Universality of Hexagonal Asynchronous Totalistic Cellular Automata

Abstract

There is increasing interest in cellular automata that update their cells asynchronously, i.e., at random times and independent of each other. Research, however, has been limited to either models of trivial phenomena, or models that require a global synchronization mechanism to induce nontrivial phenomena like computation. This paper employs local synchronization, a technique in which particular temporal sequences of states are imposed locally on cells and their direct neighbors, while the exact timing of state transitions is left undetermined. A hexagonal asynchronous totalistic cellular automaton is presented that achieves a completely asynchronous way of computation by simulating delay-insensitive circuits, a type of asynchronous circuits that are known for their robustness to variations in the timing of signals. We implement three primitive operators on the cellular automaton from which any arbitrary delay-insensitive circuit can be constructed, and show how to connect the operators such that collisions of crossing signals are avoided. The model requires six states and 55 totalistic transition rules.