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Similarly, the output bundle error for a NAND MUX system is the sum of the
errors at the two input bundles, if both the input bundles are prevalently
stimulated. This implies that certain worst-case input configurations may produce
outputs lying in the intermediate region of uncertain information. Thus, von
Neumann proposed that a logical stage is required to transform an input bundle
with a stimulation level close to zero or one into an output bundle with
stimulation level closer to the corresponding extremes [47].
Hence, the von Neumann multiplexed systems must contain a signal restora-
tion stage that reduces the degradation caused by the executive stage for certain
input bundle configurations. The signal carried by each input bundle of an
executive stage can be thought of as a probability wave that deviates from the
prevalent logic values due to signal noise at the executive stage gates. The purpose
of a restorative stage is to counteract this deviation and increase the probability of
the output bundle being in a valid stimulated or non-stimulated state. Figure 10.11
shows the restorative stages for the NAND and MAJ MUX schemes. Correlation
between the input bundles will destroy the signal restoration process, hence, the
logic circuit U (see Fig. 10.11) provides a randomizing effect to provide statistical
independence between the output bundle of the executive stage and the input to
the parallel NAND or MAJ gates that form the restorative stage.
The number of restorative stages can be increased arbitrarily, but it has been
shown in [41, 55] that the output distributions of the MUX systems stabilize very
quickly as the number of restorative stages increases while causing degradation in
speed and increase in the redundancy factor R. It can be observed from Figure
10.11, that R for a MUX system is computed as a function of the bundle size
(number of replicated gates in each MUX stage) and the number of MUX stages.
For NAND MUX, R is computed as Bundle Size (1+2 Number of Restorative
Stages), whereas, for MAJ MUX, R is evaluated as Bundle Size (1+Number of
Restorative Stages). For instance, a NAND MUX system with a bundle size of 10
and 2 restorative stages (5 MUX stages) has a R of 50, and a MAJ MUX system
with a similar configuration has a R of 30 (3 MUX stages).
Although von Neumann had proposed both NAND and MAJ MUX systems
in [47], most researchers extended his analysis of the NAND MUX architecture.
von Neumann had analyzed the NAND MUX system for N
1000 by exploiting
the Gaussian nature of the output distribution. This theory was extended to
N
W
1000 in [55], and it was proved that the output distribution is binomial for
smaller values of N. [56] introduces a modification of the von Neumann NAND
MUX technique called parallel restitution. Parallel restitution is simply a
methodology to apply the NAND MUX scheme to a large system to make it
fault-tolerant to noise. [57] was probably the first theoretical attempt at analyzing
the reliability of MAJ MUX. In that work, the authors analyzed the reliability of
MAJ MUX architectures theoretically for small R and extended this scheme by
eliminating unnecessary restorative stages.
The integration of reconfigurable architectures and NANDMUX has also been
evaluated in [16] to circumvent both permanent and transient faults. In [58-60],
different fault-tolerant techniques including NAND MUX were used on a chip
o
 
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