Figure 3.

Space-Time Dynamics Reconstruction

(A) Reconstruction of the space-time evolution of dynamic systems (Elementary Cellular Automata or ECA Morris et al., 2014). Normal space-time evolution is displayed on the left-hand side; on the right-hand side are the reconstructed space-times after row scrambling by finding the lowest algorithmic complexity configuration among all possible 9! = 362,880 row permutations (8 steps + initial configuration). All are followed by Spearman correlation values for row order.

(B) Row time inference in linear time by the generation of an algorithmic model that can run forward and backward, thus revealing the dynamics and first principles of the underlying dynamic systems without any brute force exploration or simulation.

(C) As predicted, the later in time a perturbation is performed the less disruptive (change of hypothesized generating mechanism length after perturbation) it is compared with the length of the hypothesized generating mechanism of evolution of the original system. Each pair shows the statistical rho and p values between the reconstructed and original space-time evolutions, with some models separating the system into different apparent causal elements.

(D and E) (D) Depicted is the reconstruction of one of the simplest elementary cellular automata (rule 254) and (E) one of the most random-looking ECA, both after 280 steps, illustrating the perturbation-based algorithmic calculus for model generation in two opposite behavioral cases.

(F and G) The accuracy of the reconstruction can be scaled and improved at the cost of greater computational resources by going beyond single row perturbation up to the power set (all subsets). Depicted here are reconstructions of random-looking cellular automata (30 and 73 running for 200 steps) from (F) single- (1R) and (G) double-row-knockout (2R) perturbation analysis. Errors inherited from the decomposition method (see Supplemental Information, BDM) look like “shadows” and are explained (and can be counteracted) by numerical deviations from the boundary conditions in the estimation of BDM (Zenil et al., 2018a, Zenil et al., 2016).

(H) Variations of the magnitude of the found effect are different in systems with different qualitative behavior: the simpler the less different the effects of deleterious perturbations at different times.