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Reversible Computing

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

A reversible computing system is defined as a “backwarddeterministic” system, where every computational configuration (state of the whole system) has exactly one previous configuration. Hence,a backward computation can be performed by its “inverse” computing system. Though its definition is rather simple, it is closely relatedto physical reversibility. The study of reversible computing originated from an investigation of heat generation (or energy dissipation) in computingsystems. It deals with problems how computation can be carried out efficiently and elegantly in reversible computing models, and how such systems can beimplemented in reversible physical systems. Since reversibility is one of the fundamentalmicroscopic physical laws of Nature, and future computing systems will surely become in a nano-scale size, it is very important to investigate theseproblems.

In this article, we discuss several models of reversible computing from the viewpoint of computing...

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Abbreviations

Billiard ball model:

The Billiard Ball Model (BBM) is a physical model of computation proposed by Fredkin and Toffoli [19]. It consists of idealized balls and reflectors. Balls can collide with other balls or reflectors. It is a reversible dynamical system, since it is assumed that collisions are elastic, and there is no friction. Fredkin and Toffoli showed that a reversible logic gate calledFredkin gate, which is known to be logically universal, can be embedded in BBM. Hence, a universal computer can be realized in the space of BBM.

Fredkin gate:

The Fredkin gate is a typical reversible logic gate with 3 inputs and 3 outputs whose operation is defined by the one-to-one mapping \( (c,p,q) \mapsto (c, cp + \bar{c}q, \bar{c}p + cq) \). It is known that any logic function (even if it is not a one-to-one function) can be realized by using only Fredkin gates by allowing constant inputs and garbage outputs (i.?e., useless signals). Furthermore, it is shown by Fredkin and Toffoli [19] that garbage signals can be reversibly erased, and hence any logic function is realized by a garbage-less circuit composed only of Fredkin gates.

Reversible cellular automaton:

A cellular automaton (CA) consists of a large number of finite automata called cells interconnected uniformly, and each cell changes its state depending on its neighboring cells. A reversible cellular automaton (RCA) is a one whose global function (i.?e., a transition function from the configurations to the configurations) is one-to-one. RCAs can be thought as spatio-temporal models of reversible physical systems as well as reversible computing models. (See Reversible Cellular Automata)

Reversible logic element:

A reversible logic element is a primitive from which logic circuits can be composed, and whose operation is defined by a one-to-one mapping. There are two kinds of such elements: those without memory, and with memory. Reversible elements without memory are nothing but reversible logic gates. The Fredkin gate and the Toffoli gate are well-known examples of them, which are universal. Reversible elements with 1-bit memory are also useful when constructing reversible computing systems. The rotary element is a typical one of this type, which is also known to be universal.

Reversible Turing machine:

A reversible Turing machine (RTM) is a “backward deterministic” Turing machine, and is a standard model of a reversible computing system. Bennett [4] showed that for any (irreversible) Turing machine there is an RTM that simulates the former and leaves no garbage information on the tape when it halts. Hence, Turing machines still have computation-universality even if the constraint of reversibility is added. Furthermore, there is a rather small universal reversible Turing machine.

Rotary element:

A rotary element (RE) is a reversible logic element with 1-bit memory [36]. Conceptually, it is square-shaped, and has a “rotating bar” inside. The direction of the bar is either horizontal or vertical. It has four input ports and four output ports on the four edges. If a “particle” comes from the direction parallel to the bar, then it goes straight ahead and does not affect the direction of the bar. If it comes from the direction orthogonal to the bar, then it turns rightward and the bar rotates by 90 degrees. It is known that an RE is universal, and reversible Turing machines can be built very concisely by using only REs.

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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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  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 Computing. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_456

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