ON PARALLEL vs. SEQUENTIAL THRESHOLD CELLULAR AUTOMATA

Cellular automata (CA) are an abstract model of fine-grain parallelism, as the node update operations are rather simple, and therefore comparable to the basic operations of the computer hardware. In a classical CA, all the nodes execute their operations in parallel and in perfect synchrony. We consider herewith the sequential version of CA, called SCA, and compare these SCA with the classical, parallel CA. In particular, we show that there are 1-D CA with very simple node state update rules that cannot be simulated by any comparable SCA, irrespective of the node update ordering. Consequently, the fine granularity of the basic CA operations and, therefore, the fine-grain parallelism of the classical, synchronous CA, insofar as the “interleaving semantics” is concerned, is not fine enough. We also share some thoughts on how to extend the results herein, and, in particular, we try to motivate the study of genuinely asynchronous cellular automata.