User blog:QuasarBooster/Higher order worms - more formally

I felt like outlining the worms and rules from my last blog post more precisely. As the core ideas are so similar to Beklemishev worms, there will be a heavy resemblance to the main article.

Preliminary definitions
An order-k worm \(W^k\) can be defined inductively as follows:

Any two worms of the same order can be compared using the relation \(\prec\) like so:
 * W^0=n\in\mathbb N
 * W^{k+1}=[W_1^k,W_2^k,...,W_n^k]


 * If \(A\) and \(B\) are order-0 then \(A\prec B\) iff \(A<B\).
 * Otherwise, define \(k = \min_i A_i \neq B_i\). Then \(A\prec B\) iff \(A_k\prec B_k\) . If \(k\) does not exist, \(A\prec B\) iff \(A\) contains fewer entries than \(B\).

Description
You are presented with an arbitrary worm \(W\), and your task is to reduce it to an empty list (i.e. a worm is considered 'dead' if it is 0 or an empty list). On turn \(m\) of the game, the worm is altered by the function \(\text{next}(W, m)\):


 * If \(W\) is order-0, then \(\text{next}(W, m) = W - 1\).
 * Else, if \(W_n\) is dead, then \(\text{next}(W, m) = [W_0, W_1, \ldots, W_{n-1}]\). (That is, you chop off the worm's head.)
 * Otherwise, define \(k = \max_{i < n} W_i \prec W_n\). We define the good part of the sequence to be \(g = [W_0, \ldots, W_k]\) and the bad part to be \(b = [W_{k+1}, \ldots, W_{n-1}, \text{next}(W_n, m)]\). (Note that \(W_n\) is altered as if it were on turn \(m\).) If \(k\) is nonexistent, define \(g\) to be an empty list and \(b = [W_0, \ldots, W_{n-1}, \text{next}(W_n, m)]\). 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].)

It has not yet been proven whether you can always defeat any higher-order worm. It is interesting to note that in the special case where \(W\) is order-2 and each \(W_i\) contains \(a_i\) zeroes as its only entries, then \(W\) can be defeated in precisely as many steps as it takes to defeat \(W' = [a_1,a_2,...,a_n]\).

If we assume that any worm can be reduced to an empty list, then we can create a specific fast-growing function. Define \(\text{Wormception}(n)\) to be the number of steps required to defeat a worm starting with \(W = ...[n]...\) (with \(n\) pairs of brackets). It can be shown, by comparison with order-1 worms, that the function's growth rate is at least \(f_{\varepsilon_0}(n)\).

Example
The first few steps required to compute \(\text{Wormception}(3)\) are shown below.
 * Start: [3]
 * Step 1: [2, 2], [2, 2, 2, 2], [2, 2]
 * Step 2: [2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1]
 * Step 3: [2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1, 2, 2], [2, 2, 2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 1], [2, 2], [2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 2, 0]