Did you ever see one of those auto-generated random “academic papers” like this one? When I first saw the following title, my first thought was that it is a randomly-generated “paper”:

Implications of the Turing completeness of reaction-diffusion models, informed by GPGPU simulations on an XBox 360: Cardiac arrhythmias, re-entry and the Halting problem [PDF]

Turing completeness, cardiac arrhythmias, XBox 360… those things don’t seem to have much in common. But, I had my interest piqued. I looked up the paper and read through it. And, it turns out that not only is the paper serious, what it has to say is also quite interesting.

Heart and logic circuits

I had to take author’s word on the medical aspects of the article, since I know nothing about cardiology. Apparently, understanding of electrical signals between cells in a human heart is important for research into heart arrhythmias. Makes sense.

The important insight of the article is that a logic NOR gate can be simulated using electrical signals between heart cells. Constructing a NOR gate is a powerful result, because similarly to NAND, NOR is a universal logic gate. That means that all other logic gates like AND, OR and NOT can be built out of NORs:

NOT(A) = NOR(A, A)

OR(A, B) = NOR(NOR(A, B), NOR(A, B))

AND(A, B) = NOR(NOR(A, A), NOR(B, B))

So, since you can simulate a NOR gate using cardiac cells, you can simulate an arbitrary logic circuit in heart tissue.

Heart and Turing machines

But, there has to be more to the story. The paper title mentioned Turing machines, and logic circuits are not Turing-complete. Halting problem doesn’t even make sense when applied to boolean expressions!

The missing part is the passage of time. See, a logic circuit is not Turing-complete. But, if you take a logic circuit with multiple inputs, the same number of outputs as inputs, and repeatedly apply the circuit over the results of the previous iteration, you get a Turing-complete system. This type of a system can be modeled with behavior of a heart tissue over a period of time.

The most intuitive explanation that I can think of is via Game of Life. In the Game of Life, each cell dies, becomes alive or stays alive depending on how many live neighbors it had in the previous generation. Here is one example of a Game of Life board in action:

Now, the important observation is that the Game of Life rules can be encoded using a logic circuit. For example, if the eight neighbors of a particular cell are represented as A, B … H, then the rule that the cell becomes alive if it has exactly three neighbors can be encoded as ((A and B and C and not(D) and … and not(H)) or (A and B and not(C) and D and not(E) … and not(H)) or …). This expression will be combined with an OR together with the situations under which the cell remains alive, rather than becoming alive. This will be a pretty ugly logic circuit, but it should be clear that it can be constructed.

Since Game of Life is known to be Turing-complete, then an “iterated logic circuit” is also Turing-complete, and the behavior of cardiac tissue is … Turing-complete.

Why is this useful?

Proving that a particular behavior of cardiac tissue is Turing-complete is a useful result because it shows that the cardiac tissue is in a certain sense “unpredictable”.

For example, since Game of Life is Turing-complete, the Halting problem applies. So, it is a proven fact that there is no general algorithm that can look at a particular Game of Life board and decide whether the movement will eventually stop or continue forever.

Similarly, there is no algorithm that can decide any of these properties for all Game-of-Life boards:

Whether the board will ever reach a particular configuration

Whether the number of live cells will ever exceed X

Whether the game will ever enter a cycle

etc.

Since the studied behavior of cardiac tissue is also Turing-complete, there is no general algorithm that can look at the state of cardiac tissue and decide whether the activity will ever stop, enter a regular pattern, achieve a particular configuration, etc. That is certainly a worthwhile result!

What about the XBox?

Constructing a NOR gate out of cardiac cells is computationally intensive, and the researcher used a GPU in an XBox 360 for that task.

However, the paper doesn’t conclusively show that using an XBox was a real benefit. The paper says that a C++ implementation that was originally “designed more for ease of expansion […] than for speed” ran slower on an XBox 360 CPU than an “unoptimized” shader-based implementation on an XBox 360 GPU. Comparing implementations not designed for speed is unconvincing. If the goal was to speed up the computation, why not first try to optimize the original code instead of porting it to shaders, which is undoubtedly a much more difficult task?

Also, the article doesn’t say how did XBox 360 CPU compare against an ordinary x86 machine, or against say a CUDA-based implementation on a common NVidia card. So, the paper doesn’t come close to showing that the researchers gained much by coding against the XBox 360 GPU, rather than following the current state-of-the-art approaches.

But, it is still a cool paper, and adding XBox 360 into the picture certainly attracted attention. The paper was reported in press, with titles such as these:

And, if it weren’t for the sensational articles, I wouldn’t have found out about the paper at all, so I guess I shouldn’t complain.