PROBABILISTIC LOGICS
73
because of g( l - a) s 1 - g(a) there is complete symmetry between the
a < 1 /2 region and the a > 1 /2 region. *
a
FIGURE 32 FIGURE 33
The process a a* thus brings every a nearer to that one of
0 and 1, to which it was nearer originally. This is precisely that pro
cess of restoration, which was seen in 9.2.2 to be necessary. I.e., one or
more (successive)applications of this process will have the required restor
ing effect.
Note, that this process of restoration is most effective when
o 2
a - a* = 2aJ - 3a + a has its minimum or maximum, i.e., for
6a2 - 6a + 1 = 0, i.e., for a = (3 n/~3)/6 = .788, .212. Then
a - a* = + .096. I.e., the maximum restoration is effected on error levels
at the distance of 21.2# from 0$ or 100# — these are improved (brought
nearer) by 9.6$.
9.3 Other Basic Organs
We have so far assumed that the basic components of the construction
are majority organs. From these, an analog of the majority organ — one which
picked out a majority of bundles instead of a majority of single lines — was
constructed. Since this, when viewed as a basic organ, is a universal organ,
these considerations show that it is at least theoretically possible to con
struct any network with bundles instead of single lines. However there was
no necessity for starting from majority organs. Indeed, any other basic
system whose universality was established in section b can be used instead.
The simplest procedure in such a case is to construct an (essential) equiva
lent of the (single line) majority organ from the given basic system
(cf. k.2 .2 ), and then proceed with this composite majority organ in the same
way, as was done above with the basic majority organ.
Thus, if the basic organs are those Nos. one and two in Figure 10
(cf. the relevant discussion in k . 1 .2), then the basic synthesis (that of the
majority organ, cf. above) is immediately derivable from the introductory
formula of Figure 1 ^ .
7^
VON NEUMANN
9 . 4 The Sheffer Stroke
9 .4.1 THE EXECUTIVE ORGAN. Similarly, it is possible to construct the
entire mechanism starting from the Sheffer organ of Figure 12. In this case,
however, it is simpler not to effect the passage to an (essential) equiva
lent of the majority organ (as suggested above), but to start de novo. Actu
ally, the same procedure, which was seen above to work for the majority organ,
works mutatis mutandis for the Sheffer organ, too. A brief description of
the direct procedure in this case is given in what follows:
Again, one begins by constructing a network which will perform the
task of the Sheffer organ for bundles of inputs and outputs instead of single
lines. This is shown in Figure 3^ for bundles of five wires. (The connect
ions are replaced by suitable markings, as in Figures 29 and 30.)
It is intuitively clear that if almost all lines of both input
bundles are stimulated, then almost none of the lines of the output bundle
will be stimulated. Similarly, if almost none of the lines of one Input
bundle are stimulated, then almost all lines of the output bundle will be
stimulated. In addition to this overall behavior, the following detailed
behavior is found (cf. the detailed consideration In 10.4 ). If the condition
of the organ is one of prevalent non-stimulation of the output bundle, and
hence is governed by (prevalent stimulation of) both input bundles, then the
most probable level of the output error will be (approximately) the sum of
the errors in the two governing input bundles; if on the other hand the con
dition of the organ is one of prevalent stimulation of the output bundle,
and hence is governed by (prevalent non-stimulation of) one or of both input
bundles, then the output error will be on (approximately) the same level as
the input error, if (only) one input bundle is governing (i.e., prevalently
non-stimulated), and it will be generally smaller than the input error, if
both input bundles are governing (i.e., prevalently non-stimulated). Thus
two significant inputs may produce a result lying in the intermediate zone
of uncertain information. Hence a restoring organ (for the error level) is
again needed, in addition to the executive organ.
9-^.2 THE RESTORING ORGAN. Again, the above indicates that the restoring
organ can be obtained from a special case functioning of the standard execu
tive organ, namely by obtaining all inputs from a single input bundle, and
seeing to it that the output bundle lias the same size as the original input
bundle. The principle is illustrated by Figure 35. The "black box" U Is
again supposed to effect a suitable permutation of the lines that pass
through it, for the same reasons and in the same manner as in the correspond
ing situation for the majority organ (cf. Figure 32). I.e., it must have a
"randomizing" effect.
If aN of the N incoming lines are stimulated, then the probab
ility of any Sheffer organ being stimulated (by at least one non-stimulated
input) is
PROBABILISTIC LOGICS
75
(15) a+ = 1 - a2 = h(a).
Thus approximately (i.e., with high probability provided N is large)
~ a+N outputs will be excited. Plotting the curve of a+ against a dis
closes some characteristic differences against the previous case (that one
P *3
of the majority organs, i.e., a* = 3a - 2ar = g(a), cf. 9.2.3), which
require further discussion. This curve is shown in Figure 36. Clearly a+
is an antimonotone function of a, i.e., instead of restoring an excitation
level (i.e., bringing it closer to 0 or to 1, respectively), it transforms
it into its opposite (i.e., it brings the neighborhood of 0 close to 1,
and the neighborhood of 1 close to 0). In addition it produces for a
near to 1 an a+ less near to 0 (about twice farther), but for a near
to 0 an a+ much nearer to 1 (second order 1). All these circumstances
suggest, that the operation should be iterated.
Let the restoring organ therefore consist of two of the previously
pictured organs in series, as shown in Figure 37. (The "black boxes" U 1, U 2
play the same role as their analog U plays in Figure 35 . ) This organ trans
forms an input excitation level aN into an output excitation level of ap
proximately (cf. above) ~ a++ where
a++ = ^ - a2 )2 = h(h(a)) = k(a),
i.e.,
(16) a++ = 2a2 - a*4" = k(a).
This curve of a++ against a is shown in Figure 38. This curve is very
similar to that one obtained for the majority organ (i.e.,
p O
a* = 3a - 2a° = g(a), cf. 9.2.3). Indeed: The curve intersects the diag
onal a++ = a in the interval 0 < a < 1 three times: For a = 0, a , 1 ,
— — ’ o
where aQ = (- 1 + T5)/2 = .618. (There is a fourth intersection
a = _ 1 - « = - 1.618, but this is irrelevant, since it is not in the
interval 0 < a < 1 . ) 0 < a < aQ implies 0 < a++ < a; aQ < a < 1 implies
a < a++ < 1.
In other words: The role of the error levels a ~ 0 and a ~ 1
is precisely the same as for the majority organ (cf. 9.2.3), except that the
limit between their respective areas of control lies at a = aQ instead of
at a = 1/2. I.e., the process a -» a++ brings every a nearer to either
0 or to 1, but the preference to 0 or to 1 is settled at a discrimi
nation level of 61.8$ (i.e., aQ ) instead of one of 50$ (i.e., 1/2).
Thus, apart from a certain asymmetric distortion, the organ behaves like its
counterpart considered for the majority organ — i.e., it is an effective
restoring mechanism.
VON NEUMANN
' / / / / / ✓
< —
> =
FIGURE 35
ot
a
FIGURE 36
FIGURE 37
a
PROBABILISTIC LOGICS 77
10. ERROR IK MULTIPLEX SYSTEMS
10.1 General Remarks
In section 9 the technique for constructing multiplexed automata
was described. However, the role of errcrs entered at best Intuitively and
summarily, and therefore it has still not been proved that these systems
will do what is claimed for them — namely control error. Section 10 is
devoted to a sketch of the statistical analysis necessary to show that, by
using large enough bundles of lines, any desired degree of accuracy (i.e.,
as small a probability of malfunction of the ultimate output of the network
as desired) can be obtained with a multiplexed automaton.
For simplicity, we will only consider automata which are con
structed from the Sheffer organs. These are easier to analyze since they
involve only two inputs. At the same time, the Sheffer organ is (by itself)
universal (cf. 4 .2 .1), hence every automaton is essentially equivalent to
a network of Sheffer organs.
Errors in the operation of an automaton arise from two sources.
First, the individual basic organs can make mistakes. It will be assumed
as before, that, under any circumstance, the probability of this happening
is just e . Any operation on the bundle can be considered as a random
sampling of size N (N being the size of the bundle). The number of errors
committed by the individual basic organs in any operation on the bundle is
then a random variable, distributed approximately normally with mean eN
and standard deviation Ve(1- e)N. A second source of failures arises be
cause in operating with bundles which are not all in the same state of
stimulation or non-stimulation, the possibility of multiplying error by un
fortunate combinations of lines into the basic (single line) organs is al
ways present. This interacts with the statistical effects, and in particular
with the processes of degeneration and of restoration of which we spoke In
9.2.2, 9 -2.3 and 9 -4 .2 .
10.2 The Distribution of the Response Set Size
10.2.1 EXACT THEORY. In order to give a statistical treatment of the prob
lem, consider the Figure 34, showing a network of Sheffer organs, which was
discussed in 9 .4.1 . Let again N be the number of lines in each (input or
output) bundle. Let X be the set of those i = 1, . .., N for which line
No. i in the first input bundle is stimulated at time t; let Y be the
corresponding set for the second input bundle and time t; and let Z be
the corresponding set for the output bundle, assuming the correct function
ing of all the Sheffer organs involved, and time t + 1. Let X, Y have
|N, t]N elements, respectively, but otherwise be random — I.e., equidistrib-
uted over all pairs of sets with these numbers of elements. What can then
be said about the number of elements £N of Z? Clearly |, tj , (;, are the
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.