78
VON NEUMANN
relative levels of excitation of the two input bundles and of the output
bundle, respectively, of the network under consideration. The question is
then: "What is the distribution of the (stochastic) variable £ in terms
of the (given) g, t) ?
Let W be the complementary set of Z . Let p, q, r be the
numbers of elements of X, Y, W, respectively, so that p = |N, q = tjN,
r = (i-£)N. Then the problem is to determine the distribution of the
(stochastic) variable r in terms of the (given) p, q — i.e., the prob
ability of any given r in combination with any given p, q.
W is clearly the intersection of the sets X, Y: W = X-Y. Let
U, V be the (relative) complements of W in X, Y, respectively:
U = X - W, V = Y - W, and let S be the (absolute, i.e., in the set
(1, •••, N)) complement of the sum of X and Y: S = - (X + Y). Then
W, U, V, S are pairwise disjoint sets making up together precisely the
entire set (1, ..., N), with r, p-r, q-r, N-p-q+r elements,
respectively. Apart from this they are unrestricted. Thus they offer to
gether N!/[r!(p - r)I(q - r)!(N - p - q + r )I] possible choices. Since
there are a priori N!/[pI(N - p)IJ possible choices of an X with p
elements and a priori N!/[q!(N — q )I] possible choices of a Y with q
elements, this means that the looked for probability of W having r
elements is
_/ N! / N! N!
p r F("p^rTT(q-rTi"(N"-'p-"q+r)! / p I W p T T qT(K-q7 ! /
= pi (N-p)Iql (N-q)i
r ! (p-r) 1 (q-r) f (N-p-q+r) INI
Note, that this formula also shows that p = 0 when r < 0 or
p-r<0 or q-r<0 or N-p-q+r<0, i.e., when r violates
the conditions
Max(o, p+q-N)<r< Min(p, q).
This is clear combinatorially, in view of the meaning of X, Y and W. In
terms of I, tj , (; the above conditions become
(17) 1 - Max(0, | + n — 1 ) > ? > 1 - Min(|, t] ).
Returning to the expression for p, substituting the |, t], 5
expressions for p, q, r and using Stirling1s formula for the factorials
involved, gives
( 18) p ~ - 1 = ^ e"eN ,
v 2 jtN
where
ed-e)Ti(i-n)
„ =
_
_____
SU-UTiU-ri )
_______
(5+I- 1 )(5+i)-i )(i-5)(2-5-ti-5)
PROBABILISTIC LOGICS 79
e = ( 5 + 4 - 1 ) I n (5 + 4 - 1 ) + U + n - 1 ) I n U + i) - i) +
+ (i - E) I n (1 -
5
) + (
2 - 4
- n - O I n ( 2 - 4 -
n -
£) -
- 4 I n 4 - (1 - 4 ) I n (1 - 4 ) - n I n n - (1 - i)) I n (1 - tj ).
From this
de -in + i - 1) (£ + 'n - 1)
Sir = (1 - 5) (2 - -4 - n - "?T
S2 e 1 . 1 i i
^ 2 e + 6 - 1 ?+T,-i 1 - 6 2 - 4 - n - 5
Hence 6=0, ^ = 0 f°r ? = 1 ~ ll* an<i d2e/df;2 > 0 for all
£ (In its entire interval of variability according to (17)). Consequently
e > 0 for all £ 1 - 4n (within the interval (17))- This implies., in
view of (18) that for all t , which are significantly A 1 - 4n> p tends
to 0 very rapidly as N gets large. It suffices therefore to evaluate
(18) for £ ~ 1 - 41). Now a = 1 /[4 (1 - 4 ) n(i - n)], S2e/d(;2 =
= 1 / 11 (1 - 4 ) n C1 - n ) ] for 5 = 1 - 4 t) . Hence
a ~ 4(1 - iTi T1“ TT ’
v2
( t - (1 - 4t) >r
2 4" (1 - 4 7 ri (1 - t,:
for (; ~ 1 - | t) . Therefore
(19) „
______________
1
■^2n 4 (1 - 4 ) tj (l - t)) N
U - (1 - 4n ) )2
2 4 (1 - 4 ) T) (1 -
is an acceptable approximation for p •
r is an integer-valued variable, hence ^ = i - ^ is a rational
valued variable, with the fixed denominator N. Since N is assumed to be
very large, the range of £ is very dense. It is therefore permissible to
replace it by a continuous one, and to describe the distribution of £ by a
probability-density o . p is the probability of a single value of £, and
since the values of £ are equidistant, with a separation d£ = 1/N, the
relation between o and p is best defined by ad£ = p, i.e., a = pN.
Therefore (19) becomes
(20)
*/2n n/ 4 (1 - 4 ) r, (1 - t) ) / N V i (1 - 4 ) T, (1 - n ) / N
I - (1 - 4 0
This formula means obviously the following:
£ is approximately normally distributed, with the mean 1 -
and the dispersion 41, (1 - | ) t (1 - tj )/N. Note, that the rapid decrease
of the normal distribution function (i.e., the right hand side of (20)) with
N (which is exponential *. ) is valid as long as (; is near to 1 - | t] , only
the coefficient of N (in the exponent, i. e.,
-75*( [£ - (i - I1!)] / 'Ti (1 - I ) t) (i - r)) / ~N)2 is somewhat altered: as
£ deviates from 1 - | . (This follows from the discussion of e given
above.)
The simple statistical discussion of 9 .b amounted to attributing
to 5 the unique value i - £ tj . We see now that this is approximately true:
f £ = (1 - In) + ^1 (1 — 6) n (1 - T)) / N 8,
(21) -s 5 is a stochastic variable, normally distributed, with the
I mean 0 and the dispersion 1 .
10.2.2 THEORY WITH ERRORS. We must now pass from r,£, which postulate
faultless functioning of all Sheffer organs in the network, to r f,£*
which correspond to the actual functioning of all these organs — i.e., to
a probability e of error on each functioning. Among the r organs each
of which should correctly stimulate its output, each error reduces r* by
one unit. The number of errors here is approximately normally distributed,
with the mean er and the dispersion l e (1 - e) r (cf. the remark made
in 10.1). Among the N - r organs, each of which should correctly not
stimulate its output, each error increases r 1 by one unit. The number of
errors here is again approximately normally distributed, with the mean
e (N - r), and the dispersion n/ € (1 - e ) (N - r) (cf. as above). Thus
r 1 - r is the difference of these two (independent) stochastic variables.
Hence it, too, is approximately normally distributed, with the mean
- er + e(N - r) = e(N - 2r), and the dispersion
V(^€ (l - eT^)2 + (-A (1 - e) (N - r) f = Ve (1 - e) N.
I . e ., (approximately)
r* = r + 2e (^ - r) + Ve (1 - e ) N 8 » ,
where 5 1 is normally distributed, with the mean 0 and the dispersion 1 .
Prom this
5* = £ + 2e (1 - £ ) - •V € (1 - e) / N 6' ,
and then by (21)
£ * = (1 - in) + 26 (S'! - j ) +
+ ( 1 - 2 e) s/g (1 -£)r! (1 - r))/N 6 -
- *j~~e n - r> t~'n & r •
Clearly (1 - 2 e) f (1 - | ) r (1 - r ) / N 5 - l e (1 - e ) / N 5 1, too,
is normally distributed, with the mean 0 and the dispersion
V((1 - 26) n/| (1 ) T) (1 - n ) / N f + W€ (1 - 6) / N)2 =
80 VON NEUMANN
= V ((1 - 26 )2 I (1 - I ) n (1 - „) + 6 (1 - «)) / n.
PROBABILISTIC LOGICS
81
Hence (21) becomes at last (we write again t
£ = (i-l'n) + 2 e ( m
(22)
in place of £ 1 ):
+
+ - |)Tl(l-'n) + e(l-e))/N
5*
5* is a stochastic variable, normally distributed, with the mean
0 and the dispersion 1.
10.5 The Restoring Organ
This discussion equally covers the situations that are dealt with
in Figures 35 and 37, showing networks of Sheffer organs in 9 -4 .2 .
Consider first Figure 55. We have here a single input bundle of
N lines, and an output bundle of N lines. However, the two-way split
and the subsequent "randomizing” permutation produce an input bundle of
2N lines and (to the right of U) the even lines of this bundle on one
hand, and its odd lines on the other hand, may be viewed as two input bundles
of N lines each. Beyond this point the network is the same as that one of
Figure 34, discussed in 9 .4 .1 . If the original input bundle had |N stimu
lated lines, then each one of the two derived input bundles will also have
|N stimulated lines. (To be sure of this, it is necessary to choose the
"randomizing" permutation U of Figure 35 in such a manner, that it per
mutes the even lines among each other, and the odd lines among each other.
This is compatible with its "randomizing" the relationship of the family of
all even lines to the family of all odd lines. Hence it is reasonable to
expect, that this requirement does not conflict with the desired "random
izing" character of the permutation.) Let the output bundle have (;N stimu
lated lines. Then we are clearly dealing with the same case as in (22), ex
cept that it is specialized to | = t .
Hence (22) becomes:
5 = (1 - 42 ) + 2e ( | 2 - I) +
(2 3 ) + V u i - 2 e ) 2 (4 (1 - I ) ) 2 + € (1 - e)) / N 5*
5* is a stochastic variable, normally distributed, with the
mean 0 and the dispersion 1 .
Consider next Figure 37. Three bundles are relevant here: The
input bundle at the extreme left, the intermediate bundle issuing directly
from the first tier of Sheffer organs, and the output bundle, issuing di
rectly from the second tier of Sheffer organs, i.e., at the extreme right.
Each one of these three bundles consists of N lines. Let the number of
stimulated lines in each bundle be £N, cuN, jrN, respectively. Then (23)
above applies, with its |,£ replaced first by £, a>, and second by <d, tJt:
82
VON NEUMANN
(2*0
to = (l - £ 2 ) + 2e (£ 2 - |-) +
+ V ( ( 1 - 2 e ) 2 (£(1 - O ) 2 + e (1 - e ) )/N 5 ** ,
i|f = (1 - to2 ) + 2e (ai2 - I) +
+ V ( ( 1 - 2 « ) 2 M 1 " ® ) ) 2 + e (1 - € ) )/N 8 *** ,
5 } 5*** are stochastic variables, independently and normally
distributed, with the mean 0 and the dispersion 1.
10.4 Qualitative Evaluation of the Results
In what follows, (22) and (24) will be relevant — i.e., the Sheffer
organ networks of Figures 34 and 37.
Before going into these considerations, however, we have to make
an observation concerning (22). (22) shows that the (relative) excitation
levels | , t] on the input bundles of its network generate approximately
(i.e., for large N and small e) the (relative) excitation level
= 1 - | t] on the output bundle of that network. This justifies the state
ments made in 9.4.1 about the detailed functioning of the network. Indeed:
If the two Input bundles are^ both prevalently stimulated, i.e., if | ~ 1,
t) ~ 1 then the distance of from 0 is about the sum of the distances
of | and of t) from 1: (;q = (1 - | ) + £ (1 - tj). If one of the two
input bundles, say the first one, is prevalently non-stimulated, while the
other one is prevalently stimulated, i.e., if I ~ 0, t] ~ 1 , then the dis
tance of £0 from 1 is about the distance of | from 0: 1 - = |tj.
If both input bundles are prevalently non-stimulated, i.e., if | ~ 0,
tj _ 0, then the distance of from 1 is small compared to the distances
of both | and tj from 0: 1 - £0 = £ ti .
10.5 Complete Quantitative Theory
10.5.1 GENERAL RESULTS. We can now pass to the complete statistical analysis
of the Sheffer stroke operation on bundles. In order to do this, we must
agree on a systematic way to handle this operation by a network. The system
to be adopted will be the following: The necessary executive organ will be
followed in series by a restoring organ. I.e., the Sheffer organ network of
Figure 34 will be followed in series by the Sheffer organ network of Figure 37-
This means that the formulas of (22) are to be followed by those of (24).
Thus |,t) are the excitation levels of the two input bundles, t is the
excitation level of the output bundle, and we have:
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