  Attention plays a fundamental role in both natural and artificial
intelligence systems. In deep learning, attention-based neural architectures,
such as transformer architectures, are widely used to tackle problems in
natural language processing and beyond. Here we investigate the fundamental
building blocks of attention and their computational properties. Within the
standard model of deep learning, we classify all possible fundamental building
blocks of attention in terms of their source, target, and computational
mechanism. We identify and study three most important mechanisms: additive
activation attention, multiplicative output attention (output gating), and
multiplicative synaptic attention (synaptic gating). The gating mechanisms
correspond to multiplicative extensions of the standard model and are used
across all current attention-based deep learning architectures. We study their
functional properties and estimate the capacity of several attentional building
blocks in the case of linear and polynomial threshold gates. Surprisingly,
additive activation attention plays a central role in the proofs of the lower
bounds. Attention mechanisms reduce the depth of certain basic circuits and
leverage the power of quadratic activations without incurring their full cost.
