Fredkin, E., To#oli, T.: Conservative logic, International Journal of Theoretical Physics, Vol. 21, (1982) 219-253  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for computing things that are not computable traditionally. Unimpressed by this trend, computer scientists have argued in favor of the opposite: since there is no evidence that we need more than traditional computability to explain the world, we should try to make do without this assumption, e.g. [73, 74, 14, 48]. 7 Optimal Rational Decision Makers So far we have talked about passive prediction, given the observations. Note, however, that agents interacting with an environment can also use predictions of the future to compute action sequences that maximize expected future reward. Hutter s recent AIXI ....

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

....of adiabatic reversible circuits, if they obey a consistent clocking discipline, making it possible to arbitrarily reduce dissipation at the circuits mutual interfaces. Note that the memory storage operations are achieved adiabatically because we unwrite information instead of erasing it [5, 8]. In practice, of course, for non infinitesimal slew rates of clocks, there will be dissipative losses in these adiabatic circuits from device leakage and from non ideal components such as non zero on resistance of nFETs, etc. 8.1 Clock generation We do not address the issue of clock ....

E. Fredkin and T. Toffoh. Conservative logic. International Journal of Theoretical Physics, 21(3/4):219-53, 1982.

....subject to physical laws. One needs to understand the fundamental limits imposed by these laws in order to develop asympotically nondissipative computers. Some of the basic work on the limits of computation has been addressed in the pioneering work of Bennet, LandaueL Fredkin, et al. see, e.g. [2, 3, 8]) Landauer showed that erasure of information leads to a thermodynamic minimum heat dis sipation of kT In 2 joules per bit. Bennet [2] proved that irreversible logic gates and information erasure is not fundamental to computation. Since then various computational techniques have been proposed to ....

....leads to a thermodynamic minimum heat dis sipation of kT In 2 joules per bit. Bennet [2] proved that irreversible logic gates and information erasure is not fundamental to computation. Since then various computational techniques have been proposed to avoid dissipation by unwriting information [1, 8, 9, 13, 15, 25] instead of erasing it. Some of the important facts about physical processes relevant to asymptotically nondissipative computation may be summarized as follows. The physical state of a closed system is a subset of all allowable physical configurations of the system, i.e. a subvolume of its ....

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E. Fredkin and T. Toffoli. Conservative logic. International Journal of Theoretical Physics, 21(3/4):219- 53, 1982.

....by cost, re ecting Moore s empirical law rst formulated in 1965. Within a few decades nonreversible computation will encounter fundamental heating problems associated with high density computing [4] Remarkably, however, oops can be naturally implemented using reversible computing strategies [10], since it completely resets all state modi cations due to the programs it tests. But even when we naively extrapolate Moore s law, within the next century oops will hit the Bremermann limit [6] approximately 51 operations per second on 10 bits for the ultimate laptop [29] with 1 kg of ....

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

....computation backward by inserting the output as new input, thereby obtaining the input one started with. In more formal terms, reversible computation can be characterized by one to one operations; i.e. by a reversible, bijective evolution of the computer states onto themselves [Lan61, Ben73, FT82, Ben82, Lan94, LH90] If only a finite number of such states are involved, this amounts to their permutation. For such a restricted regime, many to one operations such as deletion of bits or one to many operations such as copying are not allowed. The flow diagram depicted in Fig. 1 was ....

Fredkin, E., and Toffoli, T., Conservative logic, International Journal of Theoretical Physics, 21 (1982), 219--253.

....proved that any Turing machine can be made reversible, thus showing that general computation can take place with far less energy dissipation than in traditional machines. Subsequent work [12, 72] has explored this approach, and the general paradigm of conservative (reversible) logic was treated in [26]. However, this approach incurs overhead in circuit speed and size, and has not found its way into mainstream electronic processor design. Rather than attempting to improve designs based on gates and wires, researchers have recently broken away from this paradigm entirely, and explored more novel ....

....itself; and so on. Finally, we discuss the application of these ideas to implementing PM based alloptical computation without employing physically discrete components. Such a computing machine would be based on the propagation and collision of solitons, and could use conservative logic operations [26], since the collisions we consider preserve the total energy and number of solitons. Finding a su#ciently powerful set of operators and reusable particles in this regime would open the way for integrated computation in homogeneous nonlinear optical media [89, 42] quite a di#erent scheme from ....

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E. Fredkin and T. To#oli. Conservative logic. International Journal of Theoretical Physics, 21(3/4):219, 1982.

....included results on the theoretical minimum amount of energy lost in computation, proposed new computer architectures inspired by the limits of physics, and a paper entitled Conservative Logic, which outlined how a Turing complete machine that consumes no energy when working could be constructed. [1][2] 3] This form of computation is called reversible because it is predicated on the idea of never losing any information. The nal state of a computation can be reversed, resulting in the initial state. One important observation by physicists is that the loss of information has an insurmountable ....

Fredkin, Edward; Tooli, Tommaso. \Conservative Logic." International Journal of Theoretical Physics. Vol. 21, pp 219-53, 1982.

....any computable function. Therefore in a computation one can in principle avoid information erasure by using a logically reversible device. In subsequent years several physical models of reversible computing devices were developed; see for example the billiard ball computer of Fredkin and To oli [6]. Landauer s principle has been used by Bennett in 1981 to resolve one of the longstanding problems of physics: the paradox of Maxwell s demon. What prevents the demon from breaking the second law of thermodynamics is the fact that it must erase the record of one measurement to make room for the ....

....their initial con guration and the third contains the output. In the second step lies the key di erence between Lecerf s and Bennett s work, as without saving the output, any logically reversible computer would be of little practical use. Another model of logical reversibility, the Fredkin gate [6], is a 3 bit logic gate which is both reversible and conservative: that is input and output have the same number of bits at 1. Reversibility and conservativity are two independent properties: however we are not interested in conservativity, as it does not seem to play a role for our purposes. 3 ....

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Edward Fredkin and Tommaso Tooli. Conservative logic. International Journal of Theoretical Physics, 21:219-253, 1981.

.... of integrability to algebraic and functional equations, as well as CA, including the joint work with Bruschi and Ragnisco [10] Finally, we also mention some additional, historically significant work: the very simple universal model using ideal elastically colliding billiard balls in the plane [28, 29]; the exhaustive study of universal dynamic computations by Adamatzky [30] the very early example of pipelined computation in a one dimensional CA by Atrubin [31] Conway s universal game of Life in 2 1 dimensions [32] and perhaps the simplest known universal CA in 2 1 dimensions [33] 4 ....

....particles for our NOT gate. Note that garbage particles arise as a result of this move operation. In general, because the Manakov system is reversible, such garbage often appears in computations, and needs to be managed explicitly or used as part of computation, as with conservative logic [28]. Of course reversibility does not necessarily limit the computational power of the Manakov system, since reversible systems can be universal [50] 6 Conclusion This brings us to current work, and suggests many open questions, some theoretical, some experimental, and some a mixture of the two. ....

E. Fredkin and T. Toffoli. Conservative logic. International Journal of Theoretical Physics, 21(3/4):219--253, 1982.

....a xed length input can compute arbitrary boolean functions of that input. To oli (1977, 20] showed that reversible cellular automata can simulate irreversible ones in linear time using an extra spatial dimension. Fredkin and To oli developed much reversible boolean circuit theory (1980 1982, [26, 27, 28]) 4. GENERAL DEFINITIONS In this section we set forth some general de nitions that we will use in our proof, but that may also be useful for future proofs in reversible computing theory. Later, in section 5.2, we will give some additional, more speci c de nitions that are not anticipated to be ....

E. Fredkin and T. Tooli. Conservative logic. International Journal of Theoretical Physics, 21:219253, 1982.

....to implement f(x) as a classical circuit 2 . The study of reversibility and irreversibility in computation goes back to Szilard[7] von Neumann[8] and Landauer[4] Seminal papers by Lecerf[5] and Bennett[1] showed that reversible computation is possible in principle; later Fredkin and To oli[2] wrote a beautiful paper in which they explain how to do reversible computation in the circuit model of computation. Now that we know how to do classical computation on a quantum computer, let s switch to a discussion of the quantum search algorithm. The basic setup is the following. Alice has a ....

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

....a xedlength input can compute arbitrary boolean functions of that input. To oli (1977, 7] showed that reversible cellular automata can simulate irreversible ones in linear time using an extra spatial dimension. Fredkin and To oli developed much reversible boolean circuit theory (1980 1982, [15, 16, 17]) 4 General de nitions. In this section we set forth some general de nitions that we will use in our proof, but that may also be useful for future proofs in reversible computing theory. Later, in section 25, we will give some additional, more speci c de nitions that are not anticipated to be ....

E. Fredkin and T. Tooli. Conservative logic. International Journal of Theoretical Physics, 21:219253, 1982. 43

....of such automata: if you begin with a finite pattern on a background of zeros, then, after a long period of confusing noise, the original pattern and several disjoint, identical copies of it will reappear, on an otherwise zero background. This phenomenon was first investigated by Edward Fredkin [4], who suggested such parity automata as the simplest cellular automata capable of exhibiting the self replicating patterns originally sought by von Neumann [18] and Ulam [17] 3 There are three different directions in which these results can be generalized: 1. Rather than a Bernoulli or ....

....P(n) 2 G M w (ffl) Psi , then Card [G ] Card Theta G M w (ffl) Thus, combining (11) and (12) we see that Card Theta B Theta 0; p M Card [G ] p M ; hence, the two sets must intersect nontrivially. Let N 2 B Theta 0; p M G ; then N satisfies [3] and [4]. 2 [Claim 2] Claim 3: Let Q = dM ffl e. Then w occurs more than R times in the string: N [Q 1] N [Q 2] N [Q 3] N [M ] Proof: N satisfies condition [4] of Claim 2, and of course w occurs at most Q times in the string (N 0 N [1] N [Q] ....

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Edward Fredkin and Tomaso Toffoli. Conservative logic. International Journal of Theoretical Physics, 21(3-4):219--253, 1981.

.... more (virtual) directions along which we can send the particles of a fluid (and more directions for An than for Z n ) 10 We expect (work in progress) that, with our approach, we will be less bothered by 10 The LGA are a computational model of physics based on cellular automata [Fey82, Min82, FT82, Tof84, TM87, Sny47, Ung58, Mar84, SW86, Mac86, Tof77b, Tof77a, Vic84, Wol83, Zus69, Hil55, Svo86] For this CA approach in general, emphasis is placed on the simplicity of the local transition rules for each cellular automaton [FHP86] The processors in our methodology are not restricted to ....

Edward Fredkin and Tommaso Toffoli. Conservative logic. International Journal of Theoretical Physics, 21(3/4), 1982. 36

....to physical laws. One needs to understand the fundamental limits imposed by these laws in order to develop asymptotically non dissipative computers. Some of the basic work on the limits of computation has been addressed in the pioneering work of Bennet, Landauer, Fredkin, et al. see, e.g. [2, 3, 8]) Landauer showed that erasure of information leads to a thermodynamic minimum heat dissipation of kT ln 2 joules per bit. 1 Bennet [2] proved that irreversible logic gates and information erasure is not fundamental to computation. Since then various computational techniques have been proposed ....

....leads to a thermodynamic minimum heat dissipation of kT ln 2 joules per bit. 1 Bennet [2] proved that irreversible logic gates and information erasure is not fundamental to computation. Since then various computational techniques have been proposed to avoid dissipation by unwriting information [1, 8, 9, 13, 15, 25] instead of erasing it. Some of the important facts about physical processes relevant to asymptotically non dissipative computation may be summarized as follows. ffl The physical state of a closed system is a subset of all allowable physical configurations of the system, i.e. a subvolume of its ....

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E. Fredkin and T. Toffoli. Conservative logic. International Journal of Theoretical Physics, 21(3/4):219--53, 1982.

....Prigogine and Stengers, 58, p. 61] 4 A new perspective on this issue came recently from computation theory. 5 Today s computers erase a bit of information every time they perform a computation corresponding to a many to one operation, cf. Landauer [41] Bennett [5, 6] Fredkin and To#oli [25]. Therefore, computational operations, such as the explicit deletion of information or clearing some memory and that like, are irreversible. In spite of the fact that in the last 50 years the dissipated energy per bit of computational operation has decreased by roughly tenfold each five years, ....

Fredkin, E., and Toffoli, T. Conservative logic. International Journal of Theoretical Physics 21 (1982), 219--253.

....ON THE COMPUTATIONAL CAPABILITIES OF PHYSICAL SYSTEMS PART II: RELATIONSHIP WITH CONVENTIONAL COMPUTER SCIENCE by David H. Wolpert NASA Ames Research Center, N269 1, Moffett Field, CA 94035, dhw ptolemy.arc.nasa.gov PACS numbers: 02. 10.By, 02.60.Cb, 03.65.Bz Abstract: In the first of this pair of papers, it was proven that there cannot be a physical computer to which one can properly pose any and all computational tasks concerning the physical universe. It was then further proven that no physical computer C can correctly carry ....

....distinction between the universe and the set of all physical computers. Its existence casts an interesting light on the ideas of Fredkin, Landauer and others concerning whether the universe is a computer, whether there are information processing restrictions on the laws of physics, etc. 10, 18] In a certain sense, the universe is more powerful than any information processing system constructed within it could be. This result can alternatively be viewed as a restriction on the universe as a whole the universe cannot support the existence within it of a computer that can process ....

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Fredkin, E., and Toffoli, T. (1982). International Journal of Theoretical Physics, 21, 219.

....work concerning reversible computer architecture has been either impractical or incomplete. Ressler s work [Res79, Res81] is significant in that it is the earliest work which is directly relevant to architecting fully reversible computers, but it is flawed in its exclusive use of the Fredkin [FT82] gate and its neglect of key control flow issues. Hall s work [Hal94] while correct in many high level issues, is incomplete, suggesting no mapping between instruction set architecture (ISA) and register transfer level (RTL) implementation. Indeed, Hall bases his reversible instruction set on the ....

....Logic Once bit erasure was identified as a source of unavoidable energy dissipation, researchers investigated a number of theoretical and practical schemes to implement a reversible computing technology. The first to be proposed were a series of hypothetical mechanical constructions. Fredkin [FT82] proposed the billiard ball model which uses collisions of hard spheres and mirrors to perform reversible computations. Figure 2 2 shows a crossover gate and Feynman s two input, three output universal logic gate. Both gates are reversible and non dissipative when isolated from imperfections. ....

E. F. Fredkin and T. Toffoli. Conservative logic. International Journal of Theoretical Physics, 21(3/4):219--253, 1982.

....distinction between the universe and the set of all physical computers. Its existence casts an interesting light on the ideas of Fredkin, Landauer and others concerning whether the universe is a computer, whether there are information processing restrictions on the laws of physics, etc. [10, 19]. In a certain sense, the universe is more powerful than any information processing system constructed within it could be. This result can alternatively be viewed as a restriction on the universe as a whole the universe cannot support the existence within it of a computer that can process ....

....interpretation can be viewed as an uncertainty principle that does not involve quantum mechanics. There are a number of previous results in the literature related to these results of this paper. Many authors have shown how to construct Turing Machines out of physical systems (see for example [10, 22] and references therein) By the usual uncomputability results, there are properties of such systems that cannot be calculated on a physical Turing machine within a fixed allotment of time (assuming each step in the calculation takes a fixed non infinitesimal time) In addition, there have been a ....

Fredkin, E., and Toffoli, T. (1982). International Journal of Theoretical Physics, 21, 219.

....for viscous fluid simulation. 1.2 Background Let me briefly review some of the historical developments in the lattice gas field. An overview of the lattice gas subject has been given by Boghosian [14] Lattice gases are a special case of cellular automata, popularized in the 1980 s by Ed Fredkin [39] and by Stephen Wolfram [94, 95] An early treatment of the cellular automata subject is presented by Tommaso To#oli and Norman Margolus in their book on cellular automata machines [91] Following the cellular automata paradigm, lattice gases are ideally suited for fine grained parallel ....

.... a square lattice were investigated in the early 1970 s by the French, in particular Yves Pomeau and coworkers [47] By the late 1970 s, cellular automata research was underway at the Information Mechanics Group at MIT on reversible computation by Edward Fredkin, Tommaso To#oli, and Norman Margolus[87, 39, 61]. The idea of building special purpose machines to simulate these physics like models on a fine grained space [88, 61] originated there. A good review of the kind of cellular automata modeling done in the early 1980 s is given by Gerard Vichniac [93] During this time, Stephen Wolfram visited the ....

Edward Fredkin and Tommaso To#oli. Conservative logic. International Journal of Theoretical Physics, 21(3/4):219--253, 1982. 178

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Fredkin, E., To#oli, T.: Conservative logic, International Journal of Theoretical Physics, Vol. 21, (1982) 219-253

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

No context found.

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

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Edward Fredkin and Tommaso To#oli. Conservative logic. International Journal of Theoretical Physics, 21(3/4):219--253, 1982. 178

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Fredkin, E., and Toffoli, T. (1982). International Journal of Theoretical Physics, 21, 219.

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