I’m a bit confused about the second sentence in this snippet: “ We search for the shortest program K(X, T=<42) that produces X within a predefined time / comput

I’m a bit confused about the second sentence in this snippet: “ We search for the shortest program K(X, T=<42) that produces X within a predefined time / computational resource limit. Start with the shortest program and complexify it iteratively spanning a tree of all possible programs.” If we use the time cutoff to make the search for the Kolmogorov complexity practical, then we’re done once we have found the shortest program that produces X. If instead the author means that we can start with the shortest valid program that doesn’t necessarily produce X and then complexify it, then I believe that the logic is incorrect (a shorter valid program that produces X may arise from an otherwise invalid sequence that is not part of the tree that originated from the shortest valid program). If the statement is about finding K(X, T), ie taking the execution time into account, similarity, the optimal program might not result from an iteration of a previously valid shorter program (with the caveat that I’m not fully clear what the author means by complexify and iterate on a tree). Perhaps the suggestion is to search the space of all possible programs sorted by length; if we include invalid programs, we can think of trees of character strings starting from an empty string and spanning all possible strings that are allowed in our programming language. In any case, if we plan the search according to what I think the author means, then once we find a first hit that produces X in less than 42 seconds we are done. Or am I misunderstanding this part completely?