It has been shown previously that spin-Hall oscillators based on current-driven bi-layered film structures containing an antiferromagnet (AFM) and a normal metal can generate ultra-short (similar to 2 ps) "spike-like" pulses in response to an external current stimulus of a sufficient amplitude, thus operating as ultra-fast artificial "neurons." Here, we report the results of numerical simulations demonstrating that a single AFM "neuron" can perform the logic functions of OR, AND, MAJORITY, or Q-gates, while a circuit consisting of a small number n < 5 of AFM "neurons" can function as a FULL-ADDER or as a dynamic memory loop with variable clock frequency. The clock frequencies of such AFM-based logic devices could reach tens of GHz, which make them promising as base elements of future ultra-fast high-efficiency neuromorphic computing.