popper-logic-scientific-discovery

152
some structural components of a theory of experience
The generating period of an n-free alternative with equal distribution
must contain at least 2 n + 1 elements. The periods given as examples
can, of course, begin at different places; (C) for example can begin
with its fourth element, so that we obtain, in place of (C)
(C′)
1000011110100101 ...
There are other transformations which leave the n-freedom of a
sequence unchanged. A method of constructing generating periods
of n-free sequences for every number n will be described
elsewhere.* 3
If to the generating period of an n-free alternative we add the first n
elements of the next period, then we obtain a sequence of the length
2 n + 1 + n. This has, among others, the following property: every
arrangement of n + 1 zeros and ones, i.e. every possible n + 1-tuple,
occurs in it at least once.* 4
56 SEQUENCES OF SEGMENTS. THE FIRST
FORM OF THE BINOMIAL FORMULA
Given a finite sequence α, we call a sub-sequence of α consisting of n
consecutive elements a ‘segment of α of length n’; or, more briefly, an
‘n-segment of α’. If, in addition to the sequence α, we are given some
definite number n, then we can arrange the n-segments of α in a
sequence—the sequence of n-segments of α. Given a sequence α, we may
construct a new sequence, of n-segments of α, in such a way that we
* 3 Cf. note *1 to appendix iv. The result is a sequence of the length 2n + n − 1 such that by
omitting its last n − 1 elements, we obtain a generating period for an m-free alternative,
with m = n − 1.
* 4 The following definition, applicable to any given long but finite alternative A, with
equidistribution, seems appropriate. Let N be the length of A, and let n be the greatest
integer such that 2 n + 1 � N. Then A is said to be perfectly random if and only if the relative
number of occurrences of any given pair, triplet, . . . , m-tuplet (up to m = n) deviates
from that of any other pair, triplet, . . . , m-tuplet, by not more than, say, m/N ½
respectively. This characterization makes it possible to say of a given alternative A that it is
approximately random; and it even allows us to define a degree of approximation. A
more elaborate definition may be based upon the method (of maximizing my
E-function) described under points 8 ff. of my Third Note reprinted in appendix *ix.