Lecture Notes 2.C

# Figure 18 or 12a thus in the billiard ball model a

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Unformatted text preview: ce. C. REVERSIBLE COMPUTING 63 Figure 14 Billiard model realization of the interaction gate.” (FredFigure II.22: “Billiard ball ball model realization of theinteractiongate. kin & To↵oli, 1982) All of the above requirements are met by introducing, in addition to collisions between two balls, collisions between a ball and a fixed plane mirror. In this way, one can easily deflect the trajectory of a ball (Figure 15a), shift it sideways (Figure 15b), introduce a delay of an arbitrary number of time steps (Figure 1 Sc), and guarantee correct signalis provided by simultaneously ﬁring balls into the input ports for for trivial ¶5. Input crossover (Figure 15d). Of course, no special precautions need be taken crossover, where the1s in the the timing are such that two balls cannot possibly be present at the same the logic or argument. moment at the crossover point (cf. Figure 18 or 12a). Thus, in the billiard ball model a conservative-logic wire is realized as a potential ball path, as determined by the mirrors. ¶6. Inside the box the balls ricochet o↵ each other and ﬁxed reﬂectors, Note that, since whichhave finite the computation. and wires require a certain clearance in order to balls performs diameter, both gates function properly. As a consequence, the metric of the space in which the circuit is embedded (here, we are considering¶the After a ﬁxed time delay, the balls emerging (or not) from the output 7. Euclidean plane) is reflected in certain circuit-layout constraints (cf. P8, Section 2). Essentially, withports deﬁne the output. polynomial packing (corresponding to the Abelian-group connectivity of Euclidean space) some wires may have to be made longer than with exponential packing (corresponding to an abstract space with free-group connectivity)the number of 1s (balls) is conserved. ¶8. Obviously (Toffoli, 1977). ¶9. The computation is reversible because the laws of motion are reversible. ¶10. Interaction gate: Fig. II.22 shows the realization of the computational pr...
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