Busy beavers gone wild

When considering a class of Turing machines that halt when started from a blank tape, busy beavers for that class, as coined by Tibor Radó, are those Turing machines which eventually halt after the maximum number of steps (or after producing the maximum number of non-blank symbols). Finding for a class of Turing machines the busy beaver champion (for any of the two competitions) is of course uncomputable [19]. Nevertheless, efforts have been made to try to find the champions starting with the smallest classes of Turing machines. Finding all the different kinds of behavior of Turing machines in a class is quite exhilarating an endeavor. Rona J. Kopp [9] and Allen H. Brady [3] have thoroughly studied the four states and two symbols class and Heiner Marxen and others [15, 16] have tried for more than two decades to experimentally find the busy beaver champion for the five states and two symbols class. Heiner Marxen found a candidate but no one has been able to prove that no other machine does better. This is most certainly because the behaviors involved are too complicated. This state of affairs is an encouragement to the study of what we have called low-level reverse computability: What does it take to be able to comprehend the behavior of a certain class of Turing machines? What is needed to be able to prove these behaviors? These questions raise the links between the study of busy beavers and incompleteness.