I just returned from this year’s G4G11, where I presented my work on the computability of marble runs (with some success- while I did finish my slides, I went almost 2 and a half minutes over time despite editing and practicing my talk. PROTIP: Don’t go on a two-day vacation 4 days before a conference whose third major theme was sleep deprivation!) I promised I would post the full construction for a marble-run computer here, which not only shows that the double-switch marble run halting problem is PSPACE-Complete, but also should enable simple proofs of universal computability for many other systems. But enough with the jargon; this is going to be more of a slideshow-style post, although I’ll soon post a more in-depth explanation of my talk. (The original slides are also available here, which also includes notes on the script.)
But first, some acknowledgements. The very first image, an example of a marble machine, was taken directly from denha’s fantastic video, “Marble machine chronicle“. While denha’s Youtube channel focuses primarily on marble machines, the electronics videos are certainly interesting as well, so definitely check it out. I used Blender for most of the non-schematic animations, so thanks to the people behind the animation engine and Cycles Render. And finally, the proof would undoubtedly have been much more difficult without the ideas of Demaine, Hearn, and Demaine (choose your own ordering), not to mention the many other people who’ve done work on computational complexity theory and all the things that led up to the field. (I’m sure some would prefer I name-checked everybody involved, but then I’d have to include my kindergarten teacher and the crew behind the Broome Bridge and, well, this would be a lot longer.)
So, without further ado, here are the various images and videos from my talk, presented in substantially more than 6 minutes.
This is, as mentioned above, denha’s “Marble machine chronicle”, for those who have never heard of marble runs under that particular name before.
I made (with assistance from Peter Bickford) a short video demonstrating a few recurring elements in marble machines- specifically, the ones I would be analyzing later in the talk. The original marble machine, from the Tech Museum of Innovation, also contains a few other elements (such as loop-de-loops, chimes, and randomized switches), some of which do nothing to the computational ability of the machine, others which actually do change the problem a bit. Additionally, I consider problems with exactly one marble or pool ball, although Hilarie Orman suggested that it might be possible to simplify the construction using two or more pool balls.
This is a decoy that looks like a duck, used for a quick gag and counterexample to the Duck test. This was actually the shortest clip and the longest render for the entire project; the good news was, it rendered at 24 times the speed of a Pixar film. The bad news was that it rendered at only 24 times the speed of a Pixar film.
A short slide made to demonstrate the proof that single-switch marble machines cannot emulate a computer (unless NP=PSPACE, which is really unlikely). Up top, we have the problem we’re trying to solve, while on the lower-left is a randomly generate marble run with edges annotated and on the right is a system of equations representing said marble run. (Click through to see the rest of the images)