Existence of a synchronous deterministic CA rule that generates loop patterns

Determine whether a synchronous-updating deterministic cellular automaton rule on a two-dimensional grid with fixed zero boundary conditions can generate stable loop patterns that satisfy the loop path condition (each path cell having exactly two orthogonal one-neighbors and convex corners having an outer-diagonal zero), without getting trapped in non-desired patterns or oscillatory structures such as those seen in Conway’s Game of Life.

Background

The paper investigates generating non-intersecting loop patterns using local conditions via overlapping tiles, templates, and a probabilistic cellular automaton rule with asynchronous updating. The authors define a loop path condition requiring each path cell to have exactly two orthogonal one-neighbors and extra zeroes on convex corners, ensuring separation by a hull of zero-cells.

In discussing update schemes and rule types, the authors consider four combinations: synchronous/asynchronous and deterministic/probabilistic. While their approach succeeds using asynchronous probabilistic rules, they state that no synchronous deterministic rule has been found to produce loop patterns and explicitly identify the existence of such a rule as an open question.