Lecture Notes 2.C

# Inwhat follows whenever we speak of the realizability

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Unformatted text preview: (Figure 3b), which is suggestive of, a recise way. respectively, the trailing and the leading edge of a wire. Observe that in conservative-logic circuits the number of output ports always equals that of input ones. The junction between two adjacent unit wires can be formally treated as a node consisting of a trivial conservative-logic gate, namely, the identity gate. Inwhat follows, whenever we speak of the realizability of a function in terms of a certain set of conservative-logic primitives, the unit wire and the identity gate will be tacitly assumed to be included in this set. A conservative-logic circuit is a time-discrete dynamical system. The unit wires represent the system’s individual state variables, while the gates (including, of course, any occurrence of the identity gate) collectively represent the system’s transition function. The number N of unit wires that are present in the Figure 4. be thought of “Behavior of the dFredkin gate (a) of the unconstrained inputs, F Behavior of as the number of egrees of freedom with system. Of these N wires, the circuit may igure II.13:the Fredkin gate (a) with unconstrained inputs, and (b) with x2 constrained toat any v the remaining N0 ( N AND function. moment Nand (b) with x2 1, andalue 0, thus realizing=the thus )realizingin state 0. The quantity N1 is an 1 will be in state constrained to the value 0, - N1 will be the AND function.” additive function of theTo↵oli,’s1982) i.e., is defined for any portion of the circuit and its value for the (Fredkin & system state, whole circuit is the sum of the individual contributions from all portions. Moreover, since both the unit wire and the gates return at their outputs as many l’s as are present at their inputs, the quantity N1 is an integral of the motion of the system, i.e., is constant along any trajectory. (Analogous considerations apply to the quantity N0, but, of course, N0 and N1 are not independent integrals of the motion.) It is from this "conservation principle"...
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## This document was uploaded on 03/14/2014 for the course COSC 494/594 at University of Tennessee.

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