3,926 research outputs found
Communication Complexity and Intrinsic Universality in Cellular Automata
The notions of universality and completeness are central in the theories of
computation and computational complexity. However, proving lower bounds and
necessary conditions remains hard in most of the cases. In this article, we
introduce necessary conditions for a cellular automaton to be "universal",
according to a precise notion of simulation, related both to the dynamics of
cellular automata and to their computational power. This notion of simulation
relies on simple operations of space-time rescaling and it is intrinsic to the
model of cellular automata. Intrinsinc universality, the derived notion, is
stronger than Turing universality, but more uniform, and easier to define and
study. Our approach builds upon the notion of communication complexity, which
was primarily designed to study parallel programs, and thus is, as we show in
this article, particulary well suited to the study of cellular automata: it
allowed to show, by studying natural problems on the dynamics of cellular
automata, that several classes of cellular automata, as well as many natural
(elementary) examples, could not be intrinsically universal
Amalgams of inverse semigroups and reversible two-counter machines
We show that the word problem for an amalgam
of inverse semigroups may be undecidable even if we assume and (and
therefore ) to have finite -classes and to
be computable functions, interrupting a series of positive decidability results
on the subject. This is achieved by encoding into an appropriate amalgam of
inverse semigroups 2-counter machines with sufficient universality, and
relating the nature of certain \sch graphs to sequences of computations in the
machine
Intrinsically universal one-dimensional quantum cellular automata in two flavours
We give a one-dimensional quantum cellular automaton (QCA) capable of
simulating all others. By this we mean that the initial configuration and the
local transition rule of any one-dimensional QCA can be encoded within the
initial configuration of the universal QCA. Several steps of the universal QCA
will then correspond to one step of the simulated QCA. The simulation preserves
the topology in the sense that each cell of the simulated QCA is encoded as a
group of adjacent cells in the universal QCA. The encoding is linear and hence
does not carry any of the cost of the computation. We do this in two flavours:
a weak one which requires an infinite but periodic initial configuration and a
strong one which needs only a finite initial configuration. KEYWORDS: Quantum
cellular automata, Intrinsic universality, Quantum computation.Comment: 27 pages, revtex, 23 figures. V3: The results of V1-V2 are better
explained and formalized, and a novel result about intrinsic universality
with only finite initial configurations is give
Bulking II: Classifications of Cellular Automata
This paper is the second part of a series of two papers dealing with bulking:
a way to define quasi-order on cellular automata by comparing space-time
diagrams up to rescaling. In the present paper, we introduce three notions of
simulation between cellular automata and study the quasi-order structures
induced by these simulation relations on the whole set of cellular automata.
Various aspects of these quasi-orders are considered (induced equivalence
relations, maximum elements, induced orders, etc) providing several formal
tools allowing to classify cellular automata
Universalities in cellular automata; a (short) survey
This reading guide aims to provide the reader with an easy access to the study of universality in the field of cellular automata. To fulfill this goal, the approach taken here is organized in three parts: a detailled chronology of seminal papers, a discussion of the definition and main properties of universal cellular automata, and a broad bibliography
A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications
Cellular automata (CAs) are dynamical systems which exhibit complex global
behavior from simple local interaction and computation. Since the inception of
cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention
of several researchers over various backgrounds and fields for modelling
different physical, natural as well as real-life phenomena. Classically, CAs
are uniform. However, non-uniformity has also been introduced in update
pattern, lattice structure, neighborhood dependency and local rule. In this
survey, we tour to the various types of CAs introduced till date, the different
characterization tools, the global behaviors of CAs, like universality,
reversibility, dynamics etc. Special attention is given to non-uniformity in
CAs and especially to non-uniform elementary CAs, which have been very useful
in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin
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