A Group Theoretic Analysis of the Symmetries Underlying Base Addition and Their Learnability by Neural Networks

Area of Science:

• Artificial Intelligence
• Cognitive Science
• Machine Learning

Background:

• A key challenge in AI and cognitive modeling is creating neural networks capable of radical generalization.
• This generalization ability is rooted in discovering and implementing symmetry functions.

Purpose of the Study:

• To investigate radical generalization using symmetry through the example of base addition.
• To analyze the group theory of base addition and its carry function.

Main Methods:

• Performed a group theoretic analysis of base addition and its carry function.
• Introduced quantitative measures to characterize different carry functions.
• Trained neural networks on base addition with varied carry functions to probe inductive biases.

Main Results:

• Identified a range of alternative carry functions for base addition.
• Demonstrated that neural networks can achieve radical generalization with appropriate input formats and carry functions.
• Found a strong correlation between carry function structure and neural network learnability.

Conclusions:

• The choice of carry function is critical for efficient symmetry learning in neural networks.
• Findings have implications for understanding human cognitive function and advancing artificial intelligence.
• Properly formatted inputs and well-suited carry functions enable even simple neural networks to generalize radically.