Conservation of information sparked scientific interest once a recurring pattern was noticed in the evolutionary computing literature. In grappling with the creation of information through evolutionary algorithms, this literature consistently revealed that the information outputted by such algorithms always needed first to be programmed into them. Thus, the primary goal of thisliterature—to uncover how information could be created from scratch or de novo—was shown to be misconceived: the information was not created but instead shuffled around or smuggled in, implying that it already existed in some form or other. Information output in these situations therefore always presupposed a counterbalancing input of prior information.

Once this pattern was seen, the next logical step was to quantify the amount of information inputted and outputted, demonstrating a consistent mathematical relation between the two. This led to the proof of a number of theorems about search. In these theorems, a baseline search with probability p of success gave way to an improved search with probability q of success. Typically p wouldbe very small and close to zero, implying a practically impossible search (like searching for a needle in a haystack). By contrast, q would be much larger and close to one, implying an eminently doable search. The punchline of these theorems was that, as the improved search became itself the subject of a search (a search for a search, or S4S), the probability of finding it could not exceed p/q, rendering success of the improved search no more probable than success of the original baseline search, in effect filling one hole by digging another.

Such conservation-of-information theorems, as they came to be called, were search-space specific, adapted to different kinds of search across a range of search spaces. There was a measure-theoretic theorem in which probability measures guided search. There were also function-theoretic and fitness-theoretic theorems where mappings into the search space as well as fitness functions onthe search space respectively guided search. The key insight of this paper is that all these conservation-of-information theorems are special cases of a simple probabilistic relation based on elementary probability theory. This paper identifies the underlying rationale that makes all the previous conservation-of-information theorems work. In so doing, it provides a straightforward proof and general formulation of what may rightly be called the Law of Conservation of Information.