Toffoli, T., and Margolus, N., Invertible cellular automata: a review, Physica D, 45, 229-- 253 (1990). 40  Home/Search   Document Details and Download   Summary   Related Articles   Check

....et al. are implementable in these same languages only when the inverses are likewise deterministic. Program version is an important and useful concept within di#erent software disciplines. Some representative theoretical papers on Turing machine inversion and reversible cellular automata are [McC56, Ben73, Ben82, TN90]. Broy uses program inverses to simplify the derivation of particular types of recursive structures during program synthesis [BKB80] Inversion is used in functional programming environments, such as Lisp [Kor81] fold and unfold transformation systems [Dar81] and Hope [HH86] Programming ....

T. To#oli and N.Margolus. Invertible cellular automata: a review. Physica D, Nonlinear Phenomena, 45(1-3), 1990.

....and bijectivity by Hedlund [5] and Richardson [15] and the decidability of reversibility in dimension 1 of Amoroso and Patt [1] to its undecidability in higher dimension by Kari [6,7] in 1990. The computing power of r ca as well as their simulation powers was particularly investigated in [18]. Bennett [2] proved that reversible Turing machines could simulate any Turing Machine. In 1977, Tooeoli [16] proved that r ca of dimension d 1 are able to simulate any d ca and thus are computationally universal. To built universal r ca, Partitioned ca (pca) and Block ca (bca) were independently ....

....are a strict subset of recursive congurations (recursive mapping from Z to S) Trivially, bca and pca are ca. Although bca and pca are subclasses of ca, they are able to simulate any ca. It was proved in [3,4] that r bca can simulate r ca. This was a 1990 conjecture by Tooeoli and Margolus [18] was independently and partially proved in 1996 by Kari [8] One of the results in the present paper is that r pca can also simulate r ca. In Sect. 2, we dene the 3 ca models and prove the decidability of reversibility for pca and bca. In Sect. 3, we build simulations of ca with pca, bca with pca ....

T. Tooeoli and N. Margolus. Invertible cellular automata: A review. Physica D, 45:229253, 1990. 10

.... simply represents a permutation of the channel contents, one can easily verify that no particles get lost (assuming correct treatment of the boundaries) The second technique is to subdivide the lattice into blocks, and at each step do some exchange of the contents of the sites within one block [3, 12, 16, 20]. Since only a few cells are involved, it is easy to verify whether conservation laws are observed. In subsequent time steps, the block boundaries are changed to make information exchange across the whole lattice possible. Note that such block cellular automata are equivalent to classical cellular ....

T. Tooli and N. Margolus. Invertible cellular automata: a review. Physica D, 45:229-253, 1990.

....new dynamical theories must conform. We are interested in capturing as many principles of physics in our CA models as possible, and therefore, we strive to construct CA dynamics which are reversible. In other words, the dynamics should map the states of the system in a one to one and onto manner [90]. In addition, we would like the inverse mapping to also be a local CA rule and to resemble the forward rule as much as possible. One way to The validity of this statement requires a suitable time reversal transformation on all the variables in the system and neglects the break in this symmetry ....

Tommaso Toffoli and Norman H. Margolus. Invertible cellular automata: A review. Physica D, 45:229--253, 1990.

....neighborhood, etc. can be speci ed interactively in the simulation system. Support for block CA: Block CA are a class of cellular automata where the updating rule speci es how a block of cells change their state together, instead of specifying how one cell changes depending on the neighbors [7]. They are formally equivalent to regular CA, but are much more convenient to formulate in many cases, especially where conservation laws must be observed [8] Block CA are directly supported in JCASim. Use of icons for representing cell states: The state of a cell can be represented using ....

T. Tooli, N. Margolus, Invertible cellular automata: a review, Physica D 45 (1990) 229-253.

....dynamics, might be a find 7 . Corollary 2 is quite obvious as to be of any mathematical interest. The invertible CA, however, have a long (almost a quarter of a century) history of falls and raises and nowadays they are one of the hot spots for research in the modern mathematical physics [14], 15] A topic of 7 For comparison: the simplest configuration known by now, that is a GOE in the Conway s game of Life , gets in a rectangle 9 Theta33 and comprises 226 nonblank cells. The latter was found by a team of researchers at MIT (Roger Banks et al. by means of a specially designed ....

.... Gamma1 (G 0 i ) Delta ; 8G 0 i 2 G(G 0 0 ) Definition 15 (Trivial computational process in CPCA) We will say, that G 0 2 G defines a trivial computational process G(G 0 ) if G 0 : Pop(G 0 ) 0 is the empty configuration in a given CA 2 CP CA. 8 Toffoli and Margolus state in [14], that the class of the properly reversible (injective) cellular automata is empty. They mean reversibility on the set of all configurations (not the finite ones, as we do) but this only confirmed our impression, that the structures we have analyzed lie somewhere in the twilight zone. 12 ....

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Toffoli, Tomaso, and Norman Margolus, "Invertible Cellular Automata: A Review", Physica D 45 (1990), 229--253.

....is one to one. Such a system, as well as other reversible systems (e.g. a reversible Turing machine, reversible logic circuits, etc. has a close connection to physical reversibility, and is known to be very important when studying inevitable energy dissipation in computing processes (see e.g. [2, 10] for general survey) Besides such problems of energy consumption, these systems are also interesting from a computational viewpoint, because they have relatively rich ability of computing in spite of the reversibility constraint. Bennett [1] first proved that any (irreversible) Turing machine can ....

To#oli, T., and Margolus, N., Invertible cellular automata: a review, Physica D, 45, 229-- 253 (1990). 12

....are very basic laws in physics. So far, various reversible computing models which reflect physical reversibility have been proposed and investigated. For example, reversible Turing machines [1] reversible logic circuits [3] reversible counter machines [10] and reversible cellular automata (RCA) [5, 7, 12, 13] have been studied extensively. In spite of the strong constraint of reversibility, they turned out to have very rich ability of computing, and several versions of universality results have been shown. Such reversible systems have also been known to be very important when studying physical ....

Toffoli, T., and Margolus, N., Invertible cellular automata: a review, Physica D, 45, 229-- 253 (1990).

....P , then we obtain a vanilla rectangular structure. Partitioned rectangular structures are somewhat simpler in structure than those that are not, they are also somewhat simpler to construct. A straightforward analysis shows that the Fredkin construction for reversible cellular automata (see e.g. [9] section 5.4) gives a rectangular structure that is partitioned. Note that in general a rectangular structure is not partitioned, although the numbers of partitioned examples outweighs the nonpartitioned examples. The smallest non partitioned example is of order 6 and can be found in detail in ....

Tommaso Toffoli and Norman H. Margolus. Invertible cellular automata: a review. Physica D, 45:229--253, 1993.

.... for the existence of nonconstructible configurations is a necessary and sufficient condition [11] In the seventies the constructibility nonconstructibility problem was investigated in a series of papers in connection with the surjectivity injectivity of CA mappings [12 15] and invertibility of CA [16, 17]. To date, two important results concerning the nonconstructibility problem have been obtained: the undecidability of the surjectivity (absence of nonconstructible configurations) problem for multi dimensional CA proved in [18] and the classification of 3 site rules from the structure of the set ....

Toffoli T. and Margolus N. Invertible cellular automata: A review. Physica D 45 (1990), 229-253.

.... 1 Introduction The main interest of reversibility in computation is backtracking a phenomenon to its source and in relation with physics, isoentropic phenomena modelization and saving energy, and had have various interests in relation to physics as explained by Tooeoli and Margolus in [15]. It is well known that, given any d dimensional cellular automata (CA) it can be simulated by one (d 1) dimensional CA which is reversible [12] It is still an open problem if it can be simulated by a reversible CA of the same dimension. For example, Morita showed in [7] that this is true in ....

T. Tooeoli and N. Margolus. Invertible cellular automata: A review. Physica D, 45:229253, 1990.

....is one to one. Such a system, as well as other reversible systems (e.g. a reversible Turing machine, reversible logic circuits, etc. has a close connection to physical reversibility, and is known to be very important when studying inevitable energy dissipation in computing processes (see e.g. [2, 10] for general survey) Besides such problems of energy consumption, these systems are also interesting from a computational viewpoint, because they have relatively rich ability of computing in spite of the reversibility constraint. Bennett [1] first proved that any (irreversible) Turing machine can ....

Toffoli, T., and Margolus, N., Invertible cellular automata: a review, Physica D, 45, 229-- 253 (1990).

....did not exist any universal reversible CA. This conjecture was proven false in dimension two (and higher) in 1977 by Tooeoli [12] In 1992, Morita [10] proved that there also exists universal reversible CA in dimension one. In 1990, Tooeoli and Margolus wrote a large survey about reversible CA [15]. Physical considerations about lattice gas lead Margolus [7] to introduce a new kind of CA, block CA (BCA) together with a practical example: the Billiard ball model (BBM) They operate over the same con gurations as CA. The underlying lattice is partitioned into regularly displayed rectangular ....

T. Tooeoli and N. Margolus. Invertible cellular automata: A review. Physica D, 45:229253, 1990.

....section shows us the existence of a universal qca and answers a question about simulating qca on Quantum Turing Machines [21] The above theorems and lemmas also hold when restricted to classical ca. Theorem 6. 3 therefore proves (in the case of 1d ca) the conjecture made by Toffoli and Margolus [55] about the structural invertibility of (classical) reversible ca. An independent proof of this conjecture has been made by Kari [32] which also holds for 2d CA. Chapter 7 A Universal Quantum Cellular Automaton In this last chapter we will prove that there exists a qca which can simulate any ....

Tommaso Toffoli and Norman Margolus. Invertible cellular automata: a review. Physica D, 45:229--253, 1990. Contribution in [27].

....are invertible models [5, 8] Cellular automata represent one of the best models of parallel computation; the study of invertibility in cellular automata is of great interest in modelling physics. Several theoretical results concerning invertibility in cellular automata have been presented ([2, 9, 10, 12, 13, 15, 18]) some leading to open questions. ffl In [18] the existence of a peculiar class (residual class) of cellular automata had been predicted but, until now, no such cellular automata had been exhibited. Here we explicitly construct a cellular automaton in this class, i.e. a cellular automaton that ....

....of the best models of parallel computation; the study of invertibility in cellular automata is of great interest in modelling physics. Several theoretical results concerning invertibility in cellular automata have been presented ( 2, 9, 10, 12, 13, 15, 18] some leading to open questions. ffl In [18], the existence of a peculiar class (residual class) of cellular automata had been predicted but, until now, no such cellular automata had been exhibited. Here we explicitly construct a cellular automaton in this class, i.e. a cellular automaton that is invertible on every finite support but is ....

[Article contains additional citation context not shown here]

Toffoli, T., Margolus N., (1990) "Invertible cellular automata: a review", Physica D 666, 229--253.

.... interesting scalar QCA (the first of which seems to have been described by Feynman [10] similar discrete models for a quantum particle have been studied by several authors more recently [5,11] This is equally true in higher dimensional lattices: the one step evolution of a quantum partitioning [2,12] CA is invariant under the action of a subgroup of the translations on the lattice and may be interpreted to be composed of particle scattering matrices. Higher dimensional quantum particle automata [7,13] and their generalizations to quantum lattice gas automata [7,14] have been constructed in ....

T. Toffoli and N. H. Margolus, "Invertible cellular automata: a review", Physica D 45 (1990) 229--253.

....scheme, the data are thought of as signals that travel from site to site, while the sites themselves represent events, i.e. places where signals interact, as in Fig. 12. The latticegas scheme was arrived at independently, but in response to similar physical motivations, by a number of researchers[51]; it is widely used in fluid dynamics and materials science modeling. f t x y Figure 12: Example of lattice gas format: Rule f has 4 inputs and 4 outputs; from the state of the four arcs entering a node (current state) it computes the state of the four arcs leaving the node (new state) The idea ....

Toffoli, Tommaso, and Norman Margolus, "Invertible Cellular Automata: A Review," Physica D 45 (1990), 1--3.

....system is a backward deterministic system, i.e. roughly speaking, each computational configuration of it has at most one predecessor. Until now, various reversible systems, such as reversible Turing machines, reversible cellular automata, reversible logic gates, have been studied (see e.g. [4, 12, 13, 15] for general survey) One interesting point of a reversible system is that it is closely related to physical reversibility and the problem of energy dissipation in a computing process. It is known to be possible to construct a reversible computer that works without dissipating energy in an ideal ....

T.Toffoli, and N.Margolus, Invertible cellular automata: a review, Physica D, 45 (1990) 229--253.

....and depart on the dotted arcs; on phase 2 they interact at the dotted nodes and depart on the dashed arcs; finally, on phase 3, they interact at the hollow nodes and depart on the solid arcs, ready to start a new cycle. Computational schemes of this kind are routinely used in lattice gas models[40]. Figure 19: A uniform network representing a three phase computation cycle. On phase 1, data travel on the solid arcs, come together at the solid nodes, interact, and depart on the dotted arcs; on phase 2 they interact at the dotted nodes and depart on the dashed arcs; finally, on phase 3, they ....

Toffoli, Tommaso, and Norman Margolus, "Invertible Cellular Automata: A Review", Physica D 45 (1990), 229-- 253.

.... dynamic cellular automata (CA) is so loosely constrained that the rule space is too large to explore usefully [13] Hillman constrains his models to be reversible [14] but then faces the familiar difficulty of finding reversible CA rules without partitioning [15] or going to second order in time [16]. Lattice gas models may be expected to resolve both of these difficulties. Not only do they constitute a physically natural class of models, but particle number and momentum conservation impose tight constraints on the rule space. To construct fundamental models we should also require ....

T. Toffoli and N. H. Margolus, "Invertible cellular automata: a review", Physica D 45 (1990) 229--253.

.... physics (that had already been probed by Zuse[49] and To oli[30] ared up in the 80s with another impersonation of cellular automata, namely lattice gases[9] Actually the lattice gas scheme was arrived at independently, but in response to similar physical motivations, by many investigators (cf. [12, 29, 31, 7, 8, 18, 4, 34, 14, 15, 16]) its usefulness was underlined by Fredkin s insights into reversible computation[8, 7] and a number of original applications by Margolus[18, 34] who introduced the Margolus neighborhood [33] and coined the term partitioning cellular automata . Meanwhile, Wolfram had been investigating mainly ....

.... scheme was arrived at independently, but in response to similar physical motivations, by many investigators (cf. 12, 29, 31, 7, 8, 18, 4, 34, 14, 15, 16] its usefulness was underlined by Fredkin s insights into reversible computation[8, 7] and a number of original applications by Margolus[18, 34], who introduced the Margolus neighborhood [33] and coined the term partitioning cellular automata . Meanwhile, Wolfram had been investigating mainly one dimensional cellular automata in connection with statistical mechanics[45] and computational linguistics[44] The Information Mechanics group ....

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Toffoli, Tommaso, and Norman Margolus, \Invertible Cellular Automata: A Review," Physica D 45 (1990), 229-253.

....programmable matter. Though the two schemes are mathematically equivalent, lattice gases give one better control over certain aspects of computation that are essential to the efficient management of information at a microscopic level. Among the issues involved, which we shall not discuss here (see [22]) we may list signal creation and destruction, microscopic reversibility, and susceptibility to noise. 3.2 The CAM 8 architecture Cam 8 is an indefinitely scalable multiprocessor architecture aimed at the fine grained modeling of spatiallyextended systems; in this target area, it offers ....

Toffoli, Tommaso, and Norman Margolus, "Invertible Cellular Automata: A Review," Physica D 45 (1990), 1--3.

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Toffoli, T., and Margolus, N., Invertible cellular automata: a review, Physica D, 45, 229-- 253 (1990). 40

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T. Toffoli, N. Margolus, Invertible Cellular Automata: A Review, Physica D, Vol. 45, 1990, pp. 229--253.

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T.Toffoli, and N.Margolus, Invertible cellular automata: a review, Physica D, 45 (1990) 229--253.

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