Thursday, April 23, 2020

The Game of Life and the multiverse

John Horton Conway
As I said in the previous post in this blog, The Game of Life is a cellular automaton devised by John Conway. Let's see how it works, in a little more detail:
This cellular automaton acts on a potentially infinite two-dimensional space, divided into square cells. In each cell there is a simple automaton, or if you want, a program with two states that we can call alive and dead, or 1 and 0. The program in each cell takes as input its own state and the states of its eight neighbors. If it is alive (i.e. in state 1) and two or three of its neighbors are alive, in the next instant it will still be alive. If it is dead (in state 0) and exactly three of its neighbors are alive, in the next instant it will become alive. In any other case, it will become dead. Let's look at a figure to make it clearer:

The program in the center box is in state 1. As two of its eight neighbors are in state 1, the next instant it will continue to be in state 1 (i.e. alive).
It is usual to represent those squares whose associated program is alive (in state 1) by filling the square in black and leaving white the squares in the dead state (0), or vice versa: using black for dead squares and white for those alive.
It seems quite simple, but these rules give rise to very complex behaviors. Let’s see, for example, a disposition called a glider, because it moves over time, keeping its structure constant.

It can be seen in the figure (where the dead squares are empty) that, after four steps, the combination of zeros and ones that make up the glider has moved diagonally one position down and to the right, recovering its initial shape. If it does not find on its way other alive cells that could modify its behavior, the diagonal movement would continue indefinitely (remember that the grid is potentially infinite).
Another interesting combination is the one shown on the left, which is called the glider gun. After a certain number of steps, the initial arrangement repeats itself, but a glider has come off it and thereafter moves away indefinitely. After another identical set of steps, a new glider appears, following the steps of the first.
By combining two glider guns in various ways, an OR gate can be built (a device that, when impacted by one or two gliders at the same time, generates one output glider if any one of its two input gliders is present); or an AND gate (gliders come out only if both input gliders are present); or a NOT gate (a glider comes out if, and only if the arriving glider is not present). We know that with these three logic gates (OR, AND, NOT) one can build a computer. Therefore, in the grid of the game of life, it is potentially possible (although hardly feasible) to build a computer, capable of solving the same problems as the computer on your table.
This is just a sample of the amazingly complex structures that can be assembled with The Game of Life.
How did Soler and I prove that Tegmark's multiverse doesn't solve the fine-tuning problem?
The Swedish-American mathematician Max Tegmark invented the mathematical multiverse, which is based on the claim that every mathematically coherent structure must exists physically in some universe. In other words, there are an infinite number of universes, as the number of coherent mathematical structures is infinite. Tegmark argues that in this multiverse there would be no fine-tuning problem: as there are infinite universes, some of them must be compatible with our existence and that’s where we are. Then he adds an additional condition that makes it possible to test his theory: the universe where we are must be the most probable among all those compatible with our existence, for otherwise the fine-tuning problem would arise again.
To prove that Tegmark’s multiverse does not solve the fine-tuning problem, Soler and I put forward the following argumentation:
  1. The Game of Life is a coherent mathematical structure. Then, according to Tegmark, somewhere there is a universe that implements the Game of Life.
  2. The Game of Life allows complex structures to exist at the same level as ordinary computers. According to Tegmark’s condition, and for a hypothetical intelligent observer located in the Game of Life universe, the fine-tuning problem would not arise if that universe were the most probable among all those compatible with the existence of this type of complex structures.
  3. Analyzing the cellular automata of the Game of Life type, we have found four different versions of the rules compatible with the existence of complex structures. In addition, an infinite number of versions, also compatible with this existence, must be added, which are built assuming that the rules for state change are not constant, but oscillate over time between the four compatible rules, in all possible ways. Then the probability that a universe capable of supporting complex structures, chosen at random among all the possible universes, is based on fixed rules that do not vary with time, like the Game of Life, is zero (4 divided by infinity). Therefore, in the universe of the Game of Life, the fine tuning problem would arise.
  4. Our universe seems to be specially designed to make life possible (fine-tuning), as the universal constants have fairly critical (but not absolutely critical) values. If these universal constants were not constant, but varied over time without leaving their respective critical zones, the universe would still be compatible with our life.
  5. The number of universes with parameters that vary over time would be infinite. Therefore our universe is not the most probable among all those compatible with our existence, for the probability that we find ourselves in a universe with constant parameters is vanishingly small. Therefore, the condition established by Tegmark himself is not met. Therefore Tegmark’s multiverse doesn't solve the fine-tuning problem.
Note that, unlike other multiverses, Tegmark's multiverse lets us experiment with other universes (The Game of Life is one of them). Therefore, it is possible to reach reasonable conclusions, such as that this multiverse does not solve the problem of fine tuning.
For further details, see our paper Fine tuning explained? Multiverses and cellular automata. F.J. Soler Gil, M. Alfonseca. Journal for General Philosophy of Science, Springer, March 2013. DOI: 10.1007/s10838-013-9215-7.
The same post in Spanish
Thematic Thread on Multiverse and Fine Tuning: Previous Next
Thematic Thread on Synthetic and Artificial LifePrevious Next
Manuel Alfonseca

No comments:

Post a Comment