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Neural-Symbolic Learning and Reasoning: A Survey and Interpretation

Tarek R. Besold, Artur d'Avila Garcez, Sebastian Bader, Howard Bowman, Pedro Domingos, Pascal Hitzler, Kai-Uwe Kuehnberger, Luis C. Lamb, Daniel Lowd, Priscila Machado Vieira Lima, Leo de Penning, Gadi Pinkas, Hoifung Poon, Gerson Zaverucha

arXiv:1711.03902v1cs.AI

TL;DR

The survey addresses the challenge of combining statistical learning with symbolic reasoning in cognitive and artificial systems. It synthesises theoretical foundations, system realisations, applications, and research challenges, reporting promising results in real-world domains. The paper concludes that neural-symbolic computation links brain- and mind-oriented approaches through integrated computational models.

  • Problem

    Integrating automated learning with cognitive reasoning remains difficult, especially for real-world data containing errors, missing values, and inconsistencies.

  • Method

    The survey reviews neural-symbolic computation through theoretical foundations, integrated systems, application examples, and mechanisms combining logic, networks, learning, and reasoning.

  • Results

    Neural-symbolic systems have shown promise in applications including computational biology, fault diagnosis, simulator training and assessment, and software verification.

  • Takeaways & Limitations

    The surveyed approach supports learning from uncertain observations, probabilistic reasoning, and logic-based knowledge representation for validation in real-world applications.

  • Takeaways & Limitations

    Architectural-level neural models remain scarce, and neural explanations may become less compelling for higher-level cognitive functions.

Abstract

from arXiv · show

The study and understanding of human behaviour is relevant to computer science, artificial intelligence, neural computation, cognitive science, philosophy, psychology, and several other areas. Presupposing cognition as basis of behaviour, among the most prominent tools in the modelling of behaviour are computational-logic systems, connectionist models of cognition, and models of uncertainty. Recent studies in cognitive science, artificial intelligence, and psychology have produced a number of cognitive models of reasoning, learning, and language that are underpinned by computation. In addition, efforts in computer science research have led to the development of cognitive computational systems integrating machine learning and automated reasoning. Such systems have shown promise in a range of applications, including computational biology, fault diagnosis, training and assessment in simulators, and software verification. This joint survey reviews the personal ideas and views of several researchers on neural-symbolic learning and reasoning. The article is organised in three parts: Firstly, we frame the scope and goals of neural-symbolic computation and have a look at the theoretical foundations. We then proceed to describe the realisations of neural-symbolic computation, systems, and applications. Finally we present the challenges facing the area and avenues for further research.

1. Overview

Neural-symbolic computation addresses the challenge of integrating statistical learning with logical reasoning in cognitive and artificial systems. The survey reviews its foundations, implementations, applications, and open challenges.

  • Foundations: Human behaviour is modelled using computational logic, connectionist cognition models, and uncertainty models.These approaches target high-level reasoning, lower-level emergent dynamics, and vague or probabilistic cognitive processing, respectively.
  • Motivation: Cognitive computational systems increasingly integrate machine learning with automated reasoning.Reported applications include fault diagnosis, computational biology, simulator training and assessment, and software verification.
  • Principles: Neural networks provide parallel computation and robust learning, while logic supplies explanations and interaction with other systems.The integrated model combines continuous neural processing with discrete symbolic reasoning.
  • Motivation: Neural-symbolic integration seeks to reconcile statistical concept acquisition with symbolic concept manipulation in rational agents.The integration is framed as a key research challenge for robust computational models.
  • Applications: Combining background knowledge with data learning has been shown more effective than purely symbolic or purely connectionist systems for noisy, unstructured real-world data.Applications include business process modelling, service-oriented computing, multi-agent coordination, and multimodal processing.

2. Prolegomena of Neural-Symbolic Computation

Neural-symbolic computation studies logic and connectionism as integrated computational models, using translation, modularity, hierarchy, and learning mechanisms. Its goals include explaining cognition and producing practical systems that learn, reason, and operate robustly across applications.

  • Goals: The field aims to unify logic and connectionism, model cognition and behaviour, and improve integrated machine learning and reasoning tools.Symbolic-to-connectionist and reverse translation algorithms support neural implementations of logic, logical characterisations of neural systems, and hybrid learning systems.
  • Mechanisms: Network fibring recursively composes interconnected neurons and networks, allowing activations in one network to influence weight changes in another.It can combine agents and knowledge levels, including meta-level reasoning over object-level knowledge.
  • Organisation: Neural-symbolic systems commonly use modular and hierarchical organisation to assign tasks across networks and progressively increase representational abstraction.Modularity supports comprehensibility and maintenance, while hierarchy passes lower-level outputs to higher-level networks.
  • Architectures: Fibring and related architectures can combine network ensembles across abstraction levels, while parallel computation offers expressiveness for complex applications at comparatively low computational cost.These systems have been developed across several generations of neural architectures, including feedforward, recurrent, symmetric, and deep networks.
  • Mechanisms: Neural-symbolic computation combines representation with activation propagation, message passing, gradient-based learning, uncertainty reasoning, parallelism, and fault tolerance.These mechanisms make neural-symbolic networks computational models rather than merely representational tools.
  • Mechanisms: The integration cycle translates background knowledge into networks, learns from examples, executes reasoning, and extracts symbolic knowledge.Extraction provides explanation and supports maintenance, incremental learning, and transfer learning.
  • Applications: Translation between logic and networks, empirical case studies, and application-focused design support neural-symbolic applications.The approach targets systems that learn from experience, reason about learned knowledge, and control errors from uncertainty.

3. NSCA as Application Example for Neural-Symbolic Computing

NSCA integrates neural learning, probabilistic reasoning, and logic-based knowledge representation for cognitive-agent applications. Its driving-simulator and transport applications show learned temporal knowledge can support assessment, interpretation, and driving-style recognition, while further operational validation remains necessary.

  • Principles of Neural-Symbolic Integration Exemplified: NSCA integrates neural learning for uncertain observations, probabilistic reasoning, and logic-based representation for validation.The architecture combines robust adaptation with symbolic qualitative reasoning.
  • NSCA in a Nutshell: NSCA learned automated driver assessments from real-time simulator data and instructor scores, extracting the resulting knowledge as temporal logic rules.The system was validated in a training simulator used in real-world scenarios.
  • Mechanisms of Neural-Symbolic Integration Exemplified: The system encodes temporal rules in a recurrent temporal restricted Boltzmann machine and updates hypotheses from differences between observed and inferred beliefs.Training uses recurrent connections and combines Contrastive Divergence with backpropagation through time.
  • Mechanisms of Neural-Symbolic Integration Exemplified: NSCA represents both binary and continuous observations, including probabilities of events and real-valued measurements, while fuzzy beliefs support vague concepts such as fast and slow.Its temporal logic extends modal logic with past and future operators and equality or inequality formulae for continuous variables.
  • Mechanisms of Neural-Symbolic Integration Exemplified: Learning can refine subjective rule parameters from observed behaviour, exemplified by learning a more objective intersection-approach distance from multiple scenarios and drivers.This illustrates the paper’s stated objective of combining reasoning with learning.
  • Applications of Neural-Symbolic Integration Exemplified: In an Intelligent Transport System, NSCA recognised driving styles from real-time in-car sensor data, outperformed state-of-the-art methods, and described styles with multilevel temporal logic rules.The rules support human interpretation and expert validation.
  • NSCA in a Nutshell: The authors identify further evaluation by driving experts as necessary before validation in operational settings with many scenarios, trainees, and instructor assessments.The stated boundary concerns broader real-world validation rather than the reported simulator results.

4. Neural-Symbolic Integration in and for Cognitive Science: Building Mental Models

The paper connects neural-symbolic computation with cognitive-science accounts of mental models, reasoning, language, and compositional structure. It presents binding and higher-level cognition as central challenges while proposing mappings between cognitive behaviour, symbolic models, and neural implementations.

  • Building Mental Models: Mental models represent real or imagined situations, relationships among their parts, and possible interactions, preserving constraints relevant to reasoning.Their construction and manipulation are also linked to abductive reasoning and creativity.
  • Building Mental Models: The binding problem concerns a general neural mechanism for identifying which active neural assemblies are associated when combining representations into structured wholes.Examples include binding representations of a red Ford and a hooded figure.
  • Building Mental Models: Conjunctive coding faces scalability and novel-conjunction challenges because exhaustive localist combinations do not scale and experiences can involve previously unseen combinations.Distributed representations are proposed as more compact and scalable binding codes.
  • Building Mental Models: Neural conjunctive coding methods are presented as seemingly prerequisite for brain-based theories of mental models.This frames binding as a prerequisite for neural representation combination.
  • Cognitive Neuroscience and Neural-Symbolic Methods: Connectionist models are criticized for not naturally preserving representational compositionality, such as carrying the interpretation of “John” across different sentence positions.The issue is especially associated with classic PDP-style position-specific coding.
  • Cognitive Neuroscience and Neural-Symbolic Methods: Architectural-level neural models are scarce, and neural explanations become less compelling for higher-level functions such as deliberation, planning, and central-executive processing.The passage contrasts these areas with peripheral systems such as vision, audition, and motor action.
  • Cognitive Neuroscience and Neural-Symbolic Methods: Neural-symbolic research can map cognitive behaviour to symbolic models and then transform computational logic into artificial neural networks.The paper relates this two-step mapping to possible contributions to cognitive neuroscience.

5. Putting the Machinery to Work: Binding and First-Order Inference in a Neural-Symbolic Framework

This section addresses how neural-symbolic systems can represent and process structured symbolic knowledge, focusing on predicate logic and first-order inference in recurrent networks. It emphasizes compact variable binding while acknowledging that prior approaches have not captured full first-order logic expressiveness.

  • Putting the Machinery to Work: Complex combinatorial structures remain challenging for artificial neural networks and neuro-cognitive theories, despite their apparent plausibility in human cognition.The section turns from this general challenge to concrete neural-symbolic work.
  • Putting the Machinery to Work: A recurrent network with symmetric weights represents predicate logic through a variable-binding mechanism that compactly encodes first-order expressions in activation.Symbolic constraints are encoded in network weights so inference solutions emerge in activation.
  • Putting the Machinery to Work: Fodor and Pylyshyn identify combinatorial syntax and semantics plus structure-sensitive processing as essential characteristics for cognitive models.These requirements concern recursive construction of representations and operations based on syntactic structure.
  • Putting the Machinery to Work: Earlier approaches only partially addressed these requirements and often suffered from limited expressiveness, poor damage robustness, limited learning, high size-complexity, or ad hoc engineering.The cited critique states that none captured full first-order logic expressive power.

5.2 The Binding Problem Computationally Revisited

The binding problem remains a central computational obstacle for neural representation of structured knowledge and high-level cognition. Proposed mechanisms have advanced understanding but retain shortcomings in expressiveness, resource demands, or centralized control.

  • The Binding Problem Computationally Revisited: High-level cognitive and language modelling remains difficult, with many problems likely related to the binding problem.The section treats binding as a recurring issue across computational neural modelling.
  • The Binding Problem Computationally Revisited: A neural binding mechanism must combine predicates, functions, constants, and variables from a knowledge base so they can be manipulated together.This is described as the first step toward computationally handling such structures.
  • The Binding Problem Computationally Revisited: Existing variable-binding approaches contribute possible mechanisms but remain limited by restricted expressiveness, high space requirements, or centralized control.The passage presents these as shortcomings across the respective approaches.

5.3 Inference Specifications as Fixed Points of ANNs

This approach specifies reasoning problems as weighted logical constraints and represents their valid solutions as stable states of symmetric neural networks. Energy minimization, compilation, or learning then supports satisfiability, retrieval, unification, and inference.

  • In symmetric ANNs, fixed points reached at equilibrium with clamped inputs can represent solutions to the specified problem.
  • High-order symmetric ANNs can replace pairwise networks with hidden units, while high-order Boltzmann variants reduce simulation time and search-space size.
  • Reasoning problems are specified as weighted propositional logic formulae whose Boolean constraints encode valid inference solutions.
  • Weighted satisfiability maps truth values to a quadratic energy function that sums penalties for violated constraints.
  • A compiler accepts index-quantified penalty-logic specifications and generates ANNs for first-order retrieval, unification, and inference involving large networks.
  • Inference can settle into a global energy minimum interpreted as a proof or first-order clause chain, with competition supporting preferred explanations, inferences, plans, or actions.

5.4 A Fault Tolerant Mechanism for Dynamic Binding

The dynamic-binding mechanism uses compact spatial conjunctive binding to represent compositional structures in neural stable states. General-purpose binders support retrieval, unification, nested representations, fault tolerance, and broad graph expressiveness.

  • The binding technique constructs working memory and enables retrieval from long-term memory, unification, and inference.
  • Spatial conjunctive binding captures compositionality without dependence on timing synchronization or time-slot allocation.
  • General-purpose binders reduce unit requirements, provide fault tolerance through interchangeable ensembles, and represent any labelled directed graph structure.Each binder can bind many stored knowledge items and can point to other binders, enabling nested hierarchies.
  • Crossbar-connected GP-binders act as indirect pointers, binding objects when a binder points to both and supporting nested hierarchies.

5.5 Examples of Inferences, Cues, and Deductive Associations

The inference engine uses cues and a compiled or learned knowledge base to search for deductive associations in an attractor-based neural working memory. A worked example derives Anne’s grandparent through successive retrieval, unification, and resolution steps.

  • A compiled or learned Boltzmann-machine inference engine stores a knowledge base in synaptic connections and searches for proofs after clamping a query.
  • For the query GrandParent(V, Anne), the network binds V to Anne’s mother’s mother and finds the complete inference chain.
  • The proof retrieves and unifies Parent(mother(Anne), Anne) as its first step.
  • Resolving intermediate clauses and retrieving a second rule instance produces the required parent relation for the final deduction.
  • The final resolution establishes GrandParent(mother(mother(Anne)), Anne).
  • Only rules needed for the goal are retrieved into working memory; other stored rules remain in long-term memory.
  • Cues other than queries can guide attention or accelerate convergence, while associative-memory operation can yield several inferences with known probabilities.

5.6 Can Neurones Be Logical? Fundamental Negative and Positive Results

Attractor-based ANNs can represent and process unrestricted but memory-bound first-order inference, yet cyclic networks remain incomplete and may fail to reach global solutions. Symmetric-network search is also vulnerable to pathological oscillations and local minima.

  • Relatively compact attractor-based ANNs can represent and process unrestricted but memory-bound first-order inference.
  • Acyclic ANNs can use an activation function guaranteeing a global solution, whereas cyclic networks admit pathological scenarios that no activation function fully eliminates.
  • Because the outlined first-order networks contain many cycles, they are incomplete even with a bound on proof length.
  • Stochastic local search in Boltzmann and other symmetric networks can be trapped by local minima, so finite-time solution discovery is not guaranteed.
  • Hebbian-learning and Boltzmann-machine variants are being investigated to eliminate local minima and accelerate discovery of global minima.

6. Connectionist First-Order Logic Learning Using the Core Method

This section addresses representing and learning first-order rules in connectionist systems by embedding interpretations and approximating the immediate consequence operator. It also examines convergence, learning, and practical alternatives for first-order learning.

  • Representing and learning first-order rules within a connectionist setting is a central problem in neural-symbolic integration.
  • 6.1 Feasibility of the First-Order Core Method and the Embedding of First-Order Rules into Neural Networks: The immediate consequence operator TP is approximated because even one application can produce infinitely many atoms over the Herbrand universe.The section distinguishes approximation in space from approximation in time.
  • 6.1 Feasibility of the First-Order Core Method and the Embedding of First-Order Rules into Neural Networks: First-order interpretations are embedded into real numbers through a homeomorphic mapping, preserving structural information for neural approximation.The embedding uses a bijective level mapping and a number-system representation of interpretations.
  • 6.1 Feasibility of the First-Order Core Method and the Embedding of First-Order Rules into Neural Networks: Repeated application of the approximating function converges to a state corresponding to a program model when TP itself is contractive.The approximating functions are contractive on the embedded interpretation set Cb, though not generally on all of R.
  • 6.2 Learning in a First-Order Setting: Training data can adapt randomly initialized network weights and structure, while inductive logic programming offers a practical route for efficient first-order learning.The latter approach uses propositionalisation to make first-order learning more tractable.

7. Markov Logic Networks Combining Probabilities and First-Order Logic

This section presents Markov logic networks and related probabilistic-symbolic models that combine first-order logic with graphical representations. It then describes scalable inference opportunities, tractable sum-product networks, and relational extensions used in applications.

  • Markov logic softens weighted first-order formulae into feature templates for Markov networks, combining logical dependencies with probabilistic handling of noise and uncertainty.Each ground atom becomes a network node, while each ground formula becomes a feature.
  • MLNs face scalability challenges because grounding pairwise relations grows quadratically and transitivity relations grow cubically with domain size.A domain of 1000 people can yield 1 million possible friendships and 1 billion possible transitivity relations.
  • Lifted inference exploits symmetry by grouping variables or configurations, reducing large grounded inference problems when entities are not distinguished by evidence.The passage illustrates this with identical smoking probabilities for indistinguishable individuals.
  • MLN weight learning maximizes penalized likelihood, but gradient computation requires inference and may be ill-conditioned because formulae have unequal numbers of satisfying groundings.Short-run Markov chain Monte Carlo is used to approximate the gradient.
  • SPNs use sums and products in directed acyclic graphs to integrate learning with inference, with completeness and consistency ensuring representation of graphical-model distributions.Inference follows the network structure and can answer arbitrary probability queries in linear time.
  • Experiments report that SPN inference and learning can be faster and more accurate than standard deep networks, including better image completion on the cited task.
  • Relational sum-product networks extend SPNs to relational domains while retaining tractable inference, with applications in social networks and automated debugging.Their class specifications describe object attributes, component parts, and relations, and their structures and parameters can be learned from data.
  • MLNs and related formalisms exemplify symbolic-numerical integration and have gained applications across artificial intelligence.

8. Relating Neural-Symbolic Systems to Human-Level Artificial Intelligence

The section relates neural-symbolic integration to human-level artificial intelligence by addressing the mismatch between neural and symbolic models of human cognition. It presents reconciliation as a research direction combining neural implementations with symbolic representation and reasoning.

  • HLAI therefore requires cross-disciplinary integration among artificial intelligence, neuroscience, psychology, and computational linguistics.
  • Neural-symbolic integration starts from the complementary observations that intelligent systems have neural physical implementations and that complex tasks are modelled through symbols and logic.
  • Current artificial neural approaches remain outside the scope of symbolic knowledge representation and manipulation at the level required for HLAI.
  • Human cognition combines neural processing with successful symbolic manipulation, while current theories and computational models remain substantially mismatched.
  • The authors believe reconciling neural and symbolic approaches is central to progress in HLAI, cognitive science, and cognitive neuroscience.
  • The surveyed research formally defines and implements artificial-neural-network systems capable of learning and handling logical representations of complex domains and inferences.

9. (Most) Recent Developments and Work in Progress from the Neural-Symbolic Neighbourhood

Recent work narrows the neural-symbolic gap through architectures that add symbolic capacities to neural systems, application systems solving predominantly symbolic tasks, and research programs targeting interpretability and theoretical understanding. These developments show both practical capabilities and unresolved theoretical questions.

  • Other Paradigms in Computation and Representation Narrowing the Neural-Symbolic Gap: Conceptors equip recurrent neural networks with symbolic capacities by characterising processing modes through state-cloud geometries and stabilising selected modes.
  • Other Paradigms in Computation and Representation Narrowing the Neural-Symbolic Gap: Neural Turing Machines couple a recurrent neural controller with addressable memory, using differentiable selective reads and writes for gradient-descent training.
  • Other Application Systems Narrowing the Neural-Symbolic Gap: Connectionist vision systems combine convolutional image encoders with recurrent language decoders to generate descriptions or label image regions.
  • Other Application Systems Narrowing the Neural-Symbolic Gap: These vision systems achieved state-of-the-art image-sentence ranking and surpassed retrieval baselines, showing that conceptually symbolic semantic tasks can use purely connectionist architectures.
  • Other Application Systems Narrowing the Neural-Symbolic Gap: A distributed recurrent Q-network learned communication protocols for coordinated teams of agents operating under partial observability and uncertainty.
  • Other Application Systems Narrowing the Neural-Symbolic Gap: AlphaGo integrated neural policy and value networks with Monte Carlo tree search, reaching play comparable to the world’s best human players.
  • Other Research Efforts and Programs Narrowing the Neural-Symbolic Gap: Research programs investigate situated knowledge acquisition, interpretable long-range dependencies in recurrent networks, and differences between connectionist and symbolic complexity.
  • Other Research Efforts and Programs Narrowing the Neural-Symbolic Gap: These efforts are expected to clarify tensions between formalisms across tasks and domains and inform remedies for formal and theoretical integration challenges.

10. Challenges and Future Directions

The survey identifies representation, integrated learning and reasoning, and knowledge extraction from large networks as central challenges for neural-symbolic computation. It also emphasises that principled integration remains incomplete and requires foundational work across multiple research directions.

  • Neural-symbolic systems aim to combine connectionist fault tolerance with symbolic reasoning’s brittleness and rigidity while making symbolic reasoning efficient and effective.
  • Current efforts predominantly improve learning with reasoning methods or reasoning with learning models rather than integrating both principles symmetrically.
  • The field’s main challenges are determining representation limits, finding representations suitable for integrated learning and reasoning, and extracting knowledge from very large networks.
  • Future work includes reconciling first-order logic learning with first-order logic reasoning, semi-supervised and incremental learning, and evaluating gains from massive parallelism.
  • Additional research directions cover learning-reasoning-acting cycles, ensemble-structure representations, rule extraction, fibring functions, complexity differences, proof and type theories, analogy, abduction, and attention.
  • Knowledge about these challenges remains limited, despite growing attention to multi-agent systems, learning-reasoning-acting cycles, and theoretical comparisons of neural and logical approaches.
  • The authors reject direct bottom-up brain modelling as unproductive under current conditions and call for a foundational approach combining the field’s directions.
  • Potential application areas include combining neural image processing with maintainable and validatable symbolic text-processing systems.

11. Concluding Remarks

Neural-symbolic integration links symbolic and connectionist paradigms to address the brain–mind dichotomy and support expressive reasoning with robust learning. The survey identifies integration challenges while aiming toward a unifying theory and principled development across cognitive and computing sciences.

  • The survey connects symbolic and statistical approaches to address the perceived brain–mind dichotomy in human and artificial intelligence.
  • Neural-symbolic systems combine connectionist machines with logical abstractions.
  • Integration research balances the differing virtues and deficiencies of symbolic and connectionist paradigms for application-driven neural-symbolic methods.
  • Key challenges include computing logical models, extracting comprehensible knowledge, and balancing tractability with expressiveness.
  • The ultimate goal is neural-symbolic systems that provide expressive reasoning and robust learning while advancing toward a unifying theory.
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