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

Marcio Nicolau, Anderson R. Tavares, Zhiwei Zhang, Pedro H. C. Avelar, João Marcos Flach, Luis DC Lamb, Moshe Vardi

Keywords: Boolean functions, Deep neural networks, PAC learning, Neurosymbolic models, Learnability

TL;DR: Deep NNs can learn a wide range of Boolean functions, often generalizing better than symbolic or rule-based systems-particularly for large or complex formulas-although smaller or underconstrained formulas can be notably harder for NNs to learn.

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 neurosymbolic AI methods.

Track: Main Track

Paper Type: Long Paper

Resubmission: No

Software: https://github.com/machine-reasoning-ufrgs/mlbf and https://github.com/zzwonder/Valiants-Algorithm-for-Learning-CNF-formulas

Submission Number: 2