There has been a lot of press lately (summarized here) on the possibility that we are living inside of a computer simulation. Much of the attention has been focused on whether or not you could know if you lived in a simulation. Here, I will focus on what you could know (or compute) if you were running the simulation. Before I proceed, I’ll briefly summarize some points about the theory of computation. I’ve alluded to these ideas in several of my recent posts but have never formally introduced them. Obviously, this is very deep area so I’ll just briefly summarize some important points.
The theory of computation is basically about repeated operations on a finite set of symbols according to some rule. The paradigm is the Turing machine, which consists of a finite number of internal states and a tape on which symbols can be written or erased. The Turing machine then makes transitions from state to state based on its current state and the symbols on the tape. One of the states is for the Turing machine to halt, which signals the end of a computation. The important thing to take away is the Church-Turing thesis, which basically states that all forms of computation on finite symbol sets are basically equivalent. For example, the computer on your desk is equivalent to a Turing machine. (Actually, it is even less powerful because it is finite but I digress).
One of the things that Turing proved was that there is a universal Turing machine, which can emulate any other Turing machine. The input to a Turing machine is a set of symbols on the tape, i.e. an input string. Some of the symbols code for instructions or rules and the others code for an initial condition. Turing also showed that there are problems for which a Turing machine can never solve. The most famous is the Halting Problem, which states that there does not exist a Turing machine that can decide if another Turing machine will halt given some input. Turing actually showed that it was impossible to produce a general algorithm to decide if a given input string to a Turing machine will ever cause it to print a given symbol. In other words, there is no general algorithm to decide if a computation will have an infinite loop or perform some specific task. This doesn’t imply that you couldn’t prove that a specific program has this property, just that there isn’t a way to do it generally. The proof of the Halting problem is similar to Cantor’s diagonal proof that the set of real numbers is uncountable.
One of the consequences of Turing’s work is that the total number of possible computations is countable. You simply take all strings of length 1, then 2, etc and feed it to a Universal Turing machine. Every possible computation or numerical simulation will be included. Thus, any simulation of the universe is coded in one of these strings. Some of these inputs will lead to computations that will halt and some will run forever. Some will make the Turing machine print a particular symbol and some will not. However, there is no way to decide, which of the input strings are on any of these lists.
The question is then, given an input string, can you determine if it will produce a universe that has some property such as supporting life. There are actually two separate issues regarding this question. The first is, how would you even define life or recognize it in the computation. I will deal with this in a future post. The second is, given that you have a definition of life, then can you know ahead of time whether or not your simulation will produce it. The answer to this question is no because if it were yes then you could solve the Halting problem. This is easy to see because any definition of life must involve some pattern of symbols to be printed on the tape and there is no way to decide if an input string will ever produce a symbol much less a pattern. This doesn’t mean that a simulator couldn’t come up with a simulation of our universe, it just means that she could never come up with a general algorithm to guarantee it. So, in the infinite but countable list of possible compuations, some produce simulations of universes, perhaps even ours, but we can never know for sure which.