We present an atomic-scale mechanism based on variable-range hopping of interacting charges enabling reconfigurable logic and nonlinear classification tasks in dopant network processing units in silicon. Kinetic Monte Carlo simulations of the hopping process show temperature-dependent current-voltage characteristics, artificial evolution of basic Boolean logic gates, and fitness-dependent gate abundances in striking agreement with experiment. The simulations provide unique insights in the local electrostatic potential and current flow in the dopant network, showing subtle changes induced by control voltages that set the conditions for the logic operation. These insights will be crucial in the systematic further development of this burgeoning technology for unconventional computing. The establishment of the principles underlying the logic functionality of these devices encourages the exploration and utilization of the same principles in other materials and device geometries.