250 research outputs found
Cellular automaton supercolliders
Gliders in one-dimensional cellular automata are compact groups of
non-quiescent and non-ether patterns (ether represents a periodic background)
translating along automaton lattice. They are cellular-automaton analogous of
localizations or quasi-local collective excitations travelling in a spatially
extended non-linear medium. They can be considered as binary strings or symbols
travelling along a one-dimensional ring, interacting with each other and
changing their states, or symbolic values, as a result of interactions. We
analyse what types of interaction occur between gliders travelling on a
cellular automaton `cyclotron' and build a catalog of the most common
reactions. We demonstrate that collisions between gliders emulate the basic
types of interaction that occur between localizations in non-linear media:
fusion, elastic collision, and soliton-like collision. Computational outcomes
of a swarm of gliders circling on a one-dimensional torus are analysed via
implementation of cyclic tag systems
Soliton Cellular Automata Associated With Crystal Bases
We introduce a class of cellular automata associated with crystals of
irreducible finite dimensional representations of quantum affine algebras
U'_q(\hat{\geh}_n). They have solitons labeled by crystals of the smaller
algebra U'_q(\hat{\geh}_{n-1}). We prove stable propagation of one soliton for
\hat{\geh}_n = A^{(2)}_{2n-1}, A^{(2)}_{2n}, B^{(1)}_n, C^{(1)}_n, D^{(1)}_n
and D^{(2)}_{n+1}. For \gh_n = C^{(1)}_n, we also prove that the scattering
matrices of two solitons coincide with the combinatorial R matrices of
U'_q(C^{(1)}_{n-1})-crystals.Comment: 29 pages, 1 figure, LaTeX2
Upper Bound on the Products of Particle Interactions in Cellular Automata
Particle-like objects are observed to propagate and interact in many
spatially extended dynamical systems. For one of the simplest classes of such
systems, one-dimensional cellular automata, we establish a rigorous upper bound
on the number of distinct products that these interactions can generate. The
upper bound is controlled by the structural complexity of the interacting
particles---a quantity which is defined here and which measures the amount of
spatio-temporal information that a particle stores. Along the way we establish
a number of properties of domains and particles that follow from the
computational mechanics analysis of cellular automata; thereby elucidating why
that approach is of general utility. The upper bound is tested against several
relatively complex domain-particle cellular automata and found to be tight.Comment: 17 pages, 12 figures, 3 tables,
http://www.santafe.edu/projects/CompMech/papers/ub.html V2: References and
accompanying text modified, to comply with legal demands arising from
on-going intellectual property litigation among third parties. V3: Accepted
for publication in Physica D. References added and other small changes made
per referee suggestion
Analysis of a particle antiparticle description of a soliton cellular automaton
We present a derivation of a formula that gives dynamics of an integrable
cellular automaton associated with crystal bases. This automaton is related to
type D affine Lie algebra and contains usual box-ball systems as a special
case. The dynamics is described by means of such objects as carriers,
particles, and antiparticles. We derive it from an analysis of a recently
obtained formula of the combinatorial R (an intertwiner between tensor products
of crystals) that was found in a study of geometric crystals.Comment: LaTeX, 21 pages, 2 figure
Elementary decomposition of soliton automata
Soliton automata are the mathematical models of certain possible molecular switching devices. In this paper we work out a decomposition of soliton automata through the structure of their underlying graphs. These results lead to the original aim, to give a characterization of soliton automata in general case
Tropical R and Tau Functions
Tropical R is the birational map that intertwines products of geometric
crystals and satisfies the Yang-Baxter equation. We show that the D^{(1)}_n
tropical R introduced by the authors and its reduction to A^{(2)}_{2n-1} and
C^{(1)}_n are equivalent to a system of bilinear difference equations of Hirota
type. Associated tropical vertex models admit solutions in terms of tau
functions of the BKP and DKP hierarchies.Comment: LaTeX2e, 26page
Deterministic soliton automata with at most one cycle
AbstractSoliton valves have been proposed as molecular switching elements. Their mathematical model is the soliton graph and the soliton automaton (Dassow and Jürgensen, J. Comput. System Sci.40 (1990), 154–181). In this paper we continue the study of the logic aspects of soliton switching. There are two cases of special importance: those of deterministic and those of strongly deterministic soliton automata. The former have deterministic state transitions in the usual sense of automaton theory. The latter do not only have deterministic state transitions, but also deterministic soliton paths—a much stronger property, as it turns out. In op cit. a characterization of indecomposable, strongly deterministic soliton automata was proved and it was shown that their transition monoids are primitive groups of permutations. Roughly speaking, the main difference between deterministic and strongly deterministic soliton automata is that in the former the underlying soliton graphs may contain cycles of odd lengths while such cycles are not permitted in the soliton graphs belonging to strongly deterministic soliton automata. In the present paper, we focus on a special class of deterministic soliton automata, that of deterministic soliton automata whose underlying graphs contain at most one cycle. For this class we derive structural descriptions. Our main results concern the elimination of certain types of loops, the treatment of soliton paths with repeated edges, the structure of cycles of odd length, and the transition monoid. As an application we show that the memory element proposed in the literature (Carter, in Bioelectronics, edited by Aizawa, Research and Development Report 50, CMC Press, Denver, CO, 1984) can be transformed in into a soliton tree, thus turning a deterministic device into a logically equivalent strongly deterministic device
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