/2011.00496/) (2020)
- [Unsupervised Abstract Reasoning for Raven's Problem Matrices](https://mlsanity.com/paper/2109.10011/) (2021)
- [Raven's Progressive Matrices Completion with Latent Gaussian Process Priors](https://mlsanity.com/paper/2103.12045/) (2021)
- [A closer look at the approximation capabilities of neural networks](https://mlsanity.com/paper/2002.06505/) (2020)

## Built on by (most cited first, 9 of 9)

- [What is the Visual Cognition Gap between Humans and Multimodal LLMs?](https://mlsanity.com/paper/2406.10424/) (2024)
- [Toward Cognitive Supersensing in Multimodal Large Language Model](https://mlsanity.com/paper/2602.01541/) (2026)
- [Abstracting Concept-Changing Rules for Solving Raven's Progressive Matrix Problems](https://mlsanity.com/paper/2307.07734/) (2023)
- [A Cognitively-Inspired Neural Architecture for Visual Abstract Reasoning Using Contrastive Perceptual and Conceptual Processing](https://mlsanity.com/paper/2309.10532/) (2023)
- [Sandra -- A Neuro-Symbolic Reasoner Based On Descriptions And Situations](https://mlsanity.com/paper/2402.00591/) (2024)
- [ViTCN: Vision Transformer Contrastive Network For Reasoning](https://mlsanity.com/paper/2403.09962/) (2024)
- [Systematic Abductive Reasoning via Diverse Relation Representations in Vector-symbolic Architecture](https://mlsanity.com/paper/2501.11896/) (2025)
- [GenVP: Generating Visual Puzzles with Contrastive Hierarchical VAEs](https://mlsanity.com/paper/2503.23598/) (2025)
- [ChronoVision: Temporal Reasoning via Latent State Reconstruction](https://mlsanity.com/paper/2608.05631/) (2026)

## Abstract (arXiv)

We introduce algebraic machine reasoning, a new reasoning framework that is well-suited for abstract reasoning. Effectively, algebraic machine reasoning reduces the difficult process of novel problem-solving to routine algebraic computation. The fundamental algebraic objects of interest are the ideals of some suitably initialized polynomial ring. We shall explain how solving Raven's Progressive Matrices (RPMs) can be realized as computational problems in algebra, which combine various well-known algebraic subroutines that include: Computing the Gr"obner basis of an ideal, checking for ideal containment, etc. Crucially, the additional algebraic structure satisfied by ideals allows for more operations on ideals beyond set-theoretic operations. Our algebraic machine reasoning framework is not only able to select the correct answer from a given answer set, but also able to generate the correct answer with only the question matrix given. Experiments on the I-RAVEN dataset yield an overall 93.2% accuracy, which significantly outperforms the current state-of-the-art accuracy of 77.0% and exceeds human performance at 84.4% accuracy.