Mathematical logic and the GNN

Overview of mathematical logic

Mathematical logic becomes the study of logic using mathematical methods and techniques.’ Making Logic Part 1 Beginning Logic Reading Notes’ etc. Logic is the study of the principles of reasoning and argumentation, the concepts of truth and falsehood, etc., and mathematical logic aims to deal mathematically with the logical systems and formal systems that form the basis of logic.

Mathematical logic rigorously defines the foundations of logic using the formal methods of mathematics, and pursues the accuracy of reasoning and the efficiency of proofs. One of the representative systems of mathematical logic is called ‘propositional logic’. In propositional logic, propositions are expressed in terms of boolean values and logical operators (logical OR, logical conjunction, negation, etc.) are used to describe relationships between propositions.

Mathematical logic also has a system called ‘predicate logic’, in which individual propositions are expressed using predicates and variables, and quantifiers (such as the holomorphic quantifier and the existential quantifier) are introduced to express the range of propositions. Predicate logic is an extension of propositional logic, making it a richer and more expressive logical system.

Mathematical logic includes areas such as formal logical systems, proof theory, computability theory and model theory, which not only reveal the foundations of logic through the formal logic methods of mathematics, but also have applications in other fields such as information science, philosophy and cognitive science.

The fields of application of mathematical logic include programme justification and database query processing in computer science, logical reasoning in artificial intelligence, and grammatical analysis in natural language processing. The methods of mathematical logic are also used in logical analysis and formal logic approaches in philosophy.

Mathematical logic and the GNN

Consider combining this mathematical logic with the Graph Neural Network (GNN) technology described in ‘Graph Neural Networks’, which has the ability to process graph data consisting of nodes (objects) and edges (relations), making it a suitable approach for handling formal structures and rules in mathematical logic. This approach is suitable for dealing with the formal structures and rules of mathematical logic. This combination is expected to offer new possibilities in the areas of logical reasoning and knowledge representation. Examples of these are described below.

1. construction of knowledge graphs: mathematical logic deals with the formal representation and reasoning of knowledge, and GNNs are powerful tools for structuring that knowledge: they represent entities (nodes) and relations between them (edges) as graphs, modelling propositions and predicate logic in mathematical logic This makes it easier to visualise complex networks of knowledge. Using this GNN, it is possible to extend existing knowledge graphs and derive new relations and entities. This allows the theory of mathematical logic to be applied to real-world data.

2. logical reasoning: GNNs are also useful in the process of logical reasoning, where GNNs can be used by children to learn the ability to derive new conclusions from given knowledge. For example, they can understand how one fact relates to another and learn rules for drawing conclusions. They can also use GNNs as inference engines to generate new information using the relationships between nodes, thus enabling them to answer complex logical questions.

3. analysing complex data structures: mathematical logic deals with formal structures and systems, and GNNs are well suited for analysing these. Specifically, GNNs can be used to examine how models of mathematical logic hold for a given statement, e.g. by identifying a set of nodes that satisfy certain conditions, allowing the validity of logical propositions to be assessed, or by modelling complex logical expressions as graph structures, improving computational efficiency 3. the ability to identify a set of nodes that satisfy a particular condition, thereby speeding up the solution of problems in mathematical logic.

4. integration with natural language processing: GNNs can be integrated with natural language processing (NLP) to further extend the applications of mathematical logic. Specifically, GNNs can be used to model the meaning of sentences as a graph structure and understand logical relationships, which can then be applied to sentence inference and question-answering systems, or information extracted from natural language can be graphically represented based on GNNs to generate new knowledge. This implies the realisation of automated knowledge acquisition based on the theory of mathematical logic.

5. integrative approach: integrating mathematical logic and GNNs enables more powerful analysis and problem-solving methods. Specifically, GNNs can be used to visualise and make easier to understand formal proofs of mathematical logic, or GNNs can be used to detect inconsistencies and inconsistencies in the knowledge base and ensure consistency according to the principles of mathematical logic.

The integration of mathematical logic and GNNs offers many possibilities for knowledge representation, logical reasoning, analysis of complex data structures and integration with natural language processing. This approach is expected to result in new methods for effectively processing knowledge, discovering and solving problems, and the application of GNNs using the principles of mathematical logic is expected to foster innovation in a variety of fields.

Examples of implementations combining mathematical logic and GNNs

Examples of implementations that combine mathematical logic and Graph Neural Networks (GNN) include

1. Knowledge graph inference
   Rules based on mathematical logic (e.g. first-order predicate logic) are applied to the node and edge information of a knowledge graph, as described in ‘Knowledge information processing techniques’, and inference is enhanced by the GNN.
2. Coupling quantum logic and GNNs
   Quantum logic, also described in ‘Quantum Mechanics, Artificial Intelligence and Natural Language Processing’, is used in graph structures to infer new relations using mathematical logic rules.
3. Integration of logic programming and GNNs
   The rules defined in logic programming such as Prolog, also described in ‘Prolog and Knowledge Information Processing’, are reflected in the learning process using GNNs.

<Example 1: Knowledge Graph Completion>

Objective: To infer missing relations (edges) in the knowledge graph.

Procedure:

1. Data preparation.
   • Nodes (entities) and edges (relations) of the knowledge graph are constructed.
   • Define logical rules for each edge type (e.g. ‘parent (A, B) ∧ parent (B, C) → grandparent (A, C)’).
2. Model building.
   • Use GNNs (e.g. GCN, GAT) to learn node embeddings.
   • Add a logic inference module that uses embeddings to optimise scores to satisfy logic rules.
3. Logic rule embedding.
   • Incorporate predefined rules into loss functions (e.g. KL divergence to reflect rule constraints).

Example: Python code (PyTorch Geometric + PyTorch)

import torch
from torch_geometric.nn import GCNConv
import numpy as np

class LogicGNN(torch.nn.Module):
    def __init__(self, in_channels, out_channels):
        super(LogicGNN, self).__init__()
        self.conv1 = GCNConv(in_channels, 16)
        self.conv2 = GCNConv(16, out_channels)

    def forward(self, x, edge_index):
        x = self.conv1(x, edge_index)
        x = torch.relu(x)
        x = self.conv2(x, edge_index)
        return x

# Preparation of knowledge graph data.
node_features = torch.randn((num_nodes, feature_dim))  # node feature
edge_index = torch.tensor([[0, 1], [1, 2]])  # Edge list (e.g.)

# model initialization
model = LogicGNN(in_channels=feature_dim, out_channels=embedding_dim)
optimizer = torch.optim.Adam(model.parameters(), lr=0.01)

# Loss function: assess consistency with logic rules
def logic_loss(embeddings, rules):
    # Define rule-based losses (e.g. scores for rule violations)
    return torch.sum((rules - embeddings)**2)

# training
for epoch in range(epochs):
    optimizer.zero_grad()
    embeddings = model(node_features, edge_index)
    loss = logic_loss(embeddings, predefined_rules)
    loss.backward()
    optimizer.step()

<Example 2: Coupling quantum logic and GNNs>

Implementations that combine quantum logic with graph neural networks (GNNs) utilise the parallelism and probabilistic logic operations of quantum computers and complement data learning with conventional GNNs. This approach uses quantum states to represent graph data and allows GNNs to process quantum information for more advanced inference and learning.

Objectives.

• Quantum logic: nodes and edges are represented by quantum bits (qubits) and manipulated by quantum gates.
• GNN: performs learning based on graph structures and extracts features from quantum information.

Implementation overview

1. Use of quantum computing libraries.
   • Describes quantum logic using qiskit.
   • Encoding of graph data into quantum states.
2. Feature extraction from quantum information.
   • Quantum states are measured and used as input features for GNN.
3. Graph learning with GNN
   1. Construct GNN models using PyTorch Geometric.

Code examples.

1. encoding graph data into quantum states: first, encode the node information of the graph into quantum states.

from qiskit import QuantumCircuit, Aer, execute
import numpy as np

# Node features of the graph
node_features = np.array([1, 0, 1])  # 0/1 features per node

# Creating quantum circuits
def encode_to_quantum_state(features):
    n_qubits = len(features)
    qc = QuantumCircuit(n_qubits)

    # Encoding of node features into quantum states.
    for i, feature in enumerate(features):
        if feature == 1:
            qc.x(i)  # X-gate to |1>.

    return qc

# Encoding node features into quantum circuits.
qc = encode_to_quantum_state(node_features)
print(qc)

2. applying quantum manipulation: applying quantum gates to manipulate relationships between nodes.

# Adding gates to quantum circuits
def apply_quantum_logic(qc):
    # Relationships between nodes are represented by CNOT gates.
    qc.cx(0, 1)  # Node 0 affects node 1.
    qc.cx(1, 2)  # Node 1 affects node 2.
    return qc

qc = apply_quantum_logic(qc)
print(qc)

3. quantum measurement and generation of GNN input data: quantum states are measured and converted into GNN input data.

# quantum measurement
def measure_quantum_state(qc):
    qc.measure_all()
    simulator = Aer.get_backend('qasm_simulator')
    result = execute(qc, simulator, shots=1).result()
    counts = result.get_counts()
    measured_state = list(counts.keys())[0]  # Resulting measurements
    return np.array([int(bit) for bit in measured_state[::-1]])  # Bit strings in reverse order.

# Get measurement results
measured_features = measure_quantum_state(qc)
print("Measured Features:", measured_features)

4. building the GNN model: input the measurement results into the GNN for training.

from torch_geometric.data import Data
from torch_geometric.nn import GCNConv
import torch

# Convert measurement results to PyTorch Tensor.
x = torch.tensor(measured_features, dtype=torch.float).view(-1, 1)

# edge list
edge_index = torch.tensor([[0, 1], [1, 2]], dtype=torch.long).t()

# PyTorch Geometric data objects.
graph = Data(x=x, edge_index=edge_index)

# GNN model
class GNN(torch.nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(GNN, self).__init__()
        self.conv1 = GCNConv(input_dim, hidden_dim)
        self.conv2 = GCNConv(hidden_dim, output_dim)

    def forward(self, x, edge_index):
        x = self.conv1(x, edge_index)
        x = torch.relu(x)
        x = self.conv2(x, edge_index)
        return x

# Initialisation of the model
model = GNN(input_dim=1, hidden_dim=4, output_dim=2)

# forward pass
output = model(graph.x, graph.edge_index)
print("GNN Output:", output)

5. training: incorporate a loss function based on quantum logic to train the GNN.

optimizer = torch.optim.Adam(model.parameters(), lr=0.01)
criterion = torch.nn.CrossEntropyLoss()

# training loop
epochs = 100
for epoch in range(epochs):
    optimizer.zero_grad()
    output = model(graph.x, graph.edge_index)
    loss = criterion(output, torch.tensor([0, 1, 1]))  # Label example
    loss.backward()
    optimizer.step()

    if epoch % 10 == 0:
        print(f"Epoch {epoch}, Loss: {loss.item()}")

Results and applications

• Results: features extracted from quantum states are processed by the GNN, enabling modelling of relationships that cannot be learnt by conventional GNNs alone.
• Applications:
  • Analysis of molecular graphs in quantum chemistry.
  • Optimisation of quantum communication networks.
  • Inference of knowledge graphs with complex interdependencies.

This approach is an advanced example of incorporating the potential of quantum computation into GNNs and shows the possibility of combining data-driven learning and quantum logic.

<Example 3: Integration of Prolog rules and GNNs>

The example implementation of integrating Prolog rules and Graph Neural Networks (GNNs) aims to combine the explicit rules of logic programs with the data-driven approach of GNNs to build a system that learns from data while maintaining interpretability based on logic rules Aim.

Objective.

• Prolog rules: explicit logical relations (e.g. ancestor(X, Y) :- parent(X, Z), ancestor(Z, Y)).
• GNN: Learning relationships (e.g. parent, ancestor) from node features and edge information.

Implementation overview

1. Data preparation
   • Prepare data in graph format.
   • Extract rules defined in Prolog as logical constraints.
2. Model construction
   • Learning node embedding using GNN.
   • Prolog rules are incorporated as loss functions and used during training.
3. Training
   • Learning embedding with GNN.
   • Optimise the loss to satisfy the Prolog rule.

Code examples:

1. data preparation (graph structure + Prolog rules)

# Node and edge datasets
import torch
from torch_geometric.data import Data

# Define node features and edge lists
node_features = torch.tensor([[1], [2], [3]], dtype=torch.float)  # E.g. [ID1, ID2, ID3].
edge_index = torch.tensor([[0, 1], [1, 2]], dtype=torch.long).t()  # Edges: 0 -> 1, 1 -> 2

# PyTorch Geometric graphical data.
graph = Data(x=node_features, edge_index=edge_index)

# Prolog rule.
# ancestor(X, Y) :- parent(X, Z), ancestor(Z, Y).
# Parent relations: node 0 -> node 1, node 1 -> node 2

2. definition of GNN

from torch_geometric.nn import GCNConv

class GNN(torch.nn.Module):
    def __init__(self, input_dim, hidden_dim, output_dim):
        super(GNN, self).__init__()
        self.conv1 = GCNConv(input_dim, hidden_dim)
        self.conv2 = GCNConv(hidden_dim, output_dim)

    def forward(self, x, edge_index):
        x = self.conv1(x, edge_index)
        x = torch.relu(x)
        x = self.conv2(x, edge_index)
        return x

3. incorporating Prolog rules into the loss function:.

How to reflect rules in the loss function: add a penalty for logic constraint violation to the loss function.

def logic_loss(embeddings, edge_index, prolog_rules):
    """
    embeddings: Embedded Nodes
    edge_index: edge list
    prolog_rules: Constraints based on Prolog rules.
    """
    loss = 0
    for rule in prolog_rules:
        # 例: "parent(X, Z) ∧ ancestor(Z, Y) → ancestor(X, Y)"
        # parent (X->Z)
        for (x, z) in edge_index.t():
            for (_, y) in edge_index.t():
                if z == y:  # Assumed relationship between parent and ancestor
                    score = embeddings[x] @ embeddings[y].t()  # degree of similarity
                    loss += (1 - score).pow(2)  # Penalty for low scores.
    return loss

4. model training

# model initialization
model = GNN(input_dim=1, hidden_dim=4, output_dim=2)
optimizer = torch.optim.Adam(model.parameters(), lr=0.01)

# rule definition
prolog_rules = [
    ("parent(X, Z)", "ancestor(Z, Y)", "ancestor(X, Y)")
]

# training loop
epochs = 100
for epoch in range(epochs):
    optimizer.zero_grad()
    embeddings = model(graph.x, graph.edge_index)
    
    # Loss Calculation
    supervised_loss = torch.nn.CrossEntropyLoss()(embeddings, labels)  # Example: supervised labels
    rule_loss = logic_loss(embeddings, graph.edge_index, prolog_rules)
    loss = supervised_loss + rule_loss  # total loss

    loss.backward()
    optimizer.step()

    if epoch % 10 == 0:
        print(f"Epoch {epoch}, Loss: {loss.item()}")

reference book

This section describes reference books that link the fields of mathematical logic and graph neural networks (GNNs).

Reference books on mathematical logic

1. 『Introduction to Mathematical Logic (5th Edition)』

2. 『Computability and Logic (5th Edition)』

Reference books on graph neural networks (GNNs)

1. 『Graph Representation Learning』

2. 『Deep Learning on Graphs: A Survey』

3. 『A Practical Tutorial on Graph Neural Networks』

References linking mathematical logic and GNNs.

1.  “The Logic of Graph Neural Networks”

2.  “Augmenting Neural Networks with First-order Logic”

3. 『Knowledge Graphs: Fundamentals, Techniques, and Applications』

References on quantum logic and GNNs.

1.  “QWalkVec: Node Embedding by Quantum Walk”

2. “Quantum Computation and Quantum Information”

3. “Quantum Graph Neural Networks“