Complete Simulation of Automata Networks
Florian Bridoux, Alonso Castillo-Ramirez, Maximilien Gadouleau

TL;DR
This paper explores the concept of complete simulation in automata networks, establishing bounds on the size and time of transformations needed to simulate any finite automaton, and compares parallel and sequential update models.
Contribution
It introduces the notion of $n$-complete transformations, proves existence bounds, and analyzes their simulation time and parallelization capabilities.
Findings
No $n$-complete transformation of size $n$ exists.
An $n+1$ size transformation can be $n$-complete.
Sequential $n$-complete transformations can update all but one register in parallel.
Abstract
Consider a finite set and an integer . This paper studies the concept of complete simulation in the context of semigroups of transformations of , also known as finite state-homogeneous automata networks. For , a transformation of is \emph{-complete of size } if it may simulate every transformation of by updating one coordinate (or register) at a time. Using tools from memoryless computation, it is established that there is no -complete transformation of size , but there is such a transformation of size . By studying the the time of simulation of various -complete transformations, it is conjectured that the maximal time of simulation of any -complete transformation is at least . A transformation of is \emph{sequentially -complete of size } if it may sequentially simulate every finite sequence of…
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