of CPU is the Boolean calculation. For CPU, this calculation is the most efficient. Since the BDNN only uses Boolean calculation, thus the processing speed of which can be easily optimized on CPU. Consequently, it is possible to run BDNN of high speed on devices with only CPUs. In contrast, conventional DNN can only run fast with GPUs.

-   Clear network response: in BDNN, it is quite easy to observe the response of neurons and weights in the propagation process. This is because each neuron or weight only has two different kinds of status. This may be helpful for us to design proper learning strategy for BDNN.

In summary, with binary variables and Boolean operations, the BDNN is able to run with reasonable computational resource and storage. Although now the performance of BDNN may not be comparable with the conventional deep neural network of the same scale, it has the potential to be improved in the future.

This paper is organized as follows. In Section \[BDNN\], the principles of the BDNN is introduced as well as the hybrid binarized deep neural network (hybrid-BDNN) for non-binary input data. In Section \[training\], a training method is proposed for BDNN and hybrid-BDNN. In 