T. Toffoli "Reversible Computing", in Automata, Languages and Programming, eds. J. W. de Bakker and J. van Leeuwen (Springer, New York, 1980), p. 632; Technical Memo MIT/LCS/TM-151, MIT Lab. for Comp. Sci. (unpublished).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as low as possible. 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 ....

....background to discuss logic synthesis of wave cascade. Section 5 presents the logic synthesis algorithm. Section 6 shows preliminary experimental results. Section 7 concludes the paper. 2 Toffoli Family of Gates Feynman gate is described by equations: P = A, Q = A B. Toffoli gate [23] is described by equations: P = A, Q = B, R = AB C. Feynman gate can be generalized to the following family of gates called 1 1 family of Toffoli gates: P = A, Q = f 1 (A) B, where f 1 is an arbitrary function of one variable. There are only four functions of one variable in binary logic. ....

T. Toffoli. Reversible Computing. In Automata, Languages and Programming, Springer Verlag, 1980, pp. 632- 644.

.... 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 ....

T. Toffoli, "Reversible Computing", in Automata, Languages and Programming, Springer Verlag 1980, pp. 632644.

....computations must generate heat in the computing process, and that reversible computations have the property that if executed slowly enough, they (in the limit) can consume no energy in an adiabatic computation. Bennett [Ben73] also see Bennett, Landauer [BL85] Landauer [Lan96] Toffoli [T80]) showed that any computing machine (e.g. an abstract machine such as a Turing Machine) can be transformed to do reversible computations. Bennett s reversibility construction required extra space to store information to insure reversibility; Li, Vitanyi [LV96] give trade offs between time and ....

....a constant Boolean operation yielding 0) on a quantum gate array. Each operation in our O(n(log 4 n) log log n) time reversable computation of SCHUMACHER 0 (X) consists of certain conditional Boolean and arithmetic operations. The conditional Boolean operations suffice to be Toffoli gates [T80] which take in 3 Boolean inputs and negates the first input iff the next two bits are 1. The conditional arithmetic operations suffice to be n bit (signed) integer negation, addition and multiplication (conditional on a Boolean register) Section IV of Cleve, DiVincenzo [CD96] describes in detail ....

T. Toffoli, Reversible Computing, in Automata, Languages and Programming, eds. J. W. de Bakker and J. van Leeuwen Springer, New York, p. 632, (1980).

....cubic computational complexity in the block size. The proposed encoder and decoder are quantum mechanical inverses of each other, and constitute a very satisfying example of 2 reversible quantum computation. For fundamental references on reversible computation, see, Bennett [23, 24] and Toffoli [25]. For reversible computation in relation to quantum data compression, see, Cleve and DiVincenzo [14] II Preliminaries We begin by reviewing the definitions of quantum sources and quantum states relevant to the present coding problem. We also present precise quantum counterparts for the classical ....

T. Toffoli, "Reversible computing," in Automata, Languages, and Programming (W. de Bakker and J. van Leeuwen, eds.), pp. 632--644, Springer, New York, 1980.

....For obvious reasons, this is sometimes called a controlled NOT , or c NOT, gate. The corresponding three qubit analog T 1 23 , which NOT s the first qubit if the other two are both 1 #, is known as the To#oli gate after the person who first realized that it is universal for boolean logic [55]. This follows from the fact that, if one sets the first (target) input bit to 1, the output is the NAND of the other two inputs. More generally, the c NOT gates, together with all one qubit quantum gates, generate the entire unitary group U(2 N ) on N qubits [2] The general problem of ....

T. Toffoli, Reversible computing, in Automata, Languages and Programming, J. W. de Bakker and J. van Leeuwen, eds., Springer-Verlag, 1980, pp. 632--644.

....them. For a function f,from n bits to m bits, we get the reversible function from m n bits to m n bits: f : i 7 Gamma f(i) f r : i; j) 7 Gamma (i; f(i) Phi j) 15) Applying this method, for example, to the logical AND gate, a; b) 7 Gamma ab it will become the known Toffoli gate[186] (a; b; c) 7 Gamma (a; b; c Phi ab) which is described by the unitary matrix on three qubits: T = 0 B B B B B B B B B B B B 1 1 1 1 1 1 0 1 1 0 1 C C C C C C C C C C C C A (16) The Toffoli gate applies NOT on the last bit, conditioned that the other bits ....

....an elementary gate to operate on a small number of qubits, independent of n which can be very large. We want our computer to be able to compute any function. The set of elementary gates used should thus be universal. For classical reversible computation, there exists a single universal gate[96, 186], called the Toffoli gate, which we have already encountered. This gate computes the function a; b; c 7 Gamma a; b; ab Phi c: The claim is that any reversible function can be represented as a concatenation of the Toffoli gate on different inputs. For example, to construct the logical AND gate ....

Toffoli T 1980 Reversible computing, in Automata, Languages and Programming, Seventh Colloquium, Lecture Notes in Computer Science, Vol. 84, de Bakker J W and van Leeuwen J, eds, (Springer) 632-644

....computation can be made reversible [Lecerf 1963, Bennett 1973] In fact, reversible classical gate arrays have been studied. Much like the result that any classical computation can be done using NAND gates, there are also universal gates for reversible computation. Two of these are Toffoli gates [Toffoli 1980] and Fredkin gates [Fredkin and Toffoli 1982] these are illustrated in Table 3.1. The Toffoli gate is just a controlled controlled NOT, i.e. the last bit is negated if and only if the first two bits are 1. In a Toffoli gate, if the third input bit is set to 1, then the third output bit is the ....

....on reversible computation [Lecerf 1963, Bennett 1973] we can compute any polynomial time function F (x) as long as we keep the input x in the computer. We do this by adapting the method for computing the function F nonreversibly. These results can easily be extended to work for gate arrays [Toffoli 1980, Fredkin and Toffoli 1982] When AND, OR or NOT gates are changed to Fredkin or Toffoli gates, one obtains both additional input bits, which must be preset to specified values, and additional output bits, which contain the information needed to reverse the computation. While the additional input ....

T. Toffoli (1980) "Reversible computing," in Automata, Languages and Programming, Seventh Colloquium, Lecture Notes in Computer Science No. 84 (J. W. de Bakker and J.

....for reversible computation, that is, a gate which, when applied in succession to different triplets of bits in a gate array, could be used to simulate any arbitrary reversible computation. Two bit gates like NAND which are universal for ordinary computation are not reversible. Toffoli s version[4] of the universal reversible gate will figure prominently in the body of this paper. Quantum physics is also reversible, because the reverse time evolution specified by the unitary operator U Gamma1 = U y always exists; as a consequence, several workers recognized that reversible computation ....

....2 (m 1) Theta 2 (m 1) matrix corresponding to m (U) is 0 B B B B B B B B B 1 1 . 1 u 00 u 01 u 10 u 11 1 C C C C C C C C C A (where the basis states are lexicographically ordered, i.e. j000i; j001i; j111i) When U = 0 1 1 0 ; m (U) is the so called Toffoli gate[4] with m 1 input bits, which maps jx 1 ; xm ; yi to jx 1 ; xm ; V m k=1 x k ) Phi yi. For a general U , m (U) can be regarded as a generalization of the Toffoli gate, which, on input jx 1 ; xm ; yi, applies U to y if and only if V m k=1 x k = 1. As shown by one of us ....

[Article contains additional citation context not shown here]

T. Toffoli "Reversible Computing", in Automata, Languages and Programming, eds. J. W. de Bakker and J. van Leeuwen (Springer, New York, 1980), p. 632; Technical Memo MIT/LCS/TM-151, MIT Lab. for Comp. Sci. (unpublished).

.... been analyzed with respect to bistable magnetic devises for reversible copying canceling of records in [14] and Brownian computers [12] for Turing machines and Brownian enzymatic computers [3, 4, 6] with respect to reversible Boolean circuits by [9] for molecular (billiard ball) computers by [21], Brownian computing using Josephson devices in [16] quantum mechanic computers in [1, 2, 17] and notably by R. Feynman [7, 8] All these models seem mutually simulatable. For background information, see [5] In the last three decades there have been many partial precursors and isolated results ....

T. Toffoli. Reversible computing. In 7th International Colloquium Automata, Languages, and Programming, volume 85 of Lecture Notes in Computer Science, pages 632--644. Springer-Verlag, 1980.

.... Indeed, if one wants to build a general purpose computing structure within a cellular automaton, the most practical approach is to start with a local map that directly supports logic gates and wires, and then build the appropriate logic circuits out of these primitives[4, 77] As explained in [70, 19], in an invertible cellular automaton the gates will have to be invertible; because of this constraint, a complete, self contained logic design will have to explicitly provide, besides circuitry for the desired logic functions, additional circuitry for functions (analogous to energy supply and ....

....make similar design plans for a new machine that, if the original machine was invertible, will have exactly the inverse behavior. Thus, to rescue point (2) we have to turn our attention to design plans that do not imply dissipative primitives. For finite invertible automata this is always possible[70, 19] (see also [69] for the analogous problem in continuous systems) is it possible for cellular automata Now that the job of motivating it is done, let us reword the above question. Given an arbitrary ica, which can always be thought of as realized by a uniform combinational network of the type of ....

Toffoli, Tommaso, "Reversible Computing," Automata, Languages and Programming (de Bakker and van Leeuwen eds.), Springer-Verlag (1980), 632--644.

.... the world, including: communication, computation, construction, growth, reproduction, competition and evolution [Toffoli and Margolus, 1987, Langton, 1984, Burks, 1970, Smith, 1969] CAs also provide a means for modeling physical phenomena by reducing them to their basic, elemental laws (rules) [Toffoli, 1980, Fredkin and Toffoli, 1982, Margolus, 1984, Vichniac, 1984, Bennett and Grinstein, 1985] CAs exhibit three notable features: massive parallelism, locality of cellular interactions, and simplicity of basic components (cells) They perform computations in a distributed fashion on a spatially ....

T. Toffoli. Reversible computing. In J. W. De Bakker and J. Van Leeuwen, editors, Automata, Languages and Programming, pages 632--644. Springer-Verlag, 1980.

....the basic pseudo pure state I 1 z Gamma I 2 z Gamma 2I 1 z I 2 z . 5 The Toffoli gate We have also developed a pulse sequence, analogous to the above XOR sequence, which transforms the basic pseudospinors of a three spin system according to the truth table of the well known Toffoli gate[9]. We call this the Toffoli pulse sequence: T OF ] 3 j [ 2] 3 y Gamma [1= 4J) Gamma [ 2] 3 y Gamma [1= 4J) Gamma [ Gamma =2] 3 x Gamma [1= 4J) Gamma [ Gamma =2] 3 x (21) Although this pulse sequence assumes that the coupling constants J 13 and J 23 have the same value J , ....

Toffoli, Tommaso, "Reversible computing", Automata, Languages and Programming (J. W. de Bakker and J. van Leeuwen ed.), Springer-Verlag (1980), 632--644.

No context found.

T. Toffoli, Reversible Computing, Tech. Memo MIT/LCS/TM-151, MIT Lab. for Comp. Sci. (1980). Received January 23, 1979, in revised form October 2, 1979 and in final form February 21, 1980.

....structure within a cellular automaton, the most practical approach is to T. Toffoli, N. H. Margolus = Invertible Cellular Automata 6 start with a local map that directly supports logic gates and wires, and then build the appropriate logic circuits out of these primitives[4,77] As explained in [70,19], in an invertible cellular automaton the gates will have to be invertible; because of this constraint, a complete, self contained logic design will have to explicitly provide, besides circuitry for the desired logic functions, additional circuitry for functions (analogous to energy supply and ....

....make similar design plans for a new machine that, if the original machine was invertible, will have exactly the inverse behavior. Thus, to rescue point (2) we have to turn our attention to design plans that do not imply dissipative primitives. For finite invertible automata this is always possible[70,19] (see also [69] for the analogous problem in continuous systems) is it possible for cellular automata Now that the job of motivating it is done, let us reword the above question. Given an arbitrary ica, which can always be thought of as realized by a uniform combinational network of the type of ....

Toffoli, Tommaso, "Reversible Computing," Automata, Languages and Programming (de Bakker and van Leeuwen eds.), Springer-Verlag (1980), 632-- 644.

No context found.

T. Toffoli "Reversible Computing", in Automata, Languages and Programming, eds. J. W. de Bakker and J. van Leeuwen (Springer, New York, 1980), p. 632; Technical Memo MIT/LCS/TM-151, MIT Lab. for Comp. Sci. (unpublished).

No context found.

T. Toffoli "Reversible Computing", in Automata, Languages and Programming, eds. J. W. de Bakker and J. van Leeuwen (Springer, New York, 1980), p. 632; Technical Memo MIT/LCS/TM-151, MIT Lab. for Comp. Sci. (unpublished).

No context found.

T. Toffoli. Reversible Computing. In Automata, Languages and Programming, Springer Verlag, 1980, pp. 632- 644.

No context found.

Toffoli, T. (1980). Reversible Computing. Automata, Languages and Programing. Springer-Verlog.

No context found.

T. Toffoli, "Reversible Computing," Tech Memo MIT/LCS/TM-151, MIT Lab for Comp. Sci., 1980.

No context found.

T. Toffoli, "Reversible Computing," Lab. for Computer Science, Mass. Inst. of Technol., Cambridge, MA, Tech. Memo. MIT/LCS/TM-151, 1980.

No context found.

T. Toffoli, "Reversible Computing," Tech. Memo MIT/LCS/TM-151, MIT Lab for Comp. Sci, 1980.

No context found.

T. Toffoli, "Reversible Computing," Tech. Memo MIT/LCS/TM-151, MIT Lab for CS, `80.

No context found.

T. Toffoli, "Reversible Computing", Tech . Memo MIT/LCS/TM-151, MIT Lab for Comp. Sci, 1980.

No context found.

Toffoli T 1980 Reversible computing, in Automata, Languages and Programming, Seventh Colloquium, Lecture Notes in Computer Science, Vol. 84, de Bakker J W and van Leeuwen J, eds, (Springer) 632-644 76

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Toffoli, Tommaso, "Reversible Computing," Automata, Languages and Programming (de Bakker and van Leeuwen eds.), Springer-Verlag (1980), 632--644.

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