YATOP #1: Are busy beavers lonely creatures?

This is the first of a new series of posts that I tagged with “Yet Another (Tiny) Open Problem”. In these posts I’ll describe some well known or original problems in Theoretical Computer Science that are still unsolved (up to my knowledge). Many of them are not so relevant, so I also added a “Tiny” in the acronym. If you discover that they have been solved or partially solved or, even better, YOU SOLVED them, let me know; I’ll try to keep their open/close status up-to-date.

We begin with a weird problem that involves those cute animals known as Busy Beavers : in computational theory a busy beaver is a Turing machine that attains the maximum number of steps performed before halting or the number of nonblank symbols finally on the tape, among all Turing machines of the same size. The Turing machine must follow some rigid design specifications: it operates on a single two-way unbounded tape initially filled with $0$s and the tape alphabet is $\{0,1\}$ (0 is the blank symbol); it has $n$ states plus a halting state, and every transition is a 5-tuple: (current non-halting state, current symbol, symbol to write, direction of shift, next state). Continue reading