Beklemishev's worms are a construction described by Lev D. Beklemishev (Russian: Беклемишев Лев Дмитриевич) that result in a one-player game that takes a long time to terminate.

It is closely related to Kirby-Paris hydras.

Description

A worm is simply a list of nonnegative integers $$[W_0, W_1, \ldots, W_n]$$. In a game Beklemishev calls "the Worm battle," our hero Cedric is presented with an arbitrary worm $$W$$, and his task is to reduce it to an empty list. On turn $$m$$ of the game, the worm is altered by the function $$\text{next}(W, m)$$:

• If $$W_n = 0$$, then $$\text{next}(W, m) = [W_0, W_1, \ldots, W_{n-1}]$$. (That is, Cedric chops off the worm's head.)
• Otherwise, define $$k = \max_{i < n} W_i < W_n$$. We define the good part of the sequence to be $$g = [W_0, \ldots, W_k]$$ and the bad part to be $$b = [W_{k+1}, \ldots, W_{n-1}, W_n - 1]$$. (Note that $$W_n$$ is decremented by 1.) If $$k$$ is nonexistent, define $$g$$ to be an empty list and $$b = [W_0, \ldots, W_{n-1}, W_n - 1]$$. We then define $$\text{next}(W, m) = g + b + b + \cdots + b + b$$ with $$m+1$$ copies of $$b$$. (Here + means sequence concatenation, so for example [0, 3, 2] + [1, 4, 5] = [0, 3, 2, 1, 4, 5].)

Beklemishev proved, in a theorem he calls the Worm principle, that Cedric can always defeat the worm regardless of the initial value of $$W$$. He further showed that this fact is unprovable in Peano arithmetic.

The Worm's expansion rule is almost identical to Primitive Sequence System's expansion system.

From this, we can create a specific fast-growing function. Define $$\text{Worm}(n)$$ to be the number of steps required to defeat a worm starting with $$W = [n]$$. Then $$\text{Worm}(n)$$ is a function that dominates all functions provably recursive in Peano arithmetic, giving the function a growth rate comparable to $$f_{\varepsilon_0}(n)$$.

Examples

• Start: 
• Step 1: [0,0]
• Step 2: 
• Step 3: []

So $$\text{Worm}(1) = 3$$.

• Start: 
• Step 1: [1,1]
• Step 2: [1,0,1,0,1,0]
• Step 3: [1,0,1,0,1]
• Step 4: [1,0,1,0,0,0,0,0,0]
• Step 10: [1,0,1]
• Step 11: [1,013]
• Step 24: 
• Step 25: 
• Step 51: []

So $$\text{Worm}(2) = 51$$.

Sources

1. 
2. 
3. Beklemishev, L. (2006). The Worm principle. In Z. Chatzidakis, P. Koepke, & W. Pohlers (Eds.), Logic Colloquium '02 (Lecture Notes in Logic, pp. 75-95). Cambridge: Cambridge University Press. doi:10.1017/9781316755723.005