1,424 research outputs found
Critical droplets in Metastable States of Probabilistic Cellular Automata
We consider the problem of metastability in a probabilistic cellular
automaton (PCA) with a parallel updating rule which is reversible with respect
to a Gibbs measure. The dynamical rules contain two parameters and
which resemble, but are not identical to, the inverse temperature and external
magnetic field in a ferromagnetic Ising model; in particular, the phase diagram
of the system has two stable phases when is large enough and is
zero, and a unique phase when is nonzero. When the system evolves, at small
positive values of , from an initial state with all spins down, the PCA
dynamics give rise to a transition from a metastable to a stable phase when a
droplet of the favored phase inside the metastable phase reaches a
critical size. We give heuristic arguments to estimate the critical size in the
limit of zero ``temperature'' (), as well as estimates of the
time required for the formation of such a droplet in a finite system. Monte
Carlo simulations give results in good agreement with the theoretical
predictions.Comment: 5 LaTeX picture
Quasi-Linear Cellular Automata
Simulating a cellular automaton (CA) for t time-steps into the future
requires t^2 serial computation steps or t parallel ones. However, certain CAs
based on an Abelian group, such as addition mod 2, are termed ``linear''
because they obey a principle of superposition. This allows them to be
predicted efficiently, in serial time O(t) or O(log t) in parallel.
In this paper, we generalize this by looking at CAs with a variety of
algebraic structures, including quasigroups, non-Abelian groups, Steiner
systems, and others. We show that in many cases, an efficient algorithm exists
even though these CAs are not linear in the previous sense; we term them
``quasilinear.'' We find examples which can be predicted in serial time
proportional to t, t log t, t log^2 t, and t^a for a < 2, and parallel time log
t, log t log log t and log^2 t.
We also discuss what algebraic properties are required or implied by the
existence of scaling relations and principles of superposition, and exhibit
several novel ``vector-valued'' CAs.Comment: 41 pages with figures, To appear in Physica
State-deterministic Finite Automata with Translucent Letters and Finite Automata with Nondeterministically Translucent Letters
Deterministic and nondeterministic finite automata with translucent letters
were introduced by Nagy and Otto more than a decade ago as Cooperative
Distributed systems of a kind of stateless restarting automata with window size
one. These finite state machines have a surprisingly large expressive power:
all commutative semi-linear languages and all rational trace languages can be
accepted by them including various not context-free languages. While the
nondeterministic variant defines a language class with nice closure properties,
the deterministic variant is weaker, however it contains all regular languages,
some non-regular context-free languages, as the Dyck language, and also some
languages that are not even context-free. In all those models for each state,
the letters of the alphabet could be in one of the following categories: the
automaton cannot see the letter (it is translucent), there is a transition
defined on the letter (maybe more than one transitions in nondeterministic
case) or none of the above categories (the automaton gets stuck by seeing this
letter at the given state and this computation is not accepting).
State-deterministic automata are recent models, where the next state of the
computation determined by the structure of the automata and it is independent
of the processed letters. In this paper our aim is twofold, on the one hand, we
investigate state-deterministic finite automata with translucent letters. These
automata are specially restricted deterministic finite automata with
translucent letters.
In the other novel model we present, it is allowed that for a state the set
of translucent letters and the set of letters for which transition is defined
are not disjoint. One can interpret this fact that the automaton has a
nondeterministic choice for each occurrence of such letters to see them (and
then erase and make the transition) or not to see that occurrence at that time.
Based on these semi-translucent letters, the expressive power of the automata
increases, i.e., in this way a proper generalization of the previous models is
obtained.Comment: In Proceedings AFL 2023, arXiv:2309.0112
Beyond Generalized Multiplicities: Register Machines over Groups
Register machines are a classic model of computing, often seen as a canonical
example of a device manipulating natural numbers. In this paper, we de ne register
machines operating on general groups instead. This generalization follows the research
direction started in multiple previous works. We study the expressive power of register
machines as a function of the underlying groups, as well as of allowed ingredients (zero
test, partial blindness, forbidden regions). We put forward a fundamental connection
between register machines and vector addition systems. Finally, we show how registers
over free groups can be used to store and manipulate strings
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