162 research outputs found

### Computing by Temporal Order: Asynchronous Cellular Automata

Our concern is the behaviour of the elementary cellular automata with state
set 0,1 over the cell set Z/nZ (one-dimensional finite wrap-around case), under
all possible update rules (asynchronicity).
Over the torus Z/nZ (n<= 11),we will see that the ECA with Wolfram rule 57
maps any v in F_2^n to any w in F_2^n, varying the update rule.
We furthermore show that all even (element of the alternating group)
bijective functions on the set F_2^n = 0,...,2^n-1, can be computed by ECA57,
by iterating it a sufficient number of times with varying update rules, at
least for n <= 10. We characterize the non-bijective functions computable by
asynchronous rules.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

### On symmetric sandpiles

A symmetric version of the well-known SPM model for sandpiles is introduced.
We prove that the new model has fixed point dynamics. Although there might be
several fixed points, a precise description of the fixed points is given.
Moreover, we provide a simple closed formula for counting the number of fixed
points originated by initial conditions made of a single column of grains.Comment: Will be presented at ACRI2006 conferenc

### Sofic Trace of a Cellular Automaton

The trace subshift of a cellular automaton is the subshift of all possible
columns that may appear in a space-time diagram, ie the infinite sequence of
states of a particular cell of a configuration; in the language of symbolic
dynamics one says that it is a factor system. In this paper we study conditions
for a sofic subshift to be the trace of a cellular automaton.Comment: 10 pages + 6 for included proof

### Local Rules for Computable Planar Tilings

Aperiodic tilings are non-periodic tilings characterized by local
constraints. They play a key role in the proof of the undecidability of the
domino problem (1964) and naturally model quasicrystals (discovered in 1982). A
central question is to characterize, among a class of non-periodic tilings, the
aperiodic ones. In this paper, we answer this question for the well-studied
class of non-periodic tilings obtained by digitizing irrational vector spaces.
Namely, we prove that such tilings are aperiodic if and only if the digitized
vector spaces are computable.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

### Ultimate Traces of Cellular Automata

A cellular automaton (CA) is a parallel synchronous computing model, which
consists in a juxtaposition of finite automata (cells) whose state evolves
according to that of their neighbors. Its trace is the set of infinite words
representing the sequence of states taken by some particular cell. In this
paper we study the ultimate trace of CA and partial CA (a CA restricted to a
particular subshift). The ultimate trace is the trace observed after a long
time run of the CA. We give sufficient conditions for a set of infinite words
to be the trace of some CA and prove the undecidability of all properties over
traces that are stable by ultimate coincidence.Comment: 12 pages + 5 of appendix conference STACS'1

### 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

### Exhaustive Generation of Linear Orthogonal Cellular Automata

We consider the problem of exhaustively visiting all pairs of linear cellular automata which give rise to orthogonal Latin squares, i.e., linear Orthogonal Cellular Automata (OCA). The problem is equivalent to enumerating all pairs of coprime polynomials over a finite field having the same degree and a nonzero constant term. While previous research showed how to count all such pairs for a given degree and order of the finite field, no practical enumeration algorithms have been proposed so far. Here, we start closing this gap by addressing the case of polynomials defined over the field \F_2, which corresponds to binary CA. In particular, we exploit Benjamin and Bennett's bijection between coprime and non-coprime pairs of polynomials, which enables us to organize our study along three subproblems, namely the enumeration and count of: (1) sequences of constant terms, (2) sequences of degrees, and (3) sequences of intermediate terms. In the course of this investigation, we unveil interesting connections with algebraic language theory and combinatorics, obtaining an enumeration algorithm and an alternative derivation of the counting formula for this problem

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