The Phase of the “1” - Brain Theory from a Circuits and Systems Perspective - page 166

The above method is a variation on Deutsch's quantum algorithm [ 2 ]. For larger
numbers of simulated qubits additional global properties of multivibrator functions
become identifiable.

Symmetric and Antisymmetric Multivibrator Function Classification
Using n Simulated Qubits

Given n synchronized multivibrators, a multivibrator function operates on each
possible combination for which there is a probability. For example, for n ¼ 2, there

could be a finite probability for each combination a 1 b 1 , a 1 b 2 , a 2 b 1 , a 2 b 2 .
The class of functions g sa (symmetric and antisymmetric functions) can be
identified by their patterns in truth tables [ 3 ]. Symmetric means that the entries
below the center of the truth table is a mirror image about the center of the half
above the center. Antisymmetric means that a bitwise complement of the second
half is a mirror image about the center of the first half.
A symmetric or antisymmetric function g sa of n binary variables with dimension

2 n has the property of being symmetric or antisymmetric about the center of its
truth table and being symmetric or antisymmetric in each binary subdivision of 2 ni
entries of the table i

k ¼

1. This class of binary functions has
an even number of true entries in its truth table for n

¼ 1, 2, , n 1 for n >

1 is a limiting case
for constant and non-constant binary functions of one bit as above; any binary
function with n

> 1; n ¼

¼ 1 is either symmetric or antisymmetric. Examples for n ¼ 2 are

provided in Tables 10.3 and 10.4 .
Assume that a multivibrator function in this class has been applied to the
probability space of n multivibrators but that the actual function is unknown. As
above, it is not permitted to place an oscilloscope probe directly into a recursive
neuron because it is assumed small, delicate, and easily upset. A procedure for

Table 10.3 g sa ¼ x 1 x 2 ,

Truth table
x 1

symmetric

x 2 g as

0 0 0

0 1 1

1 0 1

1 1 0

Table 10.4 g sa ¼ x 1 ,

Truth table
x 1

antisymmetric

x 2 g as

0 0 0

0 1 0

1 0 1

1 1 1