On Bounding the Behavior of Neurons

Abstract

A neuron with binary inputs and a binary output represents a Boolean function. Our goal is to extract this Boolean function into a tractable representation that will facilitate the explanation and formal verification of a neuron’s behavior. Unfortunately, extracting a neuron’s Boolean function is in general an NP-hard problem. However, it was recently shown that prime implicants of this Boolean function can be enumerated efficiently, with only polynomial time delay. Building on this result, we first propose a best-first search algorithm that is able to incrementally tighten the inner and outer bounds of a neuron’s Boolean function. Second, we show that these bounds correspond to truncated prime-implicant covers of the Boolean function. Next, we show how these bounds can be propagated in an elementary class of neural networks. Finally, we provide case studies that highlight our ability to bound the behavior of neurons.