Worm Principle

The Worm Principle is a game introduced by Lev Beklemishev that takes a very long time to terminate.

The game starts with a sequence of numbers (f(0),f(1),...,f(n)). For the mth step of the game, we implement the following:

1. If f(n) = 0, the next sequence is (f(0),f(1),...f(n-1)).

2. Otherwise, define k to be the maximum number such that f(k) < f(n). We then define the good part of the sequence to be r = (f(0),...,f(k)) and the bad part to be s = (f(k+1),...,f(n-1),f(n)-1). (Note that f(n) is decremented by 1.) If there is no k for which f(k) < f(n), define r to be the empty string and s = (f(0),...,f(n-1),f(n)-1). We then define the new sequence to be r * s * s * ... * s, with m+1 copies of s. That is, we append m+1 copies of s to r.

Beklemishev proved that, no matter what the starting string is, the game always reaches the empty string. The number of steps required for the string (n) to terminate grows like \(f_{\varepsilon_0}(n)\).