Three paths to effectiveness

Over the past two decades, Gurevich and his colleagues have developed axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in a new framework of abstract state machines. Recently, this approach was extended to suggest axiomatic foundations for the notion of effective computation over arbitrary countable domains. This was accomplished in three different ways, leading to three, seemingly disparate, notions of effectiveness. We show that, though having taken different routes, they all actually lead to precisely the same concept. With this concept of effectiveness, we establish that there is - up to isomorphism - exactly one maximal effective model across all countable domains.