396 research outputs found
On the size of the inverse neighborhoods for one-dimensional reversible cellular automata
AbstractIn this paper we investigate the possible neighborhood size of the inverse automaton of some types of one-dimensional reversible cellular automata. Considering only the case when the local function is a size two map, we give a quadratic upper bound for the neighborhood size of the inverse automaton. We show that this bound can be lowered in some particular cases, and give an algorithm for computing these better bounds
Post-surjectivity and balancedness of cellular automata over groups
We discuss cellular automata over arbitrary finitely generated groups. We
call a cellular automaton post-surjective if for any pair of asymptotic
configurations, every pre-image of one is asymptotic to a pre-image of the
other. The well known dual concept is pre-injectivity: a cellular automaton is
pre-injective if distinct asymptotic configurations have distinct images. We
prove that pre-injective, post-surjective cellular automata are reversible.
Moreover, on sofic groups, post-surjectivity alone implies reversibility. We
also prove that reversible cellular automata over arbitrary groups are
balanced, that is, they preserve the uniform measure on the configuration
space.Comment: 16 pages, 3 figures, LaTeX "dmtcs-episciences" document class. Final
version for Discrete Mathematics and Theoretical Computer Science. Prepared
according to the editor's request
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
Procedures for calculating reversible one-dimensional cellular automata
We describe two algorithms for calculating reversible one-dimensional cellular automata of neighborhood size 2. We explain how this kind of automaton represents all the other cases. Using two basic properties of reversible automata such as uniform multiplicity of ancestors and Welch indices, these algorithms only require matrix products and transitive closures of binary relations to classify all the possible reversible automata of neighborhood size 2. We expose the features, advantages and differences with other well-known methods. Finally, we present results for reversible automata from three to six states and neighborhood size 2. © 2005 Elsevier B.V. All rights reserved
The inverse behavior of a reversible one-dimensional cellular automaton obtained by a single welch diagram
Reversible cellular automata are discrete dynamical systems based on local interactions which are able to produce an invertible global behavior. Reversible automata have been carefully analyzed by means of graph and matrix tools, in particular the extensions of the ancestors in these systems have a complete representation by Welch diagrams. This paper illustrates how the whole information of a reversible one-dimensional cellular automaton is conserved at both sides of the ancestors for sequences with an adequate length. We give this result implementing a procedure to obtain the inverse behavior by means of calculating and studying a single Welch diagram corresponding with the extensions of only one side of the ancestors. This work is a continuation of our study about reversible automata both in the local and global sense. An illustrative example is also presented
Spectral properties of reversible one-dimensional cellular automata
Reversible cellular automata are invertible dynamical systems characterized by discreteness, determinism and local interaction. This article studies the local behavior of reversible one-dimensional cellular automata by means of the spectral properties of their connectivity matrices. We use the transformation of every one-dimensional cellular automaton to another of neighborhood size 2 to generalize the results exposed in this paper. In particular we prove that the connectivity matrices have a single positive eigenvalue equal to 1; based on this result we also prove the idempotent behavior of these matrices. The significance of this property lies in the implementation of a matrix technique for detecting whether a one-dimensional cellular automaton is reversible or not. In particular, we present a procedure using the eigenvectors of these matrices to find the inverse rule of a given reversible one-dimensional cellular automaton. Finally illustrative examples are provided
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