Effect of Noise on Long-term Memory in Cellular Automata with Asynchronous Delays between the Processors

Abst ract. We consider monotonic binary cellular automa ta on the lat tice ZP as models of systems capable of long-term memory, that is, capable of admitting multiple invariant configurat ions. Long-term memory in cellular aut omata can be robust in t he presence of noise, so that in a noisy environment the automaton may admit more than one stationary distribution on configurations. We examine the effects of asyn chronous communication delays between the processors on longterm memory in cellular automata and describe when asynchronism can cause the erasure of long-term memory in the presence of noise. Our main result is a simple generalizat ion to the asynchronous computation model of a deep result of Toom charac terizing th e invariant configurat ions of monoto nic binary cellular aut omata tha t are robust to noise. Several qualitat ive consequences of asynchronism are illustr ated through examples.