Alan Turing’s statement of what is now known as the Church-Turing Thesis asserts that all forms of “mechanical” computation are equivalent to a Universal Turing Machine. The Church-Turing Thesis is not a mathematical theorem, but is instead a hypothesis that has turned out to be true as an empirical matter. That is, every model of computation that has ever been proposed has been proven to be capable of computations that are either equivalent to, or a subset of, the computations that can be performed by a UTM.
Nonetheless, as I’ve discussed in previous posts, there are good reasons to believe that nature itself is capable of generating non-computable processes. First off, it is mathematically impossible for a UTM to generate information (see the post below for a simple proof). Nonetheless, human beings do this all the time. In fact, I’m doing exactly that right now.
At the same time, Alan Turing is one of my personal heroes, so it’s not easy for me to disagree with him. He’s also a cosmically intelligent human being that I don’t disagree with, without extreme caution. As a result, I’ll hedge my bets and say that what I present below is exactly what he was hoping to be true: that there is a simple model of computation beyond the UTM, and that nature is, as common sense suggests, stranger and more complicated than even the strangest fiction.
We naturally associate computation with symbolic computation, since that’s what we’re taught to do from the time we’re born. That is, computation is what you to do numbers and symbols, and computers are tools you use to do a lot of computations in a short amount of time. This view is embedded in the Church-Turing Thesis itself, in that Turing described computation as an inherently “mechanical” process. But when you compress the input to a UTM maximally, what you really have is a “trigger”. That is, there’s some string that we provide as the input to some box, and then that box spontaneously generates some output. That input string will probably be totally unrecognizable as anything that should be associated with the output that it will ultimately generate, since, by definition, we’ve eliminated all the compressible structure in the input, leaving what will be a maximally randomized kernel.
Abstracting from this, we can define a kind of computation that is not the product of mathematical operations performed on a set of symbols. Instead, we can view the input to a computational device as a trigger for the outputs that follow. This view allows us to break free of the mathematical restraints of mechanical computation, which prevent us from generating complexity. That is, as noted above, and proven below, the complexity of the output of a UTM can never exceed the complexity of its input. If, however, we treat the input to a device as a trigger, and not the subject of some mathematical operation, then we can map a finite string to anything we’d like, including, for example, a particular configuration of a continuous surface.
Implementing this would require a continuous system that responds to inputs. Some kind of wave seems to be the best candidate, but I’m not claiming to know what physical systems we should use. Instead, I’m presenting what I believe to be a simple mathematical model of computation that is on its face non-computable, and leaving it up to the adults in the room to figure out how to implement it.
Assume that we have some input m, that we give to our computational device X.
In particular, let’s assume X is always in some state that we’ll represent as a series of integer values mapped to an uncountable space. That is, the state of X can be represented as an uncountable multi-set of integers, which for convenience we’ll imagine being represented as heights in a plane. Specifically, imagine an otherwise flat plane that has some height to it, with the height of the plane at a given point representing the value of the state of X at that point, and the aggregate set of heights representing the overall state of X at a given moment in time.
When X is given some input m, assume that its state changes from some X_0, to some other state X(m).
We could assume that the state of X changes for some period of time after receiving the input m, or switches immediately to X(m). Either way, the important part is to assume that X is, by definition, capable of being in states that require an infinite amount of information to fully represent. We could of course also allow X to be in states that require only a finite amount of information to represent, but the bit that will make X decidedly non-computable is the ability to be in a state that is in effect a chaotic wave. That is, a discontinuous set of integer values mapped onto a continuous surface, that, as a result, cannot be compressed into some finite representation.
If such a system exists, X would be, by its nature, non-computable, since, first of all, X generates states that require an uncountable number of integers to fully represent. Since a UTM can generate only finite outputs, the states of X cannot be simulated by a UTM. Further, X is capable of being in non-compressible, infinitely complex states, which means that there is no finite compression of at least some of the states of X, again implying that, even if we could “read” every point on the surface of X, there are at least some states of X that could not be compressed into a finite string that could in turn generate the complete state of X on some UTM.
As a result, if such a system exists, then we would be able to take a finite string, and trigger an output that contains an infinite amount of information, generating complexity. Further, by juxtaposing two waves directly on top of each other, we would presumably generate constructive interference, the result of which we could define as the “sum” of two inputs. By juxtaposing two slightly offset waves, we would presumably generate destructive interference, the result of which we could define as the “difference” between two inputs. As a result, if we could understand and control such a system, perhaps we could perform otherwise non-computable computations.