1,433 research outputs found
Cellular Automata as a Model of Physical Systems
Cellular Automata (CA), as they are presented in the literature, are abstract
mathematical models of computation. In this pa- per we present an alternate
approach: using the CA as a model or theory of physical systems and devices.
While this approach abstracts away all details of the underlying physical
system, it remains faithful to the fact that there is an underlying physical
reality which it describes. This imposes certain restrictions on the types of
computations a CA can physically carry out, and the resources it needs to do
so. In this paper we explore these and other consequences of our
reformalization.Comment: To appear in the Proceedings of AUTOMATA 200
Basic Ideas to Approach Metastability in Probabilistic Cellular Automata
Cellular Automata are discrete--time dynamical systems on a spatially
extended discrete space which provide paradigmatic examples of nonlinear
phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular
Automata, are discrete time Markov chains on lattice with finite single--cell
states whose distinguishing feature is the \textit{parallel} character of the
updating rule. We review some of the results obtained about the metastable
behavior of Probabilistic Cellular Automata and we try to point out
difficulties and peculiarities with respect to standard Statistical Mechanics
Lattice models.Comment: arXiv admin note: text overlap with arXiv:1307.823
Index theory of one dimensional quantum walks and cellular automata
If a one-dimensional quantum lattice system is subject to one step of a
reversible discrete-time dynamics, it is intuitive that as much "quantum
information" as moves into any given block of cells from the left, has to exit
that block to the right. For two types of such systems - namely quantum walks
and cellular automata - we make this intuition precise by defining an index, a
quantity that measures the "net flow of quantum information" through the
system. The index supplies a complete characterization of two properties of the
discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the
sense that there is a system S which locally acts like S_1 in one region and
like S_2 in some other region, if and only if S_1 and S_2 have the same index.
Second, the index labels connected components of such systems: equality of the
index is necessary and sufficient for the existence of a continuous deformation
of S_1 into S_2. In the case of quantum walks, the index is integer-valued,
whereas for cellular automata, it takes values in the group of positive
rationals. In both cases, the map S -> ind S is a group homomorphism if
composition of the discrete dynamics is taken as the group law of the quantum
systems. Systems with trivial index are precisely those which can be realized
by partitioned unitaries, and the prototypes of systems with non-trivial index
are shifts.Comment: 38 pages. v2: added examples, terminology clarifie
Universality and Decidability of Number-Conserving Cellular Automata
Number-conserving cellular automata (NCCA) are particularly interesting, both
because of their natural appearance as models of real systems, and because of
the strong restrictions that number-conservation implies. Here we extend the
definition of the property to include cellular automata with any set of states
in \Zset, and show that they can be always extended to ``usual'' NCCA with
contiguous states. We show a way to simulate any one dimensional CA through a
one dimensional NCCA, proving the existence of intrinsically universal NCCA.
Finally, we give an algorithm to decide, given a CA, if its states can be
labeled with integers to produce a NCCA, and to find this relabeling if the
answer is positive.Comment: 13 page
- …