Visual Perceptual to Conceptual First-Order Rule Learning Networks

Kun Gao; Davide Soldà; Thomas Eiter; Katsumi Inoue

cs.AIApr 9, 2026

Visual Perceptual to Conceptual First-Order Rule Learning Networks

Authors: Kun Gao, Davide Soldà, Thomas Eiter, Katsumi Inoue

Organizations: Zhongguancun Academy, Beijing, China · Vienna University of Technology (TU Wien), Vienna, Austria · National Institute of Informatics, Tokyo, Japan

Abstract

Learning rules plays a crucial role in deep learning, particularly in explainable artificial intelligence and enhancing the reasoning capabilities of large language models. While existing rule learning methods are primarily designed for symbolic data, learning rules from image data without supporting image labels and automatically inventing predicates remains a challenge. In this paper, we tackle these inductive rule learning problems from images with a framework called γILP, which provides a fully differentiable pipeline from image constant substitution to rule structure induction. Extensive experiments demonstrate that γILP achieves strong performance not only on classical symbolic relational datasets but also on relational image data and pure image datasets, such as Kandinsky patterns.

Figures & tables

Figure 1: The pipeline of the learning framework. $E$ and $E^{\prime}$ indicate the encoder function for image and text, respectively.

Model	Predecessor	Odd	Even	Lessthan	Fizz	Buzz
Gemini 2.5 Pro	1.00	1.00	0.00	0.20	0.00	0.00
GPT-o3	1.00	0.00	1.00	1.00	0.00	1.00
GPT-5	1.00	1.00	1.00	1.00	1.00	1.00
$\gamma$ ILP-ViT	1.00	1.00	1.00	1.00	-	-
$\gamma$ ILP-VAE	1.00	1.00	1.00	1.00	-	-

Table 1: Recall of the learned logic program on relational image datasets. Best results are shown in bold. A ‘–’ indicates that the rules were learned correctly but incompletely.

Panel of Figure 2 — Figure 2: Kandinsky patterns for tasks (a) two-pair (TP), (b) one-red (OR), and (c) one-triangle (OT). Constants represented by variables in the learned rules for TP are shown in (d).

Task	CNN	ViT	YL	GM	G-o4	G-4.1	G-o3	G-5	$\gamma$ ILP-VAE	$\gamma$ ILP-ViT
Types	VM	VM	VM	LLM	LLM	LLM	LLM	LLM	RM	RM
TP	0.50	0.63	0.40	0.46	0.56	0.47	0.50	0.67	0.64	0.75
OR	0.50	0.80	0.80	1.00	0.31	0.52	0.63	0.38	0.77	1.00
OT	0.50	0.90	0.80	1.00	0.53	0.40	0.37	0.78	0.77	1.00

Table 2: Classification accuracy on Kandinsky tasks: Two-pair (TP), one-red (OR), and one-triangle (OT). VM, RM, GM, YL, G, and G-o4 refer to vision models, reasoning models, Gemini 2.5 Pro, YOLO, GPT, and GPT-o4-mini.

Panel of Figure 3 — Figure 3: Constants represented by clusters for the one-red and one-triangle patterns.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Domain	Task	# Constant	# Relation
Arithmetic	Predecessor	10	3
	Odd	10	3
	Even	10	3
	Lessthan	10	3
	Fizz	7	3
	Buzz	10	3

Appendix

Table 3: The statistical information of ILP datasets.

Domain	Task	$\partial$ ILP	DFORL	GPT-5	Gemini	$\gamma$ ILP	RT
Arithmetic	Pre	1.00	1.00	1.00	1.00	1.00	6.57
	Odd	1.00	1.00	1.00	1.00	1.00	6.98
	Even	1.00	1.00	1.00	1.00	1.00	6.47
	Lessthan	1.00	1.00	1.00	1.00	1.00	12.09
	Fizz	1.00	-	1.00	1.00	-	35.55
	Buzz	1.00	-	1.00	1.00	-	133.03

Appendix

Table 4: The recall of the learned rules on the ILP datasets. For GPT and Gemini, we use the latest reasoning model GPT-5 and Gemini 2.5 Pro, respectively. “Pre”, “GP”, ”Rel”, “UE”, “AdjR”, “TC”, “GC”, “Con”, and “RT” denote Predecessor, Grandparent, Relatedness, Undirected Edge, Adjacent to Red, Two Children, Graph Coloring, Connectedness, and the average running time in second for

\gamma

ILP when it reaches its maximum accuracy, respectively. The best results are in bold if some models fail to achieve the top recall. The notation “–” indicates that the model achieves a recall between 0 and 1.

Figure 4: The MNIST sequence with label $0,5,1,5,2,5,3,5,4,5,5,5\dots$

Figure 5: The image constants represented by the clusters $R$ , $X$ , and $Y$ .

Figure 6: The constants represented by the clusters $T$ , $Y$ , and $Z$ .

Tasks	GPT-5	GPT-3o	Gemini 2.5 Pro
Two-pair	Cluster R and Cluster V represent the same shape category, but differ in color.	Everything about the paired images is identical—shape, size, cropping, background—except for the hue of the triangle.	The relationship is one of contrast in the color dimension, while maintaining consistency in the shape and overall format.
One-red	They are all solid red geometric shapes.	Every item in that set is filled with the same solid red colour; the hue is the property they all share.	color (red) is the property they all possess in an identical state, making it the primary commonality.
One-triangle	All the objects in the image share one clear common property: They are all triangles.	Every one of the thumbnails shows the same basic geometric shape: an upright, isosceles (appearance-wise almost equilateral) triangle. Color, slight aliasing and exact position vary, but the common property is simply that they are all triangles pointing upward inside the square frame.	Based on the provided image, the most immediate and common property of all the distinct triangular objects shown is that they are all isosceles triangles.

Appendix

Table 5: The semantics translation results from LLMs

Task	RIPPER-ViT	C4.5-ViT	$\gamma$ ILP-VAE	$\gamma$ ILP-ViT
Types	RM	RM	RM	RM
Two-pair	0.70	0.65	0.64	0.75
One-red	1.00	1.00	0.77	1.00
One-triangle	1.00	1.00	0.77	1.00

Appendix

Table 6: Classification accuracy on Kandinsky tasks: Two-pair, one-red, and one-triangle.

Figure 7: The constants represented by the cluster centroids #3, #6, and #7.

Figure 8: The constants represented by the cluster centroids #6 and #7.

Panel of Figure 9 — Figure 9: The constants represented by the cluster centroids #0, #1, #6, #7, and #8.

Figure 10: The constants represented by the cluster centroids #1, #2, #6, #7, and #8.

Panel of Figure 11 — Figure 11: The accuracies under different hyperparameters. DCM and LR indicate the differentiable clustering method and learning rate, respectively.

Figure 12: The stability of

\gamma

ILP in Kandinsky patterns.

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May 5, 2026J/K move · Enter open · S save

ANDRE: An Attention-based Neuro-symbolic Differentiable Rule Extractor for Inductive Logic Programming

Iman Sharifi, Peng Wei, Saber Fallah

Dept. of Mechanical and Aerospace Engineering, George Washington University, USA · Dept. of Mechanical Engineering Sciences, University of Surrey, UK

Logic Programming Differentiable Programming Neuro-Symbolic Reasoning Inductive Logic Programming Differentiable Logic

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Inductive Logic Programming (ILP) aims to learn interpretable first-order rules from data, but existing symbolic and neuro-symbolic approaches struggle to scale to noisy and probabilistic settings. Classical ILP relies on discrete combinatorial rule search and is brittle under uncertainty, while differentiable ILP methods typically depend on predefined rule templates or inaccurate fuzzy operators that suffer from vanishing gradients or poor approximation of logical structure when reasoning over probabilistic predicate valuations. This paper proposes an Attention-based Neuro-symbolic Differentiable Rule Extractor (ANDRE), a novel ILP framework that learns first-order logic programs by optimizing over a continuous rule space with attention-based logical operators. ANDRE replaces both rule templates and logical operators with fully differentiable, attention-driven conjunction and disjunction operators that approximate logical min-max semantics, enabling accurate, stable, and interpretable reasoning over probabilistic data. By softly selecting, negating, or excluding predicates within each rule, ANDRE supports flexible rule induction while preserving symbolic structure. Extensive experiments on classical ILP benchmarks, large-scale knowledge bases, and synthetic datasets with probabilistic predicates and noisy supervision demonstrate that ANDRE achieves competitive or superior predictive performance while reliably recovering correct symbolic rules under uncertainty. In particular, ANDRE remains robust to moderate label noise, substantially outperforming existing differentiable ILP methods in both rule extraction quality and stability.

Abstract

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