A perfect solution to the parity problem with elementary cellular automaton 150 under asynchronous update

Abstract

Cellular automata are locally-defined, fully discrete dynamical systems. One of the core questions in their study is whether or not they are capable of answering a global question about a configuration, by performing only local computations. Such decision problems include classical ones, such as the density classification task and the parity problem. Traditionally, all cells of a cellular automaton are updated synchronously. However, the possibility of allowing the updates to be asynchronous has raised an increasing interest. In [1], a solution to the parity problem – the determination of the parity of 1s in an arbitrary binary string – was given in terms of a synchronous one-dimensional cellular automaton with 9 neighbours. Here, we present a simpler solution to the same problem, by means of an elementary rule – binary, one-dimensional, with 3 neighbours – that works by asynchronously updating the even and odd positions of the lattice, alternately; we also precisely characterise how many times the rule must be iterated in order for the problem to be solved. Interestingly, our solution relies upon the elementary local parity rule, which apparently represents the first case of a rule able to solve a non-trivial problem both locally and globally.