E. Fredkin and T. Toffoli. Conservative Logic. In Int. J. of Theo. Phys., Vol. 21, pp. 219--253, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... fact that in principle the only thermodynamically costly computer operations are those that are logically irreversible, i.e. operations that map several distinct logical states of the computer onto a common successor, thereby throwing away information about the computer s previous state [3] 4] [11], 16] 20] The thermodynamics of computation is discussed further in Section VIII. Here we show that the minimal program size for a reversible computer to transform input into output is equal within an additive constant to the size of the minimal conversion string of Theorem 3.3. The theory ....

....stochastically into one of equiprobable successors, that step can, if properly harnessed, be used to remove bits of entropy from the computer s environment. Models have been constructed, obeying the usual conventions of classical, quantum, and thermodynamic thought experiments [1] 3] 4] 10] [11], 15] 17] 23] showing both the ability in principle to perform logically reversible computations in a thermodynamically reversible fashion (i.e. with arbitrarily little entropy production) and the ability to harness entropy increases due to data randomization within a computer to reduce ....

E. Fredkin and T. Toffoli, "Conservative logic," Int. J. Theoret. Phys., vol. 21, no. 3/4, pp. 219--253, 1982.

....error correction techniques, considered crucially important for the enterprise to continue [11, 8, 12, 13] Because of the di#culties of quantum coherence, the first quantum computer comprised only two qubits. An historical starting point that led to quantum computing was reversible computing [14]. Since microscopic physics is reversible , it is believed that quantum mechanical algorithms must be too. Reversible algorithms for simulating physics on a quantum device can serve as a guide for constructing the device. The common assumption is the quantum mechanical device itself ....

Fredkin, E., To#oli, T.: Conservative logic. Inter. J. Theor. Phys., 21, No. 3/4 (1982) 219--253

....4. A gate output can be used only once (the fanout count of each output is equal to one) If two copies of a signal are required, a copying circuit is used. 5. The resulting circuit is acyclic. In addition to the Feynman gate mentioned above, the literature discusses Toffoli [23] and Fredkin [7] gates and their construction using existing and future technologies. Three input three output gate families have been analyzed Energy may be lost for input and output operations. in [8] As a result of this analysis, several new types of binary and multiple valued reversible gates have been ....

E. Fredkin and T. Toffoli. Conservative Logic. Int. Journal of Theor. Phys., 21 (1982), pp. 219-253.

....of gate level transitions for computation or data transmission, etc. See, e.g. 1, 2] The need for extremely energy efficient computers for many present and future applications has been well articulated, and new approaches along the line of reversible or adiabatic computation are discussed in [3, 4, 5, 6, 7, 8]. Many of these and other papers also point out the very dissipative physical nature of present day computers and computations. Much of research in this second category has focused on obtaining reversible computer architectures to eliminate energy dissipation in principle. The idea is based on the ....

....as objects e.g. ones and zeros in a computer that are not to be destroyed during transformations as the computational process evolves. These objects are only moved around, i.e. conserved. This type of conservative logic is exemplified by the Fredkin gate and the billiard ball computer in [5]. During a function computation, intermediate, redundant information is generated which is eventually unwritten. The input signals, modified only by circuit s environment, are held stable suitably long after the outputs have been generated in order to facilitate the unwriting as well as to ....

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E. Fredkin and T. Toffoli, "Conservative logic," International Journal of Theoretical Physics, vol. 21, no. 3/4, pp. 219-53, 1982.

....automaton could be thought to perform. Therefore, we will use as our example computation a simple system of two particles , represented by qubits with value j1i, moving on a background of j0i qubits towards each other. The particle movements are determined by so called Billiard Ball Model [6] rules (Figure 4) a rule set used also by Margolus in [12] The rules are reversible and preserve the number of j1i s. Figure 3: The initial state of the example automaton. Guard qubits are on the left and data qubits on the right. Figure 4: The Billiard Ball rules. The upper rule represents a ....

E. Fredkin and T. Tooli. Conservative logic. In E. Fredkin, R. Landauer, and T. Tooli, editors, Proceedings of the Physics of Computation Conference, pages 219{ 253, 1982.

....includes only reversible gates realizes on all outputs only balanced functions, therefore it can realize non balanced functions only with garbage outputs. An additional constraint of reversible logic is that the fanout of every signal, including primary inputs, must be one. In the classical paper [3], Fredkin and Toffoli formulate the synthesis problem for reversible logic. Classical logic synthesis methods cannot be directly applied to design reversible logic circuits, and reversible logic specific synthesis methods do not yet exist [9] Currently, logic functions constructed with reversible ....

....be defined in many different logic systems. For the purposes of this paper, the class of reversible elements that can be modeled as binary valued logic circuits will be called classical reversible logic elements. Much work has been done in defining and characterizing classical reversible logic [2][3][11] and applying the concepts toward power conservation and specific technology implementations [4] 5] 6] This section will briefly define basic results in classical reversible logic. Consider the (3,3) conservative reversible gate with inputs , and outputs , where , and . This ....

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E. Fredkin and T. Toffoli, "Conservative Logic", Intl. J. Theoretical Physics, vol. 21, nos. 3-4, pp. 219-253, 1982.

....remain unchanged or complement themselves between generations. Naturally, our interest, from conservation point of view, includes both remote past and remote future. A key step in this line of work was the design of Reversible Automata [12] based on which conservative logic was designed. [1] Conservative logic is a comprehensive model of computation reflecting the important principle of reversibility of dynamic laws, which is the basis of conservation. In an idealistic situation, conservative logic proves, that we can build sequential circuits with zero internal power dissipation. ....

....cannot be captured. The essential primitives for information theory are storage, transmission and processing of a discrete signal. Any digital information system must have building blocks for executing these primitives. With the above in mind, Fredkin introduced four additional principles [1]. They are: Principle 4: Identity (of storage and transmission) There is no intrinsic distinction between storage and transmission of signals, and a single primitive can capture both. Principle 5: Reversibility: Since dynamic laws are reversible, all of the abstract primitives must be ....

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. E. Fredkin, T. Toffoli, "Conservative Logic", International Journal of Theoretical Physics, Vol. 21, Nos. 3/4, pp 219-253, 1982.

.... by a reversible circuit it is necessary to apply constant signals to some inputs of the circuit (in conventional circuits with complex gates using constant signals is useful but not necessary) Let us call a gate (circuit) with n inputs and m outputs (n,m) gate ( circuit) It was shown [9] that a universal reversible gate should have at least three inputs and outputs and several such gates were proposed [2 10,13 14] However, it would be interesting to know which (3,3) gates are most efficient, as well as to consider relative efficiency of reversible (3,3) gates in comparison ....

.... allow permutation of wires (not allowed in quantum gates) but we do not allow inversions (e.g. permitted when double wires are used) The following relatively simple reversible (3,3) gates have been proposed in the literature (they are listed in the chronological order) G1 Fredkin gate F0CAAC [9,17] G2 Feynman gate F0CC6A [8] G3 Peres gate 66CC78 [13] G4 Margolus gate CAB8E4 [10] G5, G6, G7, G8, G9 De Vos gates, respectively 714D2B, 8EB2D4, B4C69A D29CA6, ACE2D8 [2 6] EXPERIMENTAL RESULTS We have ran a program constructing all two gate circuits made of identical reversible ....

E. Fredkin, T. Toffoli, "Conservative Logic", Int. Journal of Theor. Phys., Vol. 21, 1982, pp. 219-253.

....We shall make use of planar nor, monotone and other restricted variants of CVP in Section 3. A proof that a problem D is P complete via a reduction from CVP is tantamount to what in other contexts has been called computational universality. The dynamics of hard spheres in classical mechanics [21] and some cellular automata rules [14, 22] have been shown to be computationally universal. Our proofs that DLA and various Ising Monte Carlo dynamics are P complete depend on showing that arbitrary logical calculations can be embedded in these dynamics. In addition to the resources of parallel ....

E. Fredkin and T. Toffoli. Conservative logic. Int. J. of Theo. Phys., 21:219, 1982.

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E. Fredkin and T. Toffoli. Conservative Logic. In Int. J. of Theo. Phys., Vol. 21, pp. 219--253, 1982.

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E. Fredkin and T. Toffoli. Conservative Logic. Int. Journal of Theor. Phys., 21 (1982), pp. 219-253.

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Edward Fredkin and Tommaso To#oli. Conservative Logic. pages 47--81, 2002.

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E. Fredkin and T. To#oli, "Conservative Logic," Int. J. of Theoretical Physics, Vol. 21, pp. 219-253, 1982.

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[4] Fredkin, E., Toffoli, T., "Conservative Logic", Fredkin, E., Toffoli, T., "Conservative Logic",

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Fredkin E and To#oli T, Conservative Logic, Int. J. Theor. Phys. 21 219--253 (1982)

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Fredkin, E., and Toffoli, T. Conservative logic. Int. J. Theoret. Phys. 21 (1982), 219-353.

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E. Fredkin and T. Toffoli. Conservative logic. Intl. J. of Theoretical Physics, vol. 21, pp. 219--253, 1982.

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E. Fredkin, and T. Toffoli, "Conservative Logic," Int. Journal of Theoretical Physics, Vol. 21, 1982, pp. 219-253.

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E. Fredkin and T. Toffoli, " Conservative logic," Internat. J. Theoret. Phys. 21 (1982), pp.219-253.

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Edward Fredkin and Tommaso Toffoli. "Conservative Logic ". International Journal of Theoretical Physics. vol. 21, pp 219-253, 1982.

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Fredkin E and Toffoli T 1982 Conservative logic, Int. J. Theor. Phys. 21 219-253

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E. Fredkin and T. Toffoli, "Conservative logic," in Proceedings of the Physics of Computation Conference, 1982, pp. 219--253.

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Fredkin, E., and Toffoli, T., Conservative logic, Int. J. Theoret. Phys., 21, 219--253 (1982).

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Fredkin, Edward and Tooli, Tommaso (1982). Conservative logic. International Journal of Theoretical Physics, 21(3/4):219-253.

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E. Fredkin and T. Toffoli, "Conservative logic," Int. J. Theoretical Phys., vol. 21, pp. 219--253, 1982.

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