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Encyclopedia of Complexity and Systems Science pp 7679–7695Cite as

Reversible Cellular Automata

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Definition of the Subject

Reversible cellular automata (RCAs) are defined as cellularautomata (CAs) with an injective global function. Every configurationof an RCA has exactly one previous configuration, and thus RCAs are“backward deterministic” CAs. The notion of reversibility originally comes from physics. It is one of the fundamentalmicroscopic physical laws of Nature. In this sense, an RCA is thoughtas an abstract model of a physically reversible space as well asa computing model. It is very important to investigate howcomputation can be carried out efficiently and elegantly ina system having reversibility. This is because future computingdevices will surely become those of a nanoscale size.

In this article, we mainly discuss on the properties of RCAsfrom the computational aspects. In spite of the strong constraint ofreversibility, RCAs have very rich ability of computing. We can seethat even very simple RCAs have universal computing ability. We canalso recognize, in some...

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Abbreviations

Cellular automaton :

A cellular automaton (CA) is a systemconsisting of a large (theoretically, infinite) number of finiteautomata, called cells, which are connected uniformly in a space.Each cell transits its state depending on the states of itself and thecells in its neighborhood. Thus the state transition of a cellis specified by a local function. Applying the local function toall the cells in the space synchronously, the transition ofa configuration (i.?e., a whole state of the cellularspace) is induced. Such a transition function is calleda global function. A CA is regarded as a kind ofdynamical system that can deal with various kinds ofspatio-temporal phenomena.

Cellular automaton with blockrules :

A CA with block rules was proposed byMargolus [18],and it is often called a CA with Margolus-neighborhood.The cellular space is divided into infinitely many blocks of the samesize (in the two-dimensional case, e.?g., \( { 2 \times 2 }\)). A local transitionfunction consisting of “block rules”, which isa mapping from a block state to a block state, isapplied to all the blocks in parallel. At the next time step, theblock division pattern is shifted by some fixed amount (e.?g.,to the north-east direction by one cell), and the same localfunction is applied to them. This model of CA is convenient to designa reversible CA. Because, if the local transition function isinjective, then the resulting CA is reversible.

Partitioned cellularautomaton :

A partitioned cellular automaton (PCA) isa framework for designing a reversible CA. It isa subclass of a usual CA where each cell is partitioned intoseveral parts, whose number is equal to the neighborhood size. Eachpart of a cell has its own state set, and can be regarded as anoutput port to a specified neighboring cell. Depending only onthe corresponding parts (not on the entire states) of the neighboringcells, the next state of each cell is determined by a localfunction. We can see that if the local function is injective, then theresulting PCA is reversible. Hence, a PCA makes it feasible toconstruct a reversible CA.

Reversible cellularautomaton :

A reversible cellular automaton (RCA) isdefined as a one whose global function is injective (i.?e.,one-to-one). It can be regarded as a kind ofa discrete model of reversible physical space. It is in generaldifficult to construct an RCA with a desired property such ascomputation-universality. Therefore, the frameworks ofa CA with Margolus neighborhood, a partitioned cellularautomaton, and others are often used to designRCAs.

Universal cellularautomaton :

A CA is called computationally universal, ifit can compute any recursive function by giving an appropriate initialconfiguration. Equivalently, it is also defined as a CA that cansimulate a universal Turing machine. Universality of RCAs can beproved by simulating other systems such as arbitrary (irreversible)CAs, reversible Turing machines, reversible counter machines, andreversible logic elements and circuits, which have already been knownto be universal.

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Authors and Affiliations

  1. Hiroshima University, Higashi-Hiroshima, Japan

    Kenichi Morita

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  1. Kenichi Morita
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Morita, K. (2009). Reversible Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_455

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