Down the Rabbit Hole

2022-12-04T05:55:06Z

1: The Turing Machine

So, Wednesday I looked at Wikipedia's front page and saw, under the "On this day" heading:

1936 – English mathematician Alan Turing published the first details of the Turing machine (model pictured), an abstract device that can simulate the logic of any computer algorithm by manipulating symbols.

It was the ""model pictured" that grabbed me. The caption was/is "A physical Turing machine model. A true Turing machine would have unlimited tape on both sides, however, physical models can only have a finite amount of tape."

I knew that -- everyone who studies computer science knows that, and a few have dreamed, as I had, of building a physical model. I even figured out how to build one out of wood, minus a few details. But there it was.

(If you're not interested in the details, you can skip this and the other indented blocks. But I digress...)
A Turing Machine is a remarkably simple device. It has a read head, a write head, a strip of tape that they operate on, and a controller with a finite number of states. It can read what's on the tape -- the classic machine uses blank, "0", and "1". (Some versions use "X" instead of "1", and some dispense with "0" and just have 1 and blank. That makes programming them a little more complicated, but not by much. Some have more symbols. It doesn't matter -- you can program around it.) The machine can move the tape backward and forward. Numbers are usually represented in Unary, so you count "1", "11", "111", ..., although with both 1 and 0 you could use binary, and some versions do.
The machine has a "state", which selects a line in the machine's program that tells it what to write, which direction to move the tape, and which state to go to next, depending on what symbol the read head is looking at. (Think of the table as a drum with pegs on it, like a music box.)
That's it. That's all you need to compute any function that can be computed by any kind of mechanical or digital computer. Of course you may need a lot of tape -- so you need to attach it to a tape factory -- and a lot of time.

The critical thing is that it's possible to design a universal Turing machine: it takes a tape, and the state table of a Turing machine (in 1's, 0's and blanks), and it uses that description to do exactly what that machine is programmed to do. Turing's big accomplishment was using the universal Turing machine to prove that there some things that a computer can't do, no matter how much time and tape you give it.

But of course I was much more fascinated by the machines, starting at the website of the model that first grabbed my attention., and proceeding to a Turing machine made of legos. I spent some time in the Turing machine gallery. But the rabbit hole went deeper than that.

2: The Universal Constructor

At about that point it ocurred to me to look at the Wikipedia page for the Von Neumann universal constructor. Because once you have a kind of machine that can simulate itself, the natural question is whether you can have a machine that can build a copy of itself.

The trivial answer to this question is "Yes, of course. Cells have been reproducing themselves for billions of years." But in the 1940s when von Neumann was thinking about this, the structure of DNA had not yet been determined -- that was 1953 -- and although it had been known since the late 1920s that DNA had something to do with heredity, nobody knew how it worked. So his insight into the machinery of reproduction was pretty remarkable.

Like Turing's insight into the machinery of computation, von Neumann's insight into the machinery of reproduction was to separate the machine -- the Universal Constructor -- from the description of what it was to construct, stored on something simple -- a tape.

Von Neumann's machine was/is a cellular automaton; it "lived" (if you can call it that) on a grid of squares, where each square can be in one of 29 different states, with rules that tell it what to do depending on the states of its neighbors. A completely working machine wasn't simulated until 1995. Its constructor had 6329 32-state cells, and a tape with a length of 145,315. It took it over 63 billion timesteps to copy itself. (Smaller and faster versions have been constructed since then).

At, say, 1000 steps/second, that would have taken over two years. It wasn't until 2008 that a program, Golly, became able to simulate it using the hashlife algorithm; it now takes only a few minutes.

Which led me even further down the rabbit hole. Because no discussion of cellular automata would be complete without Conway's Game of Life.

3: The Game of Life

It's not really a game, of course, it's a cellular automaton. Each cell in the square grid is either dead or alive. You start with an arrangement of live cells, and turn them loose according to four simple rules:

If a live cell has fewer than two live neighbors (out of the 8 cells surrounding it), it dies of loneliness.

A live cell with two or three live neighbors, stays alive.

A live cell with more than three live neighbors dies of overpopulation.

A dead cell with exactly three live neighbors becomes live.

I first encountered the game in the October 1970 issue of Scientific American, in Martin Gardner's "Mathematical Games" column. The Wikipedia article gives a good introduction.

Patterns in Life evolve in a bewildering variety of ways. Many of them die out quickly -- an isolated cell, for example. Some patterns sit there and do nothing -- they're called "still lifes". A 2x2 block of cells for an example. Some blow up unpredictably, and may or may not leave a pile of random still lifes behind. Some patterns oscillate: a horizontal row of three cells will become a vertical row in the next turn, and vice versa -- it's called a "blinker".

And some patterns move. The simplest, called a "glider", appears in this post's icon. You can crash gliders into blocks or gliders into gliders, and depending on the timing they will do different interesting things. It didn't take people long to figure out that you can build computers, including a universal Turing machine. Or a machine that prints out the digits of Pi.

Or a universal constructor.

4: The universal constructor

While I was falling into this rabbit hole, I happened to remember a passing mention of a universal constructor that can build anything at all out of exactly 15 gliders. (Strictly speaking, anything that can be constructed by crashing gliders together. Some patterns can't be made that way. But almost all the complicated and interesting ones that people are building these days can.) If this intrigues you, go read the article. Or wait until the next section, where I finally get to the bottom of the rabbit hole.

On the way down I encountered lots of weird things -- the aforementioned universal Turing machine and Pi printer, and a variety of "spaceships" that travel by, in effect, repeatedly constructing a new copy of themselves, then cleaning up the old copy. It took a while for me to get my head around that.

Then, sometime Wednesday evening, I found the book.

5: The Book of Life

It's not called "The Book of Life", of course, it's called Conway's Game of Life: Mathematics and Construction. But you get the idea. You can download the PDF.

The book ends with a pattern that simulates a Life cell. There are several different versions of this; this is the latest. It works by making copies of itself in any neighboring cells that are coming alive, then destroying itself if it's about to die. Wild.

Another fine post from The Computer Curmudgeon (also at computer-curmudgeon.com).
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