We show that any locally finite automata network with global synchronous updates can be emulated by another one , whose structure derives from that of by a simple construction, but whose updates are made asynchronously at its various component automata (e.g. possibly randomly or sequentially, with or without possible simultaneous updates at different nodes). By "emulation", we refer to the existence of a spatial-temporal covering 'local time', allowing one to project the behavior of continuously onto that of . We also show the existence of a spatial-temporal section of the asynchronous automata network's behavior which completely determines the synchronous global state of at every time step. We give the construction of the asynchronous automata network, establish its freedom from deadlocks, and construct local time functions and spatial-temporal sections relating any posssible behavior of to the single corresponding behavior of on a given input sequence starting from a given initial global state. This establishes that the behavior of any locally finite synchronous automata network actually can be emulated without the restriction of synchronous update, freeing us from the need of a global clock signal. Local information is sufficient to guarantee that the synchronous behavior of is completely determined by any asynchronous behavior of starting from a corresponding global state and given the same input sequence as . Moreover, the relative passage of corresponding local time at any two nodes in is bounded in a simple way by approximately one-third of the distance between them. As corollaries, any synchronous generalized cellular automaton or synchronous cellular automaton can be emulated by an asynchronous one of the same type. Implementation aspects of these asynchronous automata are also discussed, and open problems and research directions are indicated.

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Asynchronous automata networks can emulate any synchronous automata network

A new elementary and direct proof is given to show a consequence of the Dénes–Hermann Theorem.

On the product of all elements in a finite group.

Computation on Finite Networks of Automata

Complete simulation of automata networks

Consider a finite set $A$ and an integer $n \geq 1$. This paper studies the concept of complete simulation in the context of semigroups of transformations of $A^n$, also known as finite state-homogeneous automata networks. For $m \geq n$, a transformation of $A^m$ is \emph{$n$-complete of size $m$} if it may simulate every transformation of $A^n$ by updating one coordinate (or register) at a time. Using tools from memoryless computation, it is established that there is no $n$-complete transformation of size $n$, but there is such a transformation of size $n+1$. By studying the the time of simulation of various $n$-complete transformations, it is conjectured that the maximal time of simulation of any $n$-complete transformation is at least $2n$. A transformation of $A^m$ is \emph{sequentially $n$-complete of size $m$} if it may sequentially simulate every finite sequence of transformations of $A^n$; in this case, minimal examples and bounds for the size and time of simulation are determined. It is also shown that there is no $n$-complete transformation that updates all the registers in parallel, but that there exists a sequentally $n$-complete transformation that updates all but one register in parallel. This illustrates the strengths and weaknesses of parallel models of computation, such as cellular automata.

Complete Simulation of Automata Networks

In this paper it is shown that, in the worst case, the delay necessary to permute boolean variables on a star-connected network of n binary automata is $n - 1$.

Maurice Tchuente

Papers

Permutation Factorization on Star-Connected Networks of Binary Automata