No. We can't. If the algorithm that generates the sequence is simple, lets call it algorithm X, then a concise way of expressing the string is "the output of al

No. We can't. If the algorithm that generates the sequence is simple, lets call it algorithm X, then a concise way of expressing the string is "the output of algorithm X". That expression losslesly compresses the output, and it's just a few characters longer than the algorithm itself. Randomness theory and information theory study such things. If you add a rule such as not using Algorithm X, it's trivial to create an Algorithm Y that generates the same output and it's barely any longer than X. For example, by adding any meaningless instruction at the start of algorithm X. Of course, discovering what the shortest expression for a given sequence is is not a computable problem. So we can't always know if we've found the best form of compression for a sequence, we can only find an upper bound.