popper-logic-scientific-discovery

APPENDIX iv
A Method of Constructing Models of
Random Sequences (cf. sections 58,
64, and 66)
We assume (as in section 55) that for every given finite number n a
generating period can be constructed which is n-free (from after
effect) and which shows equal distribution. In every such period, every
combinatorially possible x-tuple (for x � n + 1) of ones and zeros will
appear at least once.* 1
(a) We construct a model sequence which is ‘absolutely free’ (from
* 1 There are various construction methods applicable to the task of constructing a generating
period for an n-free sequence with equidistribution. A simple method is the following.
Putting x = n + 1, we first construct the table of all the 2 x possible x-tuples of ones and
zeros (ordered by some lexicographic rule—say, according to magnitude). Then we
commence our period by writing down the last of these x-tuples, consisting of x ones,
checking it off our table. We then proceed according to the following rule: always add a
zero to the commencing segment if permissible; if not, add a one instead; and always check
off the table whatever is the last created x-tuple of the commencing period. (Here ‘if
permissible’ means ‘if the thereby created last x-tuple of the commencing period has not yet
occurred, and thus not yet been checked off the table’.) Proceed in this manner until all
the x-tuples of the list have been checked off: the result is a sequence of the length
2 x + x − 1, consisting of (a) a generating period, of the length 2 x = 2 n + 1 , of an n-free
alternative to which (b) the first n elements of the next period have been added. A
sequence constructed in this way may be called a ‘shortest’ n-free sequence, since it is easily