# Rethinking Arithmetic for Deep Neural Networks

## TL;DR

This paper explores the implementation of deep neural networks as Boolean circuits, demonstrating the functional completeness of binarised neural networks and discussing the implications for hardware accelerators and generalisation.

## Contribution

It introduces a formal connection between neural networks and Boolean circuits, showing binarised neural networks are functionally complete and proposing new directions for hardware and theoretical research.

## Key findings

- Binarised neural networks are functionally complete.
- Boolean circuits can effectively model neural network computations.
- LUTNet demonstrates a novel FPGA inference approach.

## Abstract

We consider efficiency in the implementation of deep neural networks. Hardware accelerators are gaining interest as machine learning becomes one of the drivers of high-performance computing. In these accelerators, the directed graph describing a neural network can be implemented as a directed graph describing a Boolean circuit. We make this observation precise, leading naturally to an understanding of practical neural networks as discrete functions, and show that so-called binarised neural networks are functionally complete. In general, our results suggest that it is valuable to consider Boolean circuits as neural networks, leading to the question of which circuit topologies are promising. We argue that continuity is central to generalisation in learning, explore the interaction between data coding, network topology, and node functionality for continuity, and pose some open questions for future research. As a first step to bridging the gap between continuous and Boolean views of neural network accelerators, we present some recent results from our work on LUTNet, a novel Field-Programmable Gate Array inference approach. Finally, we conclude with additional possible fruitful avenues for research bridging the continuous and discrete views of neural networks.