How Can We Construct Reversible Turing Machines in a Very Simple Reversible Cellular Automaton?

2021 ◽  
pp. 3-21
Author(s):  
Kenichi Morita
Keyword(s):  
Cellular Automaton ◽  
Turing Machines ◽  
Reversible Cellular Automaton

scholarly journals Two-species hardcore reversible cellular automaton: matrix ansatz for dynamics and nonequilibrium stationary state

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Marko Medenjak ◽  
Vladislav Popkov ◽  
Tomaz Prosen ◽  
Eric Ragoucy ◽  
Matthieu Vanicat
Keyword(s):  
Cellular Automaton ◽  
Time Evolution ◽  
Stationary State ◽  
Finite Lattice ◽  
Matrix Product ◽  
Density Profiles ◽  
Reversible Cellular Automaton ◽  
Product Expression ◽  
Particle Current ◽  
Exact Matrix

In this paper we study the statistical properties of a reversible cellular automaton in two out-of-equilibrium settings. In the first part we consider two instances of the initial value problem, corresponding to the inhomogeneous quench and the local quench. Our main result is an exact matrix product expression of the time evolution of the probability distribution, which we use to determine the time evolution of the density profiles analytically. In the second part we study the model on a finite lattice coupled with stochastic boundaries. Once again we derive an exact matrix product expression of the stationary distribution, as well as the particle current and density profiles in the stationary state. The exact expressions reveal the existence of different phases with either ballistic or diffusive transport depending on the boundary parameters.

CHARACTERIZING THE ABILITY OF PARALLEL ARRAY GENERATORS ON REVERSIBLE PARTITIONED CELLULAR AUTOMATA

1999 ◽  
Vol 13 (04) ◽  
pp. 523-538 ◽  
Author(s):  
KENICHI MORITA ◽  
SATOSHI UENO ◽  
KATSUNOBU IMAI
Keyword(s):  
Pattern Recognition ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Inverse System ◽  
Two Dimensional ◽  
Turing Machines ◽  
Recognition Mechanism ◽  
Parallel Array ◽  
Parallel Pattern ◽  
Monotonic Constraint

A PCAAG introduced by Morita and Ueno is a parallel array generator on a partitioned cellular automaton (PCA) that generates an array language (i.e. a set of symbol arrays). A "reversible" PCAAG (RPCAAG) is a backward deterministic PCAAG, and thus parsing of two-dimensional patterns can be performed without backtracking by an "inverse" system of the RPCAAG. Hence, a parallel pattern recognition mechanism on a deterministic cellular automaton can be directly obtained from a RPCAAG that generates the pattern set. In this paper, we investigate the generating ability of RPCAAGs and their subclass. It is shown that the ability of RPCAAGs is characterized by two-dimensional deterministic Turing machines, i.e. they are universal in their generating ability. We then investigate a monotonic RPCAAG (MRPCAAG), which is a special type of an RPCAAG that satisfies monotonic constraint. We show that the generating ability of MRPCAAGs is exactly characterized by two-dimensional deterministic linear-bounded automata.

scholarly journals Representing Reversible Cellular Automata with Reversible Block Cellular Automata

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Jérôme Durand-Lose
Keyword(s):  
System Theory ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Linear Time ◽  
Fixed Size ◽  
Mathematical System ◽  
Reversible Cellular Automaton ◽  
International Audience ◽  
Infinite Lattices ◽  
Reversible Cellular Automata

International audience Cellular automata are mappings over infinite lattices such that each cell is updated according tothe states around it and a unique local function.Block permutations are mappings that generalize a given permutation of blocks (finite arrays of fixed size) to a given partition of the lattice in blocks.We prove that any d-dimensional reversible cellular automaton can be exp ressed as thecomposition of d+1 block permutations.We built a simulation in linear time of reversible cellular automata by reversible block cellular automata (also known as partitioning CA and CA with the Margolus neighborhood) which is valid for both finite and infinite configurations. This proves a 1990 conjecture by Toffoli and Margolus <i>(Physica D 45)</i> improved by Kari in 1996 <i>(Mathematical System Theory 29)</i>.

scholarly journals Glider automata on all transitive sofic shifts

2021 ◽  
pp. 1-29
Author(s):  
JOHAN KOPRA
Keyword(s):  
Automorphism Group ◽  
Cellular Automaton ◽  
Finite Collection ◽  
Finite Point ◽  
Element Subset ◽  
Sofic Shift ◽  
Sofic Shifts ◽  
Reversible Cellular Automaton

Abstract For any infinite transitive sofic shift X we construct a reversible cellular automaton (that is, an automorphism of the shift X) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. This shows in addition that every infinite transitive sofic shift has a reversible cellular automaton which is sensitive with respect to all directions. As another application we prove a finitary version of Ryan’s theorem: the automorphism group $\operatorname {\mathrm {Aut}}(X)$ contains a two-element subset whose centralizer consists only of shift maps. We also show that in the class of S-gap shifts these results do not extend beyond the sofic case.

HOW TO SIMULATE TURING MACHINES BY INVERTIBLE ONE-DIMENSIONAL CELLULAR AUTOMATA

1995 ◽  
Vol 06 (04) ◽  
pp. 395-402 ◽  
Author(s):  
JEAN-CHRISTOPHE DUBACQ
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Turing Machine ◽  
Turing Machines ◽  
One Dimensional

The issue of testing invertibility of cellular automata has been often discussed. Constructing invertible automata is very useful for simulating invertible dynamical systems, based on local rules. The computation universality of cellular automata has long been positively resolved, and by showing that any cellular automaton could be simulated by an invertible one having a superior dimension, Toffoli proved that invertible cellular automaton of dimension d≥2 were computation-universal. Morita proved that any invertible Turing Machine could be simulated by a one-dimensional invertible cellular automaton, which proved computation-universality of invertible cellular automata. This article shows how to simulate any Turing Machine by an invertible cellular automaton with no loss of time and gives, as a corollary, an easier proof of this result.

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