Understanding Boolean Function Learnability on Deep Neural Networks: PAC Learning Meets Neurosymbolic Models

TL;DR

This paper investigates how deep neural networks learn boolean formulas, revealing that neural models often outperform symbolic systems and that formula complexity and constrainedness influence learnability, informing future neurosymbolic AI development.

Contribution

It provides empirical insights into boolean formula learnability by neural networks, bridging theoretical PAC learning with practical neurosymbolic applications.

Findings

Neural networks generalize better than rule-based and symbolic systems.

Small, shallow networks effectively approximate combinatorial optimization formulas.

Less constrained 3-CNF formulas are easier to learn than overconstrained ones.

Abstract

Computational learning theory states that many classes of boolean formulas are learnable in polynomial time. This paper addresses the understudied subject of how, in practice, such formulas can be learned by deep neural networks. Specifically, we analyze boolean formulas associated with model-sampling benchmarks, combinatorial optimization problems, and random 3-CNFs with varying degrees of constrainedness. Our experiments indicate that: (i) neural learning generalizes better than pure rule-based systems and pure symbolic approach; (ii) relatively small and shallow neural networks are very good approximators of formulas associated with combinatorial optimization problems; (iii) smaller formulas seem harder to learn, possibly due to the fewer positive (satisfying) examples available; and (iv) interestingly, underconstrained 3-CNF formulas are more challenging to learn than overconstrained ones. Such findings pave the way for a better understanding, construction, and use of interpretable neurosymbolic AI methods.