A guided tour of asynchronous cellular automata

Research on asynchronous cellular automata has received a great amount of attention these last years and has turned to a thriving field. We survey the recent research that has been carried out on this topic and present a wide state of the art where computing and modelling issues are both represented.Comment: To appear in the Journal of Cellular Automat

Deterministic computations whose history is independent of the order of asynchronous updating

Consider a network of processors (sites) in which each site x has a finite set N(x) of neighbors. There is a transition function f that for each site x computes the next state \xi(x) from the states in N(x). But these transitions (updates) are applied in arbitrary order, one or many at a time. If the state of site x at time t is \eta(x,t) then let us define the sequence \zeta(x,0), \zeta(x,1), ... by taking the sequence \eta(x,0), \eta(x,1), ..., and deleting repetitions. The function f is said to have invariant histories if the sequence \zeta(x,i), (while it lasts, in case it is finite) depends only on the initial configuration, not on the order of updates. This paper shows that though the invariant history property is typically undecidable, there is a useful simple sufficient condition, called commutativity: For any configuration, for any pair x,y of neighbors, if the updating would change both \xi(x) and \xi(y) then the result of updating first x and then y is the same as the result of doing this in the reverse order

A Universal Semi-totalistic Cellular Automaton on Kite and Dart Penrose Tilings

In this paper we investigate certain properties of semi-totalistic cellular automata (CA) on the well known quasi-periodic kite and dart two dimensional tiling of the plane presented by Roger Penrose. We show that, despite the irregularity of the underlying grid, it is possible to devise a semi-totalistic CA capable of simulating any boolean circuit on this aperiodic tiling.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

Deterministic Computations Whose History Is Independent of the Order of Asynchronous Updating

Consider a network of processors (sites) in which each site x has a finite set N(x) of neighbors. There is a transition function f that for each site x computes the next state ξ(x) from the states in N(x). But these transitions (updates) are applied in arbitrary order, one or many at a time. If the state of site x at time t is η(x; t) then let us define the sequence ζ(x; 0); ζ(x; 1), ... by taking the sequence η(x; 0),η(x; 1), ... , and deleting each repetition, i.e. each element equal to the preceding one. The function f is said to have invariant histories if the sequence ζ(x; i), (while it lasts, in case it is finite) depends only on the initial configuration, not on the order of updates. This paper shows that though the invariant history property is typically undecidable, there is a useful simple sufficient condition, called commutativity: For any configuration, for any pair x; y of neighbors, if the updating would change both ξ(x) and ξ(y) then the result of updating first x and then y is the same as the result of doing this in the reverse order. This fact is derivable from known results on the confluence of term-rewriting systems but the self-contained proof given here may be justifiable.National Science Foundation (CCR-920484

Implicit Simulations using Messaging Protocols

A novel algorithm for performing parallel, distributed computer simulations on the Internet using IP control messages is introduced. The algorithm employs carefully constructed ICMP packets which enable the required computations to be completed as part of the standard IP communication protocol. After providing a detailed description of the algorithm, experimental applications in the areas of stochastic neural networks and deterministic cellular automata are discussed. As an example of the algorithms potential power, a simulation of a deterministic cellular automaton involving 10^5 Internet connected devices was performed.Comment: 14 pages, 3 figure

Asynchronism Induces Second Order Phase Transitions in Elementary Cellular Automata

Cellular automata are widely used to model natural or artificial systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the synchrony rate. For some particular rules, varying the synchrony rate continuously produces a qualitative change in the behaviour of the cellular automaton. We investigate the nature of this change of behaviour using Monte-Carlo simulations. We show that this phenomenon is a second-order phase transition, which we characterise more specifically as belonging to the directed percolation or to the parity conservation universality classes studied in statistical physics

Computational Aspects of Asynchronous CA

This work studies some aspects of the computational power of fully asynchronous cellular automata (ACA). We deal with some notions of simulation between ACA and Turing Machines. In particular, we characterize the updating sequences specifying which are "universal", i.e., allowing a (specific family of) ACA to simulate any TM on any input. We also consider the computational cost of such simulations