The Replicator Conjecture states that for any cellular automaton where a cell's state is determined by adding the states of its neighbors, modulo the number of states, every starting pattern will replicate. Okay, that's complicated, so let's start simple.
A cellular automaton is an infinite 2-D grid of square cells, each with a state of 1 or 0, true or false, on or off, alive or dead, whatever you want to call it. Then, a rule is applied. The rule changes the state of each cell in the grid (or keeps it the same) depending on its neighbors. Each cell has 8 neighbors, 4 sharing a side with it and 4 sharing a corner.
A cellular automaton is an infinite 2-D grid of square cells, each with a state of 1 or 0, true or false, on or off, alive or dead, whatever you want to call it. Then, a rule is applied. The rule changes the state of each cell in the grid (or keeps it the same) depending on its neighbors. Each cell has 8 neighbors, 4 sharing a side with it and 4 sharing a corner.
A cell's state depends on how many neighbors it has that are "alive". There are different rules, specifying when a cell will live and when it will die.
The most well-known rule is called Conway's Game of Life, invented (discovered?) by John Conway. In this rule, a dead cell is born if it has exactly 3 live neighbors. This is written B3. A live cell survives if it has exactly 2 or 3 live neighbors. Otherwise, it dies. This is written S23. The entire rule is B3/S23. Changing these numbers can result in different types of behavior.
In some rules, all cells die within a few generations. In others, growth expands chaotically and uncontrollably. In some, like Conway's Game of Life, everything becomes small patterns of live cells that remain on the grid indefinitely.
There is a rule where everything expands, but has extremely regular behavior. In fact, every starting pattern makes copies of itself! The rule is B1357/S1357, also known as the Replicator rule.
However, cellular automata (CA's) don't have to be 2-dimensional. They can be 1-D, too, where there is a line of cells, instead of a grid. Every cell has 3 neighbors: one on its left, one on its right, and itself. This works the same way as the 2-D CA: the rule determines whether the cell will live or die from how many of its neighbors are alive. The 1-D rule B13/S13 is also a replicator; every starting pattern replicates!
There is a pattern in these rules. If a cell has an odd number of neighbors, it is alive. If it has an even number, it is dead. This is equivalent to adding all of the neighbors' states and taking the remainder when it is divided by 2. In mathematics, taking the remainder is usually referred to as "modulo". For example, 7 modulo 3 is 1, because the remainder of the operation 7 divided by 3 is 1.
There are even more possibilities if you consider that CA's don't have to have 2 different states. They can have 3 or more! Some of these rules also have the property of everything replicating, but it does not fit the definition above: the sum of the states modulo 2. Where did that 2 come from? It turns out that that number is equal to the number of states! This can be applied to the 3-state CA's by simply changing the 2 to a 3. For any number of states, just put in a different number.
So now that everything has been explained, here is the Replicator Conjecture once more: In a CA where the state of a cell is determined by adding the states of all of its neighbors, modulo the number of states, every starting pattern replicates. It does not matter where the neighbors are, how many dimensions the CA has, or how many states it has (although it has to be 2 or greater).
Keep in mind that this is still a conjecture and has not been proven (at least, as far as I know).
The most well-known rule is called Conway's Game of Life, invented (discovered?) by John Conway. In this rule, a dead cell is born if it has exactly 3 live neighbors. This is written B3. A live cell survives if it has exactly 2 or 3 live neighbors. Otherwise, it dies. This is written S23. The entire rule is B3/S23. Changing these numbers can result in different types of behavior.
In some rules, all cells die within a few generations. In others, growth expands chaotically and uncontrollably. In some, like Conway's Game of Life, everything becomes small patterns of live cells that remain on the grid indefinitely.
There is a rule where everything expands, but has extremely regular behavior. In fact, every starting pattern makes copies of itself! The rule is B1357/S1357, also known as the Replicator rule.
However, cellular automata (CA's) don't have to be 2-dimensional. They can be 1-D, too, where there is a line of cells, instead of a grid. Every cell has 3 neighbors: one on its left, one on its right, and itself. This works the same way as the 2-D CA: the rule determines whether the cell will live or die from how many of its neighbors are alive. The 1-D rule B13/S13 is also a replicator; every starting pattern replicates!
There is a pattern in these rules. If a cell has an odd number of neighbors, it is alive. If it has an even number, it is dead. This is equivalent to adding all of the neighbors' states and taking the remainder when it is divided by 2. In mathematics, taking the remainder is usually referred to as "modulo". For example, 7 modulo 3 is 1, because the remainder of the operation 7 divided by 3 is 1.
There are even more possibilities if you consider that CA's don't have to have 2 different states. They can have 3 or more! Some of these rules also have the property of everything replicating, but it does not fit the definition above: the sum of the states modulo 2. Where did that 2 come from? It turns out that that number is equal to the number of states! This can be applied to the 3-state CA's by simply changing the 2 to a 3. For any number of states, just put in a different number.
So now that everything has been explained, here is the Replicator Conjecture once more: In a CA where the state of a cell is determined by adding the states of all of its neighbors, modulo the number of states, every starting pattern replicates. It does not matter where the neighbors are, how many dimensions the CA has, or how many states it has (although it has to be 2 or greater).
Keep in mind that this is still a conjecture and has not been proven (at least, as far as I know).