Unformatted text preview: ter implements the Fredkin gate: DExercise 3.30: Verify that the billiard ball computer in Figure 3.14 computes the
Fredkin gate. Bennett, C. H. The Thermodynamics of Computation — a Review. Int. J.
In addition to reversibility, the Fredkin gate also has the interesting property that
Theo. Phys., 21, 12 (1982), 905–940.
the number of 1s is conserved between the input and output. In terms of the billiard
ball computer, this corresponds to the number of billiard balls going into the Fredkin
gate being equal to the number coming out. Thus, it is sometimes referred to as being
a conservative reversible logic gate. Such reversibility and conservative properties are
interesting to a physicist because they can be motivated by fundamental physical princi- 66 CHAPTER II. PHYSICS OF COMPUTATION Figure 8 Realization of the J-K flip-flop. ¯
Figure II.26: Implementation of J-K ﬂip-ﬂop. [from FT82]
Finally, Figure 8 shows a conservative-logic realization of the J-¬K flip-flop. (In a figure, when the explicit
value of a sink output is irrelevant to the discussion we shall generically represent this value by a question
mark.) Unlike the previous circuit, where the wires Petrosyan, Artyom, Ciliberto,a Sergio, network
Berut, Antoine, Arakelyan, Artak, act as "transmission" lines, this is sequential
with feedback, and the wire plays an effective role as a "storage" element.
Dillenschneider, Raoul and Lutz, Eric. Experimental veriﬁcation of Landauer’s principle linking information and thermodynamics. Nature 4. COMPUTATION UNIVERSALITYdoi:10.1038/nature10872 LOGIC
483, 187–189 (08 March 2012). An important result of conservative logic is that it is possible to preserve the computing capabilities of
Frank, Michael satisfying the "physical" constraints of reversibility and conservation.
ordinary digital logic while P. Introduction to Reversible Computing: Motivation, Progress, and Challenges. CF ‘05, May 4–6, 2005, Ischia, Italy. Let us consider an arbitrary sequential network constructed out of conventional logic elements, such as AND
and OR gates, inverters F., "NOT" gates), FAN-OUT nodes, and delay elements. For ,definiteness, we shall
Fredkin, E. (or To↵oli, T. Conservative logic. Int. J. Theo. Phys. 21, 3/4
use as an example the network of Figure 9-a serial adder (mod 2). By replacing in a one-to-one fashion these
elements (with the exception of the delay element-cf. footnote at the end of Section 3) with a conservativelogic realization of the same elements (as given, for example, in Figures 4b, 6a, 6b, and 6c), one obtains a
conservative-logic network that performs the same computation (Figure 10). Such a realization may involve
a nominal slow-down factor, since a path that in the original network contained only one delay element may
now traverse several unit wires. (For instance, the realization of Figure 9 has a slow-down factor of 5; note,
however, that only every fifth time slot is actually used for the given computation, and the remaining four
time slots are available for other independent computations, in a time-multiplexed mode.) Moreover, a
number of constant inputs must be provided besides the argument, and the network will yield a number of
garbage outputs besides the result. Figure 9. An ordinary sequential network computing the sum (mod 2) of a stream of binary digits. Recall
that a (+) b = ab +ab....
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