allow for a seq to be the in-out definition

[LudoTheHUN]

...so that we write the shortest program that generates that sequence. Could just generate a problem holder that converts the (finite) seq into in-out pairs. To minimise tests, could try to get first element correct, then first two, then 3 etc... There is an interesting effect there, whereby each now item in the seq requires the program search to make n steps before finding the next correct program. Since we are searching in order, there is no need to retest earlier programs (they can not satisfy the new item). Each element of the seq thus has a number associated with it, which is relative to the symbol list, testfun and the search iteration mechanism and which is the number of steps needed to get to the next correct program. If it's 0 with the next item, then the current program still holds. Lets call this number the ikzdorg number. Sequences and subsequences could be classified by the sizes of this number. Ranges of the seq where these are zero (current found program is the generator upto the next non zero ikzdorg) indicate that the Kolmogorov complexity has not increased. We could use this method to detect new information being present at some point in the seq. Brain dump: A 'finite memory' variant of this would try to generate the next element given not the whole seq upto that point, but only n previous values (at any single point). Comparing/contrasting ikzdorg numbers for the fully seq and the finite memory search could identify places where finite memory continues to have ikzdorg=0 while full seq needs to search some more (all this parameterised by the memory lookback) (memory prog presumably would be shorter?). A discrepancy here would indicate an entropy injection (into the seq... the seq is synonymous with observing an environment) but not a change in 'rules' (or 'physics') of the environment, which are being captured by the finite memory program search. ... Here, 'physics' is the program that gets you the next version of the world (a lookback of only 1 for classical worlds), so a shorter program , while full history of observations continuously needs to have more complexity as entropy is added (from the outside).... Need to consider that we are observing a subset of the world... Could be used to detect that you are in a 'closed' system?... that you are in a loop? .... What of all this for actuation, would we search for actions that we could show would require non zero ikzdorg, thus exploring new space?