January 25, 2021

Tien Thao Nguyen (LACL)

We revisit the notion of local simulation of a Cellular Automaton (CA), which allows to transform a cellular automaton into a closely related one with different local encoding of information.

This notion is useful to explore the solution space of many CA problems, in particular we investigate two well studied one dimensional CA problems, namely the Firing Squad Synchronization Problem and the Sequence Generator problem.

1. For the Firing Squad Synchronization Problem we exhibit many solutions that are minimal both in time (2n-2 for n cells) and, up to current knowledge, also in states (6 states). This improves from the solution proposed by Mazoyer in 1987; and more recently the 718 new solutions generated by Clergue, Verel and Formenti in 2018 with a cluster of machines.
Using local simulations, we show how to generate millions of such 6-state solutions.

2. The Sequence Generator problem on the CA model has been studied since at least 1960 and numerous generation algorithms have been proposed for a variety of non-regular sequences such as {2^n | n = 1, 2, 3, …}, the primes, Fibonacci sequences etc.
Using the 6-state solution of Kamikawa and Umeo of read-time sequence generator for the cubes {n^3 | n = 1, 2, 3, …}, we handcrafted a 5-state reduction for this solution and then showed how to generate several million of solutions from this 5-state solution.
It is notable that in contrast to the current ubiquity of cloud computing, we used only a modest personal computer thanks to the nice algorithmic properties of local simulations.