Title: Instantons in self-organizing logic gates,
Abstract: Self-organizing logic is a recently-suggested framework that allows the
solution of Boolean truth tables "in reverse," i.e., it is able to satisfy the
logical proposition of gates regardless to which terminal(s) the truth value is
assigned ("terminal-agnostic logic"). It can be realized if time non-locality
(memory) is present. A practical realization of self-organizing logic gates
(SOLGs) can be done by combining circuit elements with and without memory. By
employing one such realization, we show, numerically, that SOLGs exploit
elementary instantons to reach equilibrium points. Instantons are classical
trajectories of the non-linear equations of motion describing SOLGs, and
connect topologically distinct critical points in the phase space. By linear
analysis at those points, we show that these instantons connect the initial
critical point of the dynamics, with at least one unstable direction, directly
to the final fixed point. We also show that the memory content of these gates
only affects the relaxation time to reach the logically consistent solution.
Finally, we demonstrate, by solving the corresponding stochastic differential
equations, that since instantons connect critical points, noise and
perturbations may change the instanton trajectory in the phase space, but not
the initial and final critical points. Therefore, even for extremely large
noise levels, the gates self-organize to the correct solution. Our work
provides a physical understanding of, and can serve as an inspiration for, new
models of bi-directional logic gates that are emerging as important tools in
physics-inspired, unconventional computing.