Neuro-Symbolic Reasoning Theory

Neuro-symbolic AI combines neural networks (learning from data) with symbolic AI (logical reasoning) to create systems that can both learn and reason.

The Symbol Grounding Problem

Traditional AI approaches face a fundamental challenge:

Neuro-symbolic AI bridges this gap by combining both approaches.

Core Concepts

1. Differentiable Logic

Traditional logic operations (AND, OR, NOT) are discrete and non-differentiable. Differentiable logic replaces these with smooth approximations:

# Traditional logic (non-differentiable)
AND(x, y) = 1 if x=1 AND y=1, else 0

# Product t-norm (differentiable)
AND(x, y) = x * y

# Godel t-norm
AND(x, y) = min(x, y)

# Lukasiewicz t-norm
AND(x, y) = max(0, x + y - 1)

This allows gradient-based optimization through logical operations.

2. Logic Tensor Networks

Logic Tensor Networks (LTNs) represent:

• Constants: As vectors in embedding space
• Predicates: As neural networks
• Logical formulas: As differentiable computations

# Predicate: "is_mammal(x)"
is_mammal = NeuralNetwork(input_dim=embedding_dim, output_dim=1)

# Logical formula: "mammal(x) AND has_fur(y) -> warm_blooded(x)"
loss = 1 - implies(
    and_(is_mammal(x), has_fur(x)),
    is_warm_blooded(x)
)

3. Neural Theorem Proving

Use neural networks to guide proof search:

1. Encode facts and rules as embeddings
2. Train a neural network to predict useful proof steps
3. Use the network to prioritize search during inference

4. Knowledge Graph Embeddings

Represent entities and relations in continuous vector spaces:

# TransE model
score(head, relation, tail) = ||head + relation - tail||

# RotatE model
score(head, relation, tail) = ||head ⊙ relation - tail||

TensorLogic Architecture

Aprender's TensorLogic implements neuro-symbolic reasoning using tensor operations:

┌─────────────────────────────────────────────────────────────┐
│                    TensorLogic Engine                        │
├─────────────────────────────────────────────────────────────┤
│  Knowledge Base          │  Inference Engine                │
│  ┌──────────────────┐   │  ┌──────────────────────────┐   │
│  │ Facts (tensors)  │   │  │ Forward Chaining         │   │
│  │ Rules (programs) │   │  │ Backward Chaining        │   │
│  │ Weights (probs)  │   │  │ Probabilistic Inference  │   │
│  └──────────────────┘   │  └──────────────────────────┘   │
├─────────────────────────────────────────────────────────────┤
│  Tensor Operations (SIMD-accelerated via Trueno)            │
│  ┌──────────────────────────────────────────────────────┐  │
│  │ logical_join  │ logical_project │ logical_select    │  │
│  │ logical_aggregate │ matrix multiplication           │  │
│  └──────────────────────────────────────────────────────┘  │
└─────────────────────────────────────────────────────────────┘

Mathematical Foundation

Relational Composition

Given relations R(X,Y) and S(Y,Z) represented as matrices:

(R ∘ S)[i,k] = ∨_j (R[i,j] ∧ S[j,k])

For Boolean tensors, this is matrix multiplication over the Boolean semiring. For weighted tensors, use standard matrix multiplication.

Existential Quantification

Project out a variable using logical OR:

(∃Y: R(X,Y))[i] = ∨_j R[i,j]

Implemented as max along the projected dimension.

Universal Quantification

(∀Y: R(X,Y))[i] = ∧_j R[i,j]

Implemented as min along the quantified dimension.

Training Neuro-Symbolic Models

Loss Functions

1. Satisfaction Loss: Penalize unsatisfied logical constraints

   L_sat = Σ_φ (1 - satisfaction(φ))
   

2. Semantic Loss: Match predictions to logical semantics

   L_sem = KL(P_neural || P_logical)
   

3. Hybrid Loss: Combine data loss with logical constraints

   L = L_data + λ * L_logical
   

Regularization

Logical constraints act as regularization:

• Enforce consistency between predictions
• Reduce need for labeled data
• Improve generalization

Applications

1. Knowledge Graph Completion

   • Infer missing facts in knowledge graphs
   • Example: If (Alice, parent, Bob) and (Bob, parent, Charlie), infer (Alice, grandparent, Charlie)

2. Question Answering

   • Multi-hop reasoning over structured data
   • Combine entity linking with logical inference

3. Program Synthesis

   • Learn programs from input-output examples
   • Use logical constraints to prune search space

4. Explainable AI

   • Generate logical explanations for neural predictions
   • Trace inference steps through proof trees

Comparison with Pure Neural Approaches

Further Reading

• Marcus, G. (2020). "The Next Decade in AI: Four Steps Towards Robust Artificial Intelligence"
• De Raedt, L. et al. (2020). "From Statistical Relational to Neural Symbolic Artificial Intelligence"
• Lamb, L. et al. (2020). "Graph Neural Networks Meet Neural-Symbolic Computing: A Survey and Perspective"