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Reversible Cellular Automata

Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Reversible cellular automata (RCAs) are defined as cellular automata (CAs) with an injective global function. Every configuration of 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 fundamental microscopic physical laws of nature. In this sense, an RCA is thought as an abstract model of a physically reversible space as well as a computing model. It is very important to investigate how computation can be carried out efficiently and elegantly in a system having reversibility. This is because future computing devices will surely become those of a nanoscale size.

In this entry, we mainly discuss on the properties of RCAs from the computational aspects. In spite of the strong constraint of reversibility, RCAs have very rich ability of computing. We can see that even very simple RCAs have universal computing ability. We can also recognize, in...

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Abbreviations

Cellular automaton:

A cellular automaton (CA) is a system consisting of a large (theoretically, infinite) number of finite automata, called cells, which are connected uniformly in a space. Each cell changes its state depending on the states of itself and the cells in its neighborhood. Thus, the state transition of a cell is specified by a local function. Applying the local function to all the cells in the space synchronously, the transition of a configuration (i.e., a whole state of the cellular space) is induced. Such a transition function is called a global function. A CA is regarded as a kind of dynamical system that can deal with various kinds of spatiotemporal phenomena.

Cellular automaton with block rules:

A CA with block rules was proposed by Margolus (1984), and it is often called a CA with Margolus neighborhood. The cellular space is divided into infinitely many blocks of the same size (in the two-dimensional case, e.g., 2 × 2). A local transition function consisting of “block rules,” which is a mapping from a block state to a block state, is applied to all the blocks in parallel. At the next time step, the block division pattern is shifted by some fixed amount (e.g., to the north-east direction by one cell), and the same local function is applied to them. This model of CA is convenient to design a reversible CA, because if the local transition function is injective, then the resulting CA is reversible.

Partitioned cellular automaton:

A partitioned cellular automaton (PCA) is a framework for designing a reversible CA. It is a subclass of a usual CA where each cell is partitioned into several parts, whose number is equal to the neighborhood size. Each part of a cell has its own state set and can be regarded as an output port to a specified neighboring cell. Depending only on the corresponding parts (not on the entire states) of the neighboring cells, the next state of each cell is determined by a local function. We can see that if the local function is injective, then the resulting PCA is reversible. Hence, a PCA makes it feasible to construct a reversible CA.

Reversible cellular automaton:

A reversible cellular automaton (RCA) is defined as the one whose global function is injective (i.e., one-to-one). It can be regarded as a kind of a discrete model of reversible physical space. It is in general difficult to construct an RCA with a desired property such as computational universality. Therefore, the frameworks of a CA with Margolus neighborhood, a partitioned cellular automaton, and others are often used to design RCAs.

Universal cellular automaton:

A CA is called computationally universal if it can compute any recursive function by giving an appropriate initial configuration. Equivalently, it is also defined as a CA that can simulate a universal Turing machine. Universality of RCAs can be proved by simulating other systems such as arbitrary (irreversible) CAs, reversible Turing machines, reversible counter machines, and reversible logic elements and circuits, which have already been known to be universal.

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Authors and Affiliations

  1. Department of Information Engineering, Hiroshima University, 739-8527, Higashi-Hiroshima, Japan

    Kenichi Morita

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Correspondence to Kenichi Morita .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Morita, K. (2015). Reversible Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_455-5

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  • DOI: https://doi.org/10.1007/978-3-642-27737-5_455-5

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Chapter history

  1. Latest

    Reversible Cellular Automata
    Published:
    31 January 2018

    DOI: https://doi.org/10.1007/978-3-642-27737-5_455-7

  2. Reversible Cellular Automata
    Published:
    26 September 2017

    DOI: https://doi.org/10.1007/978-3-642-27737-5_455-6

  3. Original

    Reversible Cellular Automata
    Published:
    05 November 2015

    DOI: https://doi.org/10.1007/978-3-642-27737-5_455-5

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