Is it possible to really know when a program is simply repeating the same pattern over and over again?
Conway's Game of Life is 100% deterministic, which means that no matter when you encounter a pattern, you always know exactly what the next evolution of the pattern will be. On top of this, a given input in one generation will always result in one specific output for the next generation, regardless of when that input is received.
So to find periods in the evolution of the state, all you'd have to do is detect when/if a duplicate state appears; at that moment, you know you've found a cycle. I'm going to write my example code in C++, but any language which has a "Hash Table" or similar data structure can use the same basic algorithms.
//We're expressly defining a grid as a 50x50 grid.
typedef std::array<std::array<bool, 50>, 50> Conway_Grid;
struct Conway_Hash {
size_t operator()(Conway_Grid const& grid) const {
size_t hash = 0;
for(int i = 0; i < grid.size(); i++) {for(int j = 0; j < grid[i].size(); j++) {
if(grid[i][j])
hash += (i * 50 + j);
//I make no guarantees as to the quality of this hash function...
}}
return hash;
}
};
struct Conway_Equal {
bool operator()(Conway_Grid const& g1, Conway_Grid const& g2) const {
for(int i = 0; i < grid.size(); i++) {for(int j = 0; j < grid[i].size(); j++) {
if(g1[i][j] != g2[i][j])
return false;
}}
return true;
}
};
typedef int Generation;
std::unordered_map<Conway_Grid, Generation, Conway_Hash, Conway_Equal> cache;
Conway_Grid get_next_gen(Conway_Grid const& grid) {
Conway_Grid next{};
for(int i = 1; i < grid.size() - 1; i++) {for(int j = 1; j < grid[i].size() - 1; j++) {
int neighbors = 0;
for(int x = i - 1; x <= i + 1; x++) { for(int y = j - 1; y <= j + 1; y++) {
if(x == i && y == j) continue;
if(grid[x][y]) neighbors++;
}}
if(grid[i][j] && (neighbors == 2 || neighbors == 3))
next[i][j] = true;
else if(!grid[i][j] && (neighbors == 3))
next[i][j] = true;
}}
return next;
}
int main() {
Conway_Grid grid{};//Initialized all to false
grid[20][20] = true;
grid[21][20] = true;
grid[22][20] = true;//Blinker
for(Generation gen = 0; gen < 1'000; gen++) { //We'll search a thousand generations
auto it = cache.find(grid);
if(it != cache.end()) {//"Is the grid already in the cache?"
std::cout << "Period found at generation " << gen;
std::cout << ", which was started on generation " << it->second;
std::cout << ", which means the period length is " << gen - it->second << '.' << std::endl;
break;
}
cache[grid] = gen; //"Inserts the current grid into the cache"
grid = get_next_gen(grid); //"Updates the grid to its next generation"
}
return 0;
}
Note that this code actually works for any period length, not just a length less than 4. In the above code, for a blinker (three cells in a row), we get the following result:
Period found at generation 2, which was started on generation 0, which means the period length is 2.
As a sanity check, I decided to import a Gosper Glider Gun to make sure that it worked just as well.
grid[31][21] = true;
grid[29][22] = true;
grid[31][22] = true;
grid[19][23] = true;
grid[20][23] = true;
grid[27][23] = true;
grid[28][23] = true;
grid[41][23] = true;
grid[42][23] = true;
grid[18][24] = true;
grid[22][24] = true;
grid[27][24] = true;
grid[28][24] = true;
grid[41][24] = true;
grid[42][24] = true;
grid[7][25] = true;
grid[8][25] = true;
grid[17][25] = true;
grid[23][25] = true;
grid[27][25] = true;
grid[28][25] = true;
grid[7][26] = true;
grid[8][26] = true;
grid[17][26] = true;
grid[21][26] = true;
grid[23][26] = true;
grid[24][26] = true;
grid[29][26] = true;
grid[31][26] = true;
grid[17][27] = true;
grid[23][27] = true;
grid[31][27] = true;
grid[18][28] = true;
grid[22][28] = true;
grid[19][29] = true;
grid[20][29] = true;
Gosper's Glider Gun doesn't normally have a period, since it creates an infinite number of gliders over time, and the pattern never repeats. But because the grid is bounded, and we simply erase cells on the border of the grid, this pattern will eventually create a repeating pattern, and sure enough, this program finds it:
Period found at generation 119, which was started on generation 59, which means the period length is 60.
(This is doubly good, because the period of just the gun is supposed to be 60)
Note that this is almost certainly not the best solution to this problem, as this solution saves each generated grid in memory, and for larger grids, this will both eat up RAM and CPU cycles. But it's the simplest solution, and you'll likely be able to find a similar solution for whichever programming language you're using.
