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A Survey on Neural-symbolic Learning Systems

Dongran Yu, Bo Yang, Dayou Liu, Hui Wang, Shirui Pan

arXiv:2111.08164v3cs.LG

TL;DR

Neural and symbolic systems have complementary strengths, but neural networks can be limited in reasoning and explanation. This survey proposes a three-part taxonomy and reviews techniques, applications, and future directions. It concludes that neural-symbolic learning systems merit further research and discussion.

  • Problem

    Neural networks perform well on perception but can struggle with unseen situations and explicit explanations, while integrating neural and symbolic approaches remains a design challenge.

    Source: §1 · p. 2

  • Method

    The survey proposes three categories of neural-symbolic systems and reviews their techniques, symbol representations, applications, and future directions.

    Source: §1 · p. 5

  • Results

    The survey presents a taxonomy of three structured categories, describes their techniques, surveys applications, and discusses future directions.

    Source: §6 · p. 23

  • Takeaways & Limitations

    The authors conclude that systematic, comprehensive research surveys in this field have value for theory and application and warrant further discussion.

    Source: §6 · p. 23

    Some learning-reasoning approaches rely on predefined logic programming or rules, restricting generalizability to other tasks.

    Source: §3.3 · p. 20

Abstract

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In recent years, neural systems have demonstrated highly effective learning ability and superior perception intelligence. However, they have been found to lack effective reasoning and cognitive ability. On the other hand, symbolic systems exhibit exceptional cognitive intelligence but suffer from poor learning capabilities when compared to neural systems. Recognizing the advantages and disadvantages of both methodologies, an ideal solution emerges: combining neural systems and symbolic systems to create neural-symbolic learning systems that possess powerful perception and cognition. The purpose of this paper is to survey the advancements in neural-symbolic learning systems from four distinct perspectives: challenges, methods, applications, and future directions. By doing so, this research aims to propel this emerging field forward, offering researchers a comprehensive and holistic overview. This overview will not only highlight the current state-of-the-art but also identify promising avenues for future research.

1. Introduction

The introduction motivates combining neural perception with symbolic reasoning to address their respective limitations. It presents a survey that classifies neural-symbolic systems, reviews techniques and applications, and discusses future directions.

1. Introduction

  • The introduction places AI’s history across reasoning, knowledge-focused, and machine-learning periods, noting milestones including ImageNet with CNNs and AlphaGo.It presents these milestones as examples of the power of machine-learning approaches, predominantly driven by neural networks.Source: §1 · p. 2
  • Neural networks excel at perception but can struggle with unseen situations and explicit explanations, motivating integration with symbolic reasoning.Symbolic systems can use deductive reasoning to generate information and explain reasoning processes.Source: §1 · pp. 2–3
  • Neural-symbolic systems are described as faster at reasoning than pure symbolic systems, stronger in generalization than standalone neural systems, and gray-box systems that expose computation.The introduction links these advantages to neural-network integration, symbolic knowledge as training data, and explicit reasoning traces or evidence chains.Source: §1 · pp. 3–4
  • The introduction frames integration as a challenge because symbolic systems use discrete representations and search, while neural systems use continuous features and learned mappings.The desired combination should balance the two approaches to fit the problem requirements.Source: §1 · p. 4
  • The survey addresses a gap in prior reviews by covering popular methods and specific techniques, alongside classifications and applications.The introduction says some existing reviews focus on particular perspectives or remain primarily theoretical.Source: §1 · p. 4
  • The paper proposes a taxonomy with three categories: learning for reasoning, reasoning for learning, and learning-reasoning.It also promises descriptions of representative methods and symbol types such as logic knowledge and knowledge graphs.Source: §1 · p. 5
  • The survey covers neural-symbolic applications and proposes four potential future research directions.It positions itself as complementing prior literature with an overview of technologies, developments, and application areas.Source: §1 · pp. 4–5

2. Categorization and Frameworks

The survey defines neural-symbolic systems as hybrids of neural learning and symbolic reasoning, organized by how these components are integrated. It distinguishes sequential, parallel, and bidirectional interaction patterns.

2. Categorization and Frameworks

  • Neural-symbolic systems map data and symbols to ground-truth solutions, combining neural learning with symbolic reasoning.The survey classifies integration by how deep neural networks and symbolic knowledge or reasoning are combined; Table 2 summarizes approaches across six dimensions.Source: §2 · p. 5, Table 2 · p. 6, §2.3 · p. 8

2.1. Learning for reasoning

  • Neural networks support symbolic reasoning by narrowing search or extracting symbols from data for downstream reasoning [113] [40] [107].In this sequential arrangement, neural components provide features or symbols to a symbolic reasoning module.Source: §2.1 · pp. 5–7

2.2. Reasoning for learning

  • Symbolic knowledge guides or constrains neural training, with the systems operating in parallel.Knowledge may be encoded as a loss regularizer; the survey describes this integration as supporting performance, generalization, and interpretability.Source: §2.2 · pp. 7–8

2.3. Learning-reasoning

  • Neural and symbolic outputs feed each other iteratively, balancing their roles; unlike the first two loosely coupled taxonomies, this approach is tightly coupled.Neural predictions can be reasoned over symbolically, and the reasoning results can return to the neural component for refinement.Source: §2.3 · p. 8

3. Methods of neural-symbolic learning systems

This section surveys methods for connecting neural learning and symbolic reasoning, including approaches that accelerate reasoning, convert unstructured inputs into symbolic representations, impose logic-based constraints, transfer semantic or symbolic knowledge into neural models, and integrate learning with reasoning. It also highlights representative frameworks and the design considerations and limitations shaping these methods.

3.1. Learning for reasoning

  • Learning for reasoning uses neural networks either to accelerate symbolic search or to abstract unstructured data into symbolic representations for reasoning.The survey identifies neural networks as both accelerators of symbolic reasoning and transformers of inputs such as images and text into useful symbolic representations.Source: §3.1 · p. 9, §3.1.2 · pp. 11–12

3.1.1. Accelerating symbolic reasoning

  • pLogicNet and ExpressGNN combine neural networks with variational EM to approximate knowledge-graph inference; ExpressGNN uses a GNN and tunable entity embeddings.Both target triplet completion by treating it as inference over hidden variables; ExpressGNN’s changes address identical embeddings for isomorphic nodes.Source: §3.1.1 · p. 10
  • NLIL extends differentiable ILP to learn expressive tree and conjunctive logic rules from data and explain observed patterns.It uses predicate operations and logical connectives to expand rule expressiveness beyond chains.Source: §3.1.1 · p. 11

3.1.2. Abstracting unstructured data into symbols

  • NS-CL [81] jointly learns visual concepts, words, and sentence parsing without explicit supervision, then uses object-based scene representations and symbolic programs for visual question answering.Its perception module represents image objects, its parser creates executable programs, and a quasi-symbolic executor derives answers.Source: §3.1.2 · p. 11

3.2. Reasoning for learning

  • Reasoning-for-learning methods comprise regularization models that constrain training with symbolic knowledge and knowledge-transfer models that move semantic knowledge across domains.The survey presents these approaches as ways to incorporate prior knowledge or use it to guide learning in another domain.Source: §3.2 · p. 12

3.2.1. Regularization models

  • Regularization methods encode logic as training constraints, including teacher-guided distillation, semantic-based regularization, and semantic loss using an arithmetic circuit.HDNN uses a rule-encoding teacher to guide a student; SBR penalizes constraint violations, while SL encodes rules with a Sentential Decision Diagram.Source: §3.2.1 · pp. 12–13
  • LENSR aligns embeddings of prior and predicted logic, while CA-ZSL uses knowledge-graph-informed CRF potentials to predict labels for unseen objects.LENSR models local dependencies, which may limit expressiveness; modeling a global dependency graph is proposed to improve expressiveness and capture uncertainty.Source: §3.2.1 · pp. 13–14

3.2.2. Knowledge transfer models

  • Zero-shot methods SEKB-ZSL and DGP transfer knowledge-graph semantic classifier weights to visual classifiers, while DGP reduces GCN over-smoothing and uses attention to weight node connections.KGTN extends class-correlation transfer to few-shot classification using a gated graph neural network to learn embeddings linking seen and unseen classes.Source: §3.2.2 · pp. 15–16
  • Symbolic knowledge is combined with neural networks by using intermediate representations such as graphs or trees, soft truth degrees, loss regularization, or knowledge encoded in network structure.Logic rules commonly constrain loss functions, while knowledge graphs provide relational information.Source: §3.2.2 · p. 17

3.3. Learning-reasoning

  • Learning-reasoning approaches interact rather than operate in isolation, but reliance on predefined logic limits generalizability and motivates data-driven learning of symbolic knowledge.Designs should account for uncertainty, global knowledge fit, differing knowledge importance, and interpretability.Source: §3.3 · pp. 17, 20
  • DeepProbLog uses neural predicates to turn neural-network outputs into facts for probabilistic logic, then uses arithmetic circuits and the gradient semiring for inference and end-to-end training.SDDs are used to construct the directed graph that is transformed into an arithmetic circuit.Source: §3.3 · pp. 17–18
  • ABL iteratively revises classifier pseudo-labels through Prolog-based abductive reasoning, while WS-NeSyL uses sampled rules and perceptive and cognitive decoders to supervise reasoning.ABL is weakly supervised; WS-NeSyL uses a back-search algorithm to sample rules and trains iteratively until convergence.Source: §3.3 · pp. 18–19
  • BPGR models how much symbolic knowledge is triggered with MLN and couples visual reasoning to symbolic error correction in a two-level probabilistic graph trained by variational EM.Its high-level structure represents visual reasoning results, while the low-level structure contains ground atoms of logic rules.Source: §3.3 · p. 19, Figure 15 · p. 19

4. Applications

The survey reviews neural-symbolic applications in visual detection, knowledge-graph reasoning, classification, question answering, and reinforcement learning.

4.1. Object/visual-relationship detection

  • In visual detection, Logic Tensor Networks encode constraints for reasoning over noisy images, while expert knowledge and Markov Logic Networks support remote-sensing and object-relation tasks.The surveyed approaches also use logic rules to describe data characteristics and improve interpretability in image recognition.Source: §4.1 · p. 20

4.2. Knowledge graph reasoning

  • Knowledge-graph methods score First-Order Logic statements for link prediction, explore multi-hop paths, or use k-hop subgraphs with GNN reasoning [133] [89].DeepPath uses reinforcement learning to evaluate sampled paths and reduce search space; GraIL reasons over subgraphs around head and tail entities.Source: §4.2 · p. 20

4.3. Classification/ few-shot classification

  • Classification approaches combine learners and reasoners, use commonsense or logical knowledge for few-shot and zero-shot learning, and apply logic rules to dialogue recognition [83] [114].The surveyed dialogue method recognizes unseen-class labels without additional training data.Source: §4.3 · p. 21

4.4. Intelligent question answering

  • Question-answering methods use neural modules for symbolic programs, differentiable reasoning over text, memory-based iterative reasoning, or knowledge graphs and scene graphs.The surveyed approaches cover text and visual question answering, including arithmetic, sorting, and counting operations.Source: §4.4 · p. 21

4.5. Reinforcement learning

  • Reinforcement-learning approaches use symbolic priors to map sensor data into symbols and learn policies, or combine symbolic planning with hierarchical reinforcement learning [39].PEORL feeds learned experience back to symbolic knowledge, while SDRL divides work among planner, controller, and meta-controller.Source: §4.5 · pp. 21–22

5. Future directions

The paper identifies future directions in efficient inference, automatic rule construction, symbolic representation learning, and expansion into new application fields.

5.1. Efficient methods

  • Neural networks could help address symbolic inference costs, including exponential growth in Markov Logic Network groundings; approximate inference can improve speed at the cost of accuracy [33].The survey frames computationally difficult symbolic tasks as a direction for neural methods.Source: §5.1 · p. 22

5.2. Automatic construction of symbolic knowledge

  • Because logic rules often require time-consuming, unscalable expert effort, automatically learning rules that express prior knowledge from data remains an important research direction [34].The passage says automatic rule learning remains largely unexplored despite existing ILP-based knowledge-extraction methods.Source: §5.2 · p. 22

5.4. Application field expansion

  • The survey describes extending neural-symbolic learning to new domains as a promising direction, noting exploration in the COVID-19 pandemic and advanced robotics, including medical-literature information extraction.Source: §5.4 · p. 23

6. Conclusion

The conclusion presents a framework and taxonomy for neural-symbolic learning systems, surveys their methods and applications, and discusses future directions.

6. Conclusion

  • The paper proposes a taxonomy with three structured categorizations, describes their techniques and applications, and discusses future directions for neural-symbolic learning systems.The authors argue that systematic, comprehensive surveys of the field merit further research and discussion.Source: §6 · p. 23

A. Preliminaries

The preliminaries focus on logic knowledge and knowledge graphs, with logic knowledge divided into propositional and first-order logic.

A. Preliminaries

  • The section treats symbolic knowledge as logic knowledge and knowledge graphs, and divides logic knowledge into propositional and first-order logic.Source: A. Preliminaries · p. 27

A.1.1. Propositional logic

Propositional logic represents declarative statements as true or false and connects propositions with logical operators. It is commonly compiled into directed acyclic graph forms including CNF, d-DNNF, and SDD.

A.1.1. Propositional logic

  • Propositional logic statements are declarative sentences that are either true or false, depending on whether they are consistent with the fact.Source: §A.1.1 · p. 27
  • The connectors between propositions are ∧, ∨, ¬, and ⇒.Source: §A.1.1 · p. 27
  • In the formula described, P represents the antecedent or condition, while Q represents the consequent or conclusion.Source: §A.1.1 · p. 27
  • Propositional logic is usually compiled as a directed acyclic graph; representative forms include CNF, d-DNNF, and SDD, with SDD a subset of d-DNNFs.For example, the paper presents graph forms for Smoke ⇒ Cough.Source: §A.1.1 · p. 27

A.1.2. First-order logic

The section introduces first-order logic for describing complex problems and outlines its elements and rule form. It also presents Markov logic networks as a knowledge representation for FOL.

A.1.2. First-order logic

  • Propositional logic is hard to use for describing complex problems, for which predicate logic is required; the paper introduces first-order logic (FOL).Source: §A.1.2 · p. 27
  • FOL uses constants, variables, functions, and predicates, which respectively represent domain objects, object-ranging values, mappings between objects, and relations or attributes.Source: §A.1.2 · p. 28
  • In the described FOL rule form, the body comprises multiple atoms B1(x) through Bn(x), and the head H(x) is the result derived from that body.Source: §A.1.2 · p. 28
  • A Markov logic network (MLN) represents FOL knowledge as an undirected graph whose nodes are variables, with a joint distribution based on rule weights and counts of true rule values.The section says t-norm fuzzy logic is used to calculate logical connectives.Source: §A.1.2 · p. 28
  • The example uses rules (F1, 1.5) and (F2, 1.1) with constant set C = {A, B} to generate a ground MLN.Source: §A.1.2 · p. 28

A.1.3. Knowledge graph

A knowledge graph is a directed, labeled graph whose nodes denote semantic symbols and whose edges express relationships. The section also describes vector-space representations of these symbols.

A.1.3. Knowledge graph

  • A knowledge graph is a directed, labeled graph: nodes represent semantic symbols or concepts, and edges express relationships between node pairs.Source: §A.1.3 · p. 28
  • The graph is formally represented as G = (H, R, T), with head entities H, tail entities T, and relationships R; in (cat, attribute, paw), cat and paw are entities and attribute is the relationship.Source: §A.1.3 · p. 28
  • Knowledge-graph representations encode discrete entities, attributes, and relationships into a low-dimensional vector space as distributed representations.The section lists methods including R-GCN, TransE, RotatE, and GCN.Source: §A.1.3 · p. 29

A.2. Neural networks

The section describes CNNs, GNNs, and RNNs through their architectures and processing roles. CNNs extract input features, GNNs aggregate neighboring-node information, and RNNs use recurrent connections across time steps.

A.2. Neural networks

  • CNN convolutional layers extract input features, usually from images, and fully connected layers then produce classification or regression information; the final layer is usually softmax.Source: §A.2 · p. 29, Figure 19 · p. 30
  • GNN graph convolutional layers aggregate neighboring nodes’ feature information to form hidden representations for tasks such as node and graph classification.Source: §A.2 · p. 30, Figure 20 · p. 30
  • RNNs feed output activations back to the input through recurrent connections, transitioning the hidden layer through states over successive time steps; their inputs are usually sequences.LSTM is identified as a variant of RNN.Source: §A.2 · p. 30, Figure 21 · p. 30
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