### Turing’s Deep Field: Visualizing the Computational Universe

Thursday, January 5th, 2012I generated this image in the course of an investigation of the distribution of runtimes of programs in relation to the lengths of mathematical proofs, the results of which are being published in my paper bearing the title “Computer Runtimes and the Length of Proofs with an Algorithmic Probabilistic Application to Optimal Waiting Times in Automatic Theorem Proving” Volume 7160 of the Lecture Notes in Computer Science series (LNCS), a festschrift for Cristian Calude.

The preprint is now available online in the ArXiv here. The paper shows that as theoretically predicted, most machines either stop quickly or never halt. It also suggests how a theoretical result for halting times may be used to predict the number of random theorems (dis)proven in a random axiom system –see the section I’ve called “Gödel meets Turing in the computational universe”.

The plot was the winning image in this year’s **Kroto Institute Image Competition**, in the Computational Imagery category titled “Runtime Space in a Peano Curve”, it shows the calculation of the distribution of runtimes from simulating (4n+2)^(4n) = 10000 Turing machines with 2 symbols and n=2 states (of a total of more than 10^12 simulated Turing machines with up to n=4 states) following a quasi-lexicographical order in a Peano curve preserving–as far as possible–the distance between 2 machines arranged in a 2-dimensional array from a 1-dimensional enumeration of Turing machines.

In the image each point or cluster of points represents a Turing machine or a group of Turing machines, and the color is determined by a spectrum encoding their halting runtimes–the lighter the square the sooner the machine entered the halting state. White cells represent machines that are proven to halt in infinite time. Red cells show the Turing machines that take longer to halt (popularly called Busy Beavers). Knowing the values of the Busy Beaver functions allows us to identify the machines that never halt (depicted in white). The image is noncomputable, meaning that the process cannot be arbitrarily extended because of the undecidability of the halting problem (i.e. there is no procedure for ascertaining the color of the following pixels to zoom out the picture and cover a larger fraction of the computational universe). Put it in the words of crystilogic,

Turing machines with an arbitrary number of states can encode any possible mathematical problem and are therefore perfect compressors of the known, the yet to be known, and even the unknowable (due to the undecidability of the halting problem).

This is how the image looks like as displayed in the stairway of the Kroto Research Institute in the UK:

Some postcards with the winning image were printed and they can be sent to scholar or enthusiasts upon request sending an email to hectorz[at] labores.eu.

I want to dedicate this prize to my former thesis advisor Jean-Paul Delahaye who suggested me the Peano arrangement as a packing for my visual results of halting times. And also to Cristian Calude who co-advised me all along my PhD thesis and who encouraged me to publish this paper and what better place to do so than for his festschrift.

I’m releasing the images under an open licence in honour of the Turing Year, so that it may be used for any artistic illustration of some of Turing’s main results (the halting problem) or for any other purpose in any medium.

Postcards of the winning image are also available upon request. Just send an email requesting 1 or more postcards to hectorz [at] labores.eu or to let him know (if you can) that you will be using any of the images (or if you need better resolution versions).