Neuro-Symbolic (NeSy) AI could be regarded as an analogy to human dual-process cognition, modeling the intuitive System 1 with neural networks and the algorithmic System 2 with symbolic reasoning. However, for complex learning targets, NeSy systems often generate outputs inconsistent with domain knowledge and it is challenging to rectify them. Inspired by the human Cognitive Reflection, which promptly detects errors in our intuitive response and revises them by invoking the System 2 reasoning, we propose to improve NeSy systems by introducing Abductive Reflection (ABL-Refl) based on the Abductive Learning (ABL) framework. ABL-Refl leverages domain knowledge to abduce a reflection vector during training, which can then flag potential errors in the neural network outputs and invoke abduction to rectify them and generate consistent outputs during inference. ABL-Refl is highly efficient in contrast to previous ABL implementations. Experiments show that ABL-Refl outperforms state-of-the-art NeSy methods, achieving excellent accuracy with fewer training resources and enhanced efficiency.

Neuro-Symbolic (NeSy) AI could be regarded as an analogy to human dual-process cognition, modeling the intuitive System 1 with neural networks and the algorithmic System 2 with symbolic reasoning. However, for complex learning targets, NeSy systems often generate outputs inconsistent with domain knowledge. Inspired by the human Cognitive Reflection, which promptly detects errors in our intuitive response and revises them by invoking the System 2 reasoning, we propose to improve NeSy systems by introducing Abductive Reflection (ABL-Refl) based on the Abductive Learning (ABL) framework. ABL-Refl leverages domain knowledge to abduce a reflection vector during training, which can then flag potential errors in the neural network outputs and invoke abduction to rectify them and generate consistent outputs during inference. Experiments show that ABL-Refl outperforms state-of-the-art NeSy methods, achieving excellent accuracy with fewer training resources and enhanced efficiency.

Abstraction is a critical technique in general problem-solving, allowing complex tasks to be decomposed into smaller, manageable sub-tasks. While traditional symbolic planning relies on predefined primitive symbols to construct structured abstractions, its reliance on formal representations limits applicability to real-world tasks. On the other hand, reinforcement learning excels at learning end-to-end policies directly from sensory inputs in unstructured environments but struggles with compositional generalization in complex tasks with delayed rewards. In this paper, we propose Abductive Abstract Reinforcement Learning (A2RL), a novel neuro-symbolic RL framework bridging the two paradigms based on Abductive Learning (ABL), enabling RL agents to learn abstractions directly from raw sensory inputs without predefined symbols. A2RL induces a finite state machine to represent high-level, step-by-step procedures, where each abstract state corresponds to a sub-algebra of the original Markov Decision Process (MDP). This approach not only bridges the gap between symbolic abstraction and sub-symbolic learning but also provides a natural mechanism for the emergence of new symbols. Experiments show that A2RL can mitigate the delayed reward problem and improve the generalization capability compared to traditional end-to-end RL methods.

Large Language Models (LLMs) have shown promising results across various tasks, yet their reasoning capabilities remain a fundamental challenge. Developing AI systems with strong reasoning capabilities is regarded as a crucial milestone in the pursuit of Artificial General Intelligence (AGI) and has garnered considerable attention from both academia and industry. Various techniques have been explored to enhance the reasoning capabilities of LLMs, with neuro-symbolic approaches being a particularly promising way. This paper comprehensively reviews recent developments in neuro-symbolic approaches for enhancing LLM reasoning. We first present a formalization of reasoning tasks and give a brief introduction to the neuro-symbolic learning paradigm. Then, we discuss neuro-symbolic methods for improving the reasoning capabilities of LLMs from three perspectives: Symbolic->LLM, LLM->Symbolic, and LLM+Symbolic. Finally, we discuss several key challenges and promising future directions. We have also released a GitHub repository including papers and resources related to this survey: https://github.com/LAMDASZ-ML/Awesome-LLM-Reasoning-with-NeSy.

Making neural visual generative models controllable by logical reasoning systems is promising for improving faithfulness, transparency, and generalizability. We propose the Abductive visual Generation (AbdGen) approach to build such logic-integrated models. A vector-quantized symbol grounding mechanism and the corresponding disentanglement training method are introduced to enhance the controllability of logical symbols over generation. Furthermore, we propose two logical abduction methods to make our approach require few labeled training data and support the induction of latent logical generative rules from data. We experimentally show that our approach can be utilized to integrate various neural generative models with logical reasoning systems, by both learning from scratch or utilizing pre-trained models directly. The code is released at https://github.com/future-item/AbdGen.

Optical Character Recognition (OCR) of historical document images remains a challenging task because of the distorted input images, extensive number of uncommon characters, and the scarcity of labeled data, which impedes modern deep learning-based OCR techniques from achieving good recognition accuracy. Meanwhile, there exists a substantial amount of expert knowledge that can be utilized in this task. However, such knowledge is usually complicated and could only be accurately expressed with formal languages such as first-order logic (FOL), which is difficult to be directly integrated into deep learning models. This paper proposes KESAR, a novel Knowledge-Enhanced Document Segmentation And Recognition method for historical document images based on the Abductive Learning (ABL) framework. The segmentation and recognition models are enhanced by incorporating background knowledge for character extraction and prediction, followed by an efficient joint optimization of both models. We validate the effectiveness of KESAR on historical document datasets. The experimental results demonstrate that our method can simultaneously utilize knowledge-driven reasoning and data-driven learning, which outperforms the current state-of-the-art methods.

Neuro-symbolic hybrid systems are promising for integrating machine learning and symbolic reasoning, where perception models are facilitated with information inferred from a symbolic knowledge base through logical reasoning. Despite empirical evidence showing the ability of hybrid systems to learn accurate perception models, the theoretical understanding of learnability is still lacking. Hence, it remains unclear why a hybrid system succeeds for a specific task and when it may fail given a different knowledge base. In this paper, we introduce a novel way of characterising supervision signals from a knowledge base, and establish a criterion for determining the knowledge’s efficacy in facilitating successful learning. This, for the first time, allows us to address the two questions above by inspecting the knowledge base under investigation. Our analysis suggests that many knowledge bases satisfy the criterion, thus enabling effective learning, while some fail to satisfy it, indicating potential failures. Comprehensive experiments confirm the utility of our criterion on benchmark tasks.

Integrating complementary strengths of raw data and logical rules to improve the learning generalization has been recently shown promising and effective, e.g., abductive learning is one generic framework that can learn the perception model from data and reason between rules simultaneously. However, the performance would be seriously decreased when inaccurate logical rules appear, which may be even worse than baselines using only raw data. Efforts on this issue are highly desired while remain to be limited. This paper proposes a simple and effective safe abductive learning method to alleviate the harm caused by inaccurate rules. Unlike the existing methods which directly use all rules without correctness checks, it utilizes them selectively by constructing a graphical model with an adaptive reasoning process to prevent performance hazards. Theoretically, we show that induction and abduction are mutually beneficial, and can be rigorously justified from a classical maximum likelihood estimation perspective. Experiments on diverse tasks show that our method can tolerate at least twice as many inaccurate rules as accurate ones and achieve highly competitive performance while other methods can't. Moreover, the proposal can refine inaccurate rules and works well in extended weakly supervised scenarios.

Recently there are great efforts on leveraging machine learning and logical reasoning. Many approaches start from a given knowledge base, and then try to utilize the knowledge to help machine learning. In real practice, however, the given knowledge base can often be incomplete or even noisy, and thus, it is crucial to develop the ability of knowledge refinement or enhancement. This paper proposes to enable the Abductive learning (ABL) paradigm to have the ability of knowledge refinement/enhancement. In particular, we focus on the problem that, in contrast to closed-environment tasks where a fixed set of symbols are enough to represent the concepts in the domain, in open-environment tasks new concepts may emerge. Ignoring those new concepts can lead to significant performance decay, whereas it is challenging to identify new concepts and add them to the existing knowledge base with potential conflicts resolved. We propose the ABL_nc approach which exploits machine learning in ABL to identify new concepts from data, exploits knowledge graph to match them with entities, and refines existing knowledge base to resolve conflicts. The refined/enhanced knowledge base can then be used in the next loop of ABL and help improve the performance of machine learning. Experiments on three neuro-symbolic learning tasks verified the effectiveness of the proposed approach.

Most systems integrating data-driven machine learning with knowledge-driven reasoning usually rely on a specifically designed knowledge base to enable efficient symbolic inference. However, it could be cumbersome for the nonexpert end-users to prepare such a knowledge base in real tasks. Recent years have witnessed the success of large-scale knowledge graphs, which could be ideal domain knowledge resources for real-world machine learning tasks. However, these large-scale knowledge graphs usually contain much information that is irrelevant to a specific learning task. Moreover, they often contain a certain degree of noise. Existing methods can hardly make use of them because the large-scale probabilistic logical inference is usually intractable. To address these problems, we present ABductive Learning with Knowledge Graph (ABL-KG) that can automatically mine logic rules from knowledge graphs during learning, using a knowledge forgetting mechanism for filtering out irrelevant information. Meanwhile, these rules can form a logic program that enables efficient joint optimization of the machine learning model and logic inference within the Abductive Learning (ABL) framework. Experiments on four different tasks show that ABL-KG can automatically extract useful rules from large-scale and noisy knowledge graphs, and significantly improve the performance of machine learning with only a handful of labeled data.

Abductive Learning is a framework that combines machine learning with first-order logical reasoning. It allows machine learning models to exploit complex symbolic domain knowledge represented by first-order logic rules. However, it is challenging to obtain or express the ground-truth domain knowledge explicitly as first-order logic rules in many applications. The only accessible knowledge base is implicitly represented by groundings, i.e., propositions or atomic formulas without variables. This paper proposes Grounded Abductive Learning (GABL) to enhance machine learning models with abductive reasoning in a ground domain knowledge base, which offers inexact supervision through a set of logic propositions. We apply GABL on two weakly supervised learning problems and found that the model's initial accuracy plays a crucial role in learning. The results on a real-world OCR task show that GABL can significantly reduce the effort of data labeling than the compared methods.

For many reasoning-heavy tasks with raw inputs, it is challenging to design an appropriate end-to-end pipeline to formulate the problem-solving process. Some modern AI systems, e.g., Neuro-Symbolic Learning, divide the pipeline into sub-symbolic perception and symbolic reasoning, trying to utilise data-driven machine learning and knowledge-driven problem-solving simultaneously. However, these systems suffer from the exponential computational complexity caused by the interface between the two components, where the sub-symbolic learning model lacks direct supervision, and the symbolic model lacks accurate input facts. Hence, they usually focus on learning the sub-symbolic model with a complete symbolic knowledge base while avoiding a crucial problem: where does the knowledge come from? In this paper, we present Abductive Meta-Interpretive Learning (MetaAbd) that unites abduction and induction to learn neural networks and logic theories jointly from raw data. Experimental results demonstrate that MetaAbd not only outperforms the compared systems in predictive accuracy and data efficiency but also induces logic programs that can be re-used as background knowledge in subsequent learning tasks. To the best of our knowledge, MetaAbd is the first system that can jointly learn neural networks from scratch and induce recursive first-order logic theories with predicate invention.

To utilize the raw inputs and symbolic knowledge simultaneously, some recent neuro-symbolic learning methods use abduction, i.e., abductive reasoning, to integrate sub-symbolic perception and logical inference. While the perception model, e.g., a neural network, outputs some facts that are inconsistent with the symbolic background knowledge base, abduction can help revise the incorrect perceived facts by minimizing the inconsistency between them and the background knowledge. However, to enable effective abduction, previous approaches need an initialized perception model that discriminates the input raw instances. This limits the application of these methods, as the discrimination ability is usually acquired from a thorough pre-training when the raw inputs are difficult to classify. In this paper, we propose a novel abduction strategy, which leverages the similarity between samples, rather than the output information by the perceptual neural network, to guide the search in abduction. Based on this principle, we further present ABductive Learning with Similarity (ABLSim) and apply it to some difficult neuro-symbolic learning tasks. Experiments show that the efficiency of ABLSim is significantly higher than the state-of-the-art neuro-symbolic methods, allowing it to achieve better performance with less labeled data and weaker domain knowledge.

In many practical tasks, there are usually two kinds of common information: cheap unlabeled data and domain knowledge in the form of symbols. There are some attempts using one single information source, such as semi-supervised learning and abductive learning. However, there is little work to use these two kinds of information sources at the same time, because it is very difficult to combine symbolic logical representation and numerical model optimization effectively. The learning becomes even more challenging when the domain knowledge is insufficient. In this paper, we present an attempt-Semi-Supervised ABductive Learning (SS-ABL) framework. In this framework, semi-supervised learning is trained via pseudo labels of unlabeled data generated by abductive learning, and the background knowledge is refined via the label distribution predicted by semi-supervised learning. The above framework can be optimized iteratively and can be naturally interpretable. The effectiveness of our framework has been fully verified in the theft judicial sentencing of real legal documents. In the case of missing sentencing elements and mixed legal rules, our framework is apparently superior to many existing baseline practices, and provides explanatory assistance to judicial sentencing.

Perception and reasoning are two representative abilities of intelligence that are integrated seamlessly during human problem-solving processes. In the area of artificial intelligence (AI), the two abilities are usually realised by machine learning and logic programming, respectively. However, the two categories of techniques were developed separately throughout most of the history of AI. In this paper, we present the abductive learning targeted at unifying the two AI paradigms in a mutually beneficial way, where the machine learning model learns to perceive primitive logic facts from data, while logical reasoning can exploit symbolic domain knowledge and correct the wrongly perceived facts for improving the machine learning models. Furthermore, we propose a novel approach to optimise the machine learning model and the logical reasoning model jointly. We demonstrate that by using abductive learning, machines can learn to recognise numbers and resolve unknown mathematical operations simultaneously from images of simple hand-written equations. Moreover, the learned models can be generalised to longer equations and adapted to different tasks, which is beyond the capability of state-of-the-art deep learning models.

Statistical machine learning is widely used in image classification. However, most techniques (1) require many images to achieve high accuracy and (2) do not provide support for reasoning below the level of classification, and so are unable to support secondary reasoning, such as the existence and position of light sources and other objects outside the image. This paper describes an Inductive Logic Programming approach called Logical Vision which overcomes some of these limitations. LV uses Meta-Interpretive Learning (MIL) combined with low-level extraction of high-contrast points sampled from the image to learn recursive logic programs describing the image. In published work LV was demonstrated capable of high-accuracy prediction of classes such as regular polygon from small numbers of images where Support Vector Machines and Convolutional Neural Networks gave near random predictions in some cases. LV has so far only been applied to noise-free, artificially generated images. This paper extends LV by (a) addressing classification noise using a new noise-telerant version of the MIL system Metagol, (b) addressing attribute noise using primitive-level statistical estimators to identify sub-objects in real images, (c) using a wider class of background models representing classical 2D shapes such as circles and ellipses, (d) providing richer learnable background knowledge in the form of a simple but generic recursive theory of light reflection. In our experiments we consider noisy images in both natural science settings and in a RoboCup competition setting. The natural science settings involve identification of the position of the light source in telescopic and microscopic images, while the RoboCup setting involves identification of the position of the ball. Our results indicate that with real images the new noise-robust version of LV using a single example (i.e. one-shot LV) converges to an accuracy at least comparable to a thirty-shot statistical machine learner on both prediction of hidden light sources in the scientific settings and in the RoboCup setting. Moreover, we demonstrate that a general background recursive theory of light can itself be invented using LV and used to identify ambiguities in the convexity/concavity of objects such as craters in the scientific setting and partial obscuration of the ball in the RoboCup setting.

In many real tasks there are human knowledge expressed in logic formulae as well as data samples described by raw features (e.g., pixels, strings). It is popular to apply SRL or PILPtechniques to exploit human knowledge through learning of symbolic data, or statistical learning techniques to learn from the raw data samples; however, it is often desired to directly exploit these logic formulae on raw data processing, like human beings utilizing knowledge to guide perception. In this paper, we propose an approach, LASIN, which combines Logical Abduction and Statistical Induction. The LASIN approach generates candidate hypotheses based on the abduction of first-order formulae, and then, the hypotheses are exploited as constraints for statistical induction. We apply theLASIN approach to the learning of representation of written primitives, where a primitive is a basic component in human writing. Our results show that the discovered primitives are reasonable for human perception, and these primitives, if used in learning tasks such as classification and domain adaptation, lead to better performances than simply applying feature learning based on raw data only.