Lecture Notes 2.C

For the u x x2 v ends moment we shall not concern

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Unformatted text preview: ngle at time t. grom the unit wire table Fate, defined by theone obtains by composition more general wires of arbitrary length. Thus, a wire of length i (i • 1) represents a space-time signal path1whosey1 y2 are separated by an interval of i time units. For the u x x2 v ends moment we shall not concern ourselves with0 specific spatial layout of such a path (cf. constraint P8). 0 0 the 0 0 0 Observe that the unit wire is invertible, c0 0 1 0 1 0 i.e., it conserves in the output the number of 0's and l's onservative ( 010 001 that are present at the input), and is mapped into its inverse by the transformation t -t. 0 1 1 0 1 1 (2) 00 100 2.4. Conservative-Logic Gates; 1 CHAPTER II. PHYSICS OF COMPUTATION The Fredkin Gate. Having introduced a primitive whose role 54 1 0 1 to1represent in a stylized way physical computing events. 01 is to represent signals, we now need primitives 110 110 111 111 and graphically represented as in Figure 2a. This computing element can be visualized as a device that performs conditional crossover of two data signals according to the value of a control signal (Figure 2b). F for unit wire. wire. When this value is Figure II.9: Symbol igure 1.parallel paths; when& To↵oli, 1982) Observe that the 1 the two data signals follow The unit(Fredkin 0, they cross over. Fredkin gate is nonlinear and coincides with its own inverse. A conservative-logic gate is any Boolean function that is invertible and conservative (cf. Assumptions P5 and P7 above). It is well known that, under the ordinary rules of function composition (where fan-out is allowed), the two-input NAND gate constitutes a universal primitive for the set of all Boolean functions. In conservative logic, an analogous role is played by a single signal-processing primitive, namely, the Fredkin gate, defined by the table u x1 x2 v y1 y2 000 000 001 010 F “(a) Symbol0 and (b) (b) operation of the Fredkin gate.” (Fredkin 10 0 1 Figure II.10: igure 2. (a) Symbol and 0operation of the Fredkin gate. 0 1 1 0 1 1 (2) & To↵oli...
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