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Here are some functions I made up using Conway's Game of Life.

Informal definition

  • Die(n) = the maximal number of steps that a configuration of at most n cells can make before dying out.
  • SS(n) = the maximal number of steps that a configuration of at most n cells can make when stabilizing. (when there is no configuration that stabilizes, this returns 0)
  • SC(n) = the maximal number of cells that a configuration of at most n cells can have when stabilizing. (when there is no configuration that stabilizes, this returns 0)
  • NGC(n) = the maximal number of cells that a configuration of at most n cells can have though its number of cells is still bounded above by some number K for all generations.
  • NGS(n) = the maximal number of steps that a configuration of at most n cells can make though its number of cells is still bounded above by some number K for all generations.
  • Osc(n) = the maximal period of an oscilliator with at most n cells.

The starting patterns must fit in an N2 by N2 square.

Formal definitions

Let C(P,t) be the number of cells alive at time t (natural number or 0), with a starting pattern P.

Let H(P) be the width of the pattern P. (horizontal)

Let V(P) be the length of the pattern P. (vertical)

Let C be a cell. And A(C,Q,t) is the condition of the cell returns 1 if alive and 0 if dead.

Now, let Q be a pattern that statistifies the requirement \(C(Q,0)≤N,H(Q)≤N^2,V(Q)≤N^2\).

The following functions are functions from \(\mathbb{N}\) to \(\mathbb{N}\):

\(\text{Die}(N) = \text{max}\{t|\exists Q: [C(Q,t)=0 \wedge C(Q,t-1) \neq 0]\}\)

\(\text{NGC}(N) =\text{max}\{C(Q,t)|\exists K \forall T: C(Q,T) < K\}\)

\(\text{SC}(N)=\text{max}\{C(Q,t)|\forall C: A(C,Q,t) = A(C,Q,t+1)\}\)

\(\text{SS}(N)=\text{max}\{t|\exists Q: [\forall C: A(C,Q,t) = A(C,Q,t+1)\wedge \exists C: A(C,Q,t-1) \neq A(C,Q,t)]\} \)

The other 2 will or will not come...

1, 2, 3 cells

Die(1), SS(1), SC(1), NGC(1), NGS(1) and Osc(1)

1 cell dies out after one step, so Die(1) = 1. No pattern with 1 cell stabilizes, so SS(1) = SC(1) = 0. NGC(1) is also 0, since a died out doesn't grow in number of cells. But a died out pattern stopped growing, so NGS(1) = 1. (just as Die(1)). Also, no oscilliator with 1 cell exists, but the empty board, so Osc(1) = 1.

Die(2), SS(2), SC(2), NGC(2), NGS(2) and Osc(2)

2 cells dies out after one step, so Die(2) = 1. No pattern with 2 cells stabilizes, so SS(2) = SC(2) = 0. NGC(2) is also 0, since a died out doesn't grow in number of cells, but has 0 cells. But a died out pattern stopped growing, so NGS(2) = 1. Also, no oscilliator with 1 cell exists, but the empty board, so Osc(2) = 1.

Die(3) and NGS(3)

Step 1   Step 2   Step 3

  x        x
 x x       x

So this proves NGS(3) ≥ Die(3) ≥ 2. By analysing all configurations it can be proven that NGS(3) = Die(3) = 2.

SS(3) and SC(3) and NGC(3)

SS(3) = 1 and NGC(3) = SC(3) = 4.

Step 1   Step 2

  x       xx
 xx       xx

Osc(3)

Osc(3) = 2.

Step 1   Step 2

           x
 xxx       x
           x

Some sort of an idea

We can build some sort of a Turing Machine on this....

We have a machine with N states. Every state can specify a rule. (The rule is presented as S/B, being S the cells that survive if they have that many neighbours, and B being the cells that are born when they have that many neighbours. So Conway's life is 23/3.). Furthermore, a new state is specified. Also, a begin situation with N cells is specified. 

Now, Life(N) is the maximal number of steps that a configuration with N states and N cells can make before stopping growing in number of cells. I have no idea whether its well defined or not.

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