This posting on the Java core library mailing list proposes to replace the current quicksort with a new, 2-pivot quicksort.

Ordinary quicksort in Haskell looks something like this:

quicksort [] = []
quicksort (pivot:rest) =
   quicksort [x| x ← rest, x ≤ pivot] ++
   [pivot] ++
   quicksort [x| x ← rest, x > pivot]

Note the similarity of this to the notation for ‘invariants’ used in the email:

[ <= p | >= p ]

Similarly, for the 2-pivot invariants (I have corrected a typo in this):

[ < P1 | P1 <= & <= P2 | > P2 ]

We can derive the code:

quicksort2 [] = []
quicksort2 (only:[]) = [only]
quicksort2 (first:second:rest) =
   quicksort2 [x|x ← rest, x < p1] ++
   [p1] ++
   quicksort2 [x|x ← rest, p1 ≤ x && x ≤ p2] ++
   [p2] ++
   quicksort2 [x|x ← rest, p2 < x]
  where (p1,p2) = if first ≤ second then (first,second) else (second,first)

Which does, in fact, run faster than the ordinary 1-pivot quicksort.

n-pivot quicksort

Of course, this being Haskell we want to take things to their logical conclusions, and allow any number of pivots to be used. My formulation follows:

quicksortN _ [] = []
quicksortN _ (x:[]) = [x]
quicksortN n xs
    | n > length xs = quicksortN ((n `div` 2) `max` 1) xs
    | otherwise = part True pivots
    where
      (takexs,dropxs) = splitAt n xs
      pivots = quicksortN 1 takexs
      part False (p2:[])      = quicksortN n [x|x ← dropxs, p2 < x]
      part False (p1:p2:rest) = quicksortN n [x|x ← dropxs, p1 < x && x ≤ p2] ++ [p2] ++ part False (p2:rest)
      part True  (p1:[])      = quicksortN n [x|x ← dropxs, x ≤ p1] ++ [p1] ++ quicksortN n [x|x ← dropxs, p1 < x] -- remember me? :)
      part True  (p1:p2:rest) = quicksortN n [x|x ← dropxs, x ≤ p1] ++ [p1] ++ part False (p1:p2:rest)

Now there are only 3 things that I can see that are left ‘tweakable’;

• how to reduce ‘n’ when it is greater than length of the list
  • dividing by 2 here
• how to sort the pivots
  • using a recursive call to the 1-pivot version of itself here
• whether or not to reduce the number of pivots as sorting continues
  • ‘no’ chosen here

I have no idea whether these are sensible defaults

Which n should I choose?

Finally, some code for the timings. You’ll need timeit to run it.

main = do
 gen ← getStdGen
 let n = 10^6
 let r = take n $ randoms gen :: [Int]
 print $ sum r -- force list
 timeIt . putStrLn . ("quicksort = "++) . show . sum $ quicksort r
 timeIt . putStrLn . ("quicksort2 = "++) . show . sum $ quicksort2 r
 mapM_ (\x → timeIt . putStrLn . (("quicksortN "++ show x++" = ")++) . show . sum $ quicksortN x r)
    [1..20]

Sample output:

$ ./sort
858124546
quicksort = 858124546
CPU time:   4.60s
quicksort2 = 858124546
CPU time:   3.66s
quicksortN 1 = 858124546
CPU time:   5.19s
quicksortN 2 = 858124546
CPU time:   3.77s
quicksortN 3 = 858124546
CPU time:   3.28s
quicksortN 4 = 858124546
CPU time:   3.27s
quicksortN 5 = 858124546
CPU time:   3.06s
quicksortN 6 = 858124546
CPU time:   3.04s
quicksortN 7 = 858124546
CPU time:   3.04s
quicksortN 8 = 858124546
CPU time:   3.07s
quicksortN 9 = 858124546
CPU time:   3.10s
quicksortN 10 = 858124546
CPU time:   3.14s
quicksortN 11 = 858124546
CPU time:   3.22s
quicksortN 12 = 858124546
CPU time:   3.31s
quicksortN 13 = 858124546
CPU time:   3.30s
quicksortN 14 = 858124546
CPU time:   3.37s
quicksortN 15 = 858124546
CPU time:   3.54s
quicksortN 16 = 858124546
CPU time:   3.41s
quicksortN 17 = 858124546
CPU time:   3.52s
quicksortN 18 = 858124546
CPU time:   3.65s
quicksortN 19 = 858124546
CPU time:   3.63s
quicksortN 20 = 858124546
CPU time:   3.77s

As expected, generalized quicksort performs slower than the equivalent hard-coded quicksorts for specific values of n, but higher values of n than 1 or 2 can perform better.

Preamble; choice of implementation

I decided to go with the latest and greatest version of Scheme: R⁶RS. There are currently two implementations available under Ubuntu Linux: Ikarus and Ypsilon. I installed both so I wouldn’t be swayed by any tempting extensions to the standard. Despite this, I ended up using Ikarus for most testing, as it ran quite a lot faster, although Ypsilon gave much better stack traces.

I also used the streams library for dealing with infinite lists (SRFI 41), which is included with both implementations.

Note: I’ll be providing all the code so that you should be able to copy-paste it into a new file and run it.

(import (rnrs)
    (rnrs r5rs (6)) ; provides 'delay' & 'force'
    (streams))

The Machine

Turing sought to capture the essence of computation. For this purpose he constructed an idealized machine, which can read and write symbols to an infinitely long piece of tape. We are going to model these idealized machines and see what they can do.

First, we want some types to represent the a-machines (the a is for automatic). Each machine has a set of states (m-configurations) it can be in, each of which contains a mapping from a set of symbols to a list of actions and a new m-configuration. I decided that the state would be my basic unit of construction, but twiddled the meaning of m-configuration a tiny bit so that instead of having each configuration contain a mapping from symbols, I would instead have a list of configurations, each one with a list of which symbols activate it. I ended up with the following:

; type for a configuration
(define-record-type m-cfg
            (fields symbols operations next))

This defines a new record type called m-cfg with the fields symbols, operations and next. The define-record-type form defines a constructor (make-m-cfg) and accessors for each field (m-cfg-*). Rather than have a ‘machine’ type, I decided that this would just be left implicit; if we know what the current state is then we can follow the links to next whenever we want to.

The Tape

Now, each machine operates upon an infinitely long ‘tape’. To model this, I use two streams, which are infinite lists. One is the infinite length of the tape to the left of the machine, and the other is the infinite length of the tape to the right of the machine. I decided that the first item in the right list would be the current item that the machine is reading.

; type for a tape (modeled as two stacks)
(define-record-type tape
            (fields left right min max index))

You will note that there are also the fields min, max, and index. These are solely used to track how much of the tape the machine has “visited”. Without this information, we would not know how much of the tape to show when we want to look at it; and since it is infinitely long, this could be a problem!

Representing operations

The operations that a machine can perform consist of:

• Move right
• Move left
• Print symbol
• Erase symbol
• Halt

I implemented these as an enumeration, just in case, but I didn’t actually end up utilizing any of the enumeration features.

; the operations available
(define-enumeration op
  (right left erase print halt)
  op-set)

There is a small difficulty with this representation: the ‘print’ operation needs to be able to take an argument. I decided that operations would always be passed around as lists, and that only ‘print’ would have a second element in the list: its argument. Here is a little shorthand to make this representation easier:

; these need to be lists, because print takes an argument
(define L (list (op left)))
(define R (list (op right)))
(define P (lambda (c) (list (op print) c)))
(define E (list (op erase)))
(define H (list (op halt)))

Thus, we can represent a list of operations like this: (list R R (P #\A) L E H)—that’s right, right, print ‘A’ (Scheme’s syntax for characters is a little weird), left, erase, halt.

Moving around on the tape

Here is a simple tape; it is completely empty. Note that I use the symbol 'empty to represent empty places on the tape. stream-constant makes an infinite stream of the value(s) supplied.

(define empty-tape
  (make-tape (stream-constant 'empty) (stream-constant 'empty) 0 0 0))

Next we need code to actually implement the operations described above. It is fairly straightforward, but we also have to keep track of the index and max/min points on the tape:

; Tape manipulation:
(define (current-symbol tape)
  (stream-car (tape-right tape)))
 
(define (move-item from to)
  (values (stream-cdr from)
      (stream-cons (stream-car from) to)))
 
(define (move-right tape)
  (let-values (
      [(right left) (move-item (tape-right tape) (tape-left tape))]
      [(min max) (update-min-max (tape-min tape) (tape-max tape) (+ 1 (tape-index tape)))])
    (make-tape left right min max (+ 1 (tape-index tape)))))
 
(define (move-left tape)
  (let-values (
      [(left right) (move-item (tape-left tape) (tape-right tape))]
      [(min max) (update-min-max (tape-min tape) (tape-max tape) (- (tape-index tape) 1))])
    (make-tape left right min max (- (tape-index tape) 1))))
 
(define (print tape symbol)
  (make-tape
    (tape-left tape)
    (stream-cons symbol (stream-cdr (tape-right tape)))
    (tape-min tape)
    (tape-max tape)
    (tape-index tape)))
 
(define (erase tape)
  (print tape 'empty))
 
(define (update-min-max min max i)
  (cond 
    [(< i min) (values i max)]
    [(> i max) (values min i)]
    [else      (values min max)]))

Notice that erase is actually redundant because we can just print 'empty. We also want a “dispatcher” of sorts that takes a value representing an operation and performs that operation. This is where passing the argument around with the ‘print’ came in useful:

; performs an operation on a tape, returns new tape
(define (perform-op tape oper)
  (case (car oper)
    [(op left) (move-left tape)]
    [(op right) (move-right tape)]
    [(op print) (print tape (cadr oper))]
    [(op erase) (erase tape)]
    [(op halt) #f])) ; ungraceful halt!

Running the machine

Now that we have the operations implemented, we can almost run a state against a tape. First we need to figure out just which of the configurations of the state to run. This procedure receives a list of configurations (but they are all part of the same ‘state’) and a symbol, and picks the first configuration that has a matching symbol. Note that the configurations can also have the symbol 'any, which matches anything.

; finds the correct rule to follow for this symbol
(define (find-cfg machine symbol)
  (find (lambda (cfg) (or
                (find [lambda (s) (eqv? symbol s)]
                  (m-cfg-symbols cfg))
                (find [lambda (s) (eqv? 'any s)]
                  (m-cfg-symbols cfg))))
        machine))

Now that we have a way to find out which rule to perform, and how to perform it, we can run it against a tape. This procedure advances the machine to the next state, performing all the operations needed. It returns the new state and a new tape.

; runs a machine forward one step
(define (run-machine tape machine)
  (let ([cfg (find-cfg (force machine) (current-symbol tape))])
    (list (fold-left perform-op tape (m-cfg-operations cfg))
          (m-cfg-next cfg))))

Displaying the tape

Being able to run the machine against a tape isn’t much good if we can’t see the result, so here’s a procedure to print out what it looks like. This is where we need the indexes we kept track of on the tape, so we know when to stop printing.

(define (print-tape tape)
  ; move as far left as possible
  (let ([leftTape (go-far-left tape)])
    ; now go right
    (do ([t leftTape])
      ((> (tape-index t) (tape-max tape)) t)
      (when (eqv? (tape-index tape) (tape-index t))
              (display "["))
      (cond
          [(eqv? (current-symbol t) 'empty) (display ".")]
          [else (display (current-symbol t))])
      (when (eqv? (tape-index tape) (tape-index t))
              (display "]"))
      (set! t (move-right t)))))
 
; moves to the far left of the tape (as far as has been travelled)
(define (go-far-left tape)
  (do ([t tape])
    ((= (tape-index t) (tape-min tape)) t)
     (set! t (move-left t))))

I print out a ‘.’ for each blank, and surround the current symbol with square brackets (actually I’ve changed that to be underlining in this blog post).

Turing Machines!

Now we can test it! Here is my definition of Turing’s first published machine:

; Turing's first published machine!
(define m1 (letrec 
  ([b (delay (list
       (make-m-cfg (list 'any) (list (P #\0) R) c)))]
   [c (delay (list
       (make-m-cfg (list 'any) (list R) e)))]
   [e (delay (list
       (make-m-cfg (list 'any) (list (P #\1) R) f)))]
   [f (delay (list
       (make-m-cfg (list 'any) (list R) b)))])
  b))

Now you see why we needed to import ‘delay’… since Scheme is a strictly-evaluated language, we can’t just letrec each state in terms of the others, so I wrap each one up in delay, then force it in one place; just before we use it in the run-machine procedure.

Other than that it is fairly straightforward, each state has a list of m-configurations, each of which has a list of what symbols it accepts, the actions to take, and the next state to move to. After we letrec, we have the initial state defined as the ‘machine’—in this case, b.

We can run this machine like so:

(define (go t m)
 (do ([tm (list t m)])
  ((eqv? tm #f) (car tm))
  (print-tape (car tm))
  (newline)
  (set! tm (run-machine (car tm) (cadr tm)))))
 
(go empty-tape m1)

This gives us the following output:

.
0.
0..
0.1.
0.1..
0.1.0.
0.1.0..
0.1.0.1.
0.1.0.1..
0.1.0.1.0.
...

This looks correct: P0, R, R, P1, R, R, etc.

Machine redux

The next machine Turing gives is the same as the first one, only in a different form:

; the same machine, only smaller
(define m2 (letrec
         ([b (delay (list
           (make-m-cfg (list 'empty) (list (P #\0)) b)
           (make-m-cfg (list #\0) (list R R (P #\1)) b)
           (make-m-cfg (list #\1) (list R R (P #\0)) b)))])
         b))

It has only one state, but changes depending on what the current symbol is. This produces the same output as the first machine, but in fewer steps:

.
0
0.1
0.1.0
0.1.0.1
0.1.0.1.0
0.1.0.1.0.1
0.1.0.1.0.1.0
0.1.0.1.0.1.0.1
0.1.0.1.0.1.0.1.0
...

You may be wondering why Turing leaves blanks between each printed symbol. He used the convention that only the ‘even’ squares (termed F-squares) would be the output. The ‘odd’ squares (termed E-squares) would be used as a scratch-pad.

Machine-à-trois

The third machine is a little more interesting. Whereas the first two printed out 01010101..., this prints 01011011101111011111...:

; Turing's third machine
(define m3 (letrec
         ([b (delay (list
                   (make-m-cfg (list 'any) (list (P #\ǝ) R (P #\ǝ) R (P #\0) R R (P #\0) L L) o)))]
          [o (delay (list
                  (make-m-cfg (list #\1) (list R (P #\x) L L L) o)
                  (make-m-cfg (list #\0) (list) q)))]
          [q (delay (list
                  (make-m-cfg (list #\0 #\1) (list R R) q)
                  (make-m-cfg (list 'empty) (list (P #\1) L) p)))]
          [p (delay (list
                  (make-m-cfg (list #\x) (list E R) q)
                  (make-m-cfg (list #\ǝ) (list R) f)
                  (make-m-cfg (list 'empty) (list L L) p)))]
          [f (delay (list
                  (make-m-cfg (list 'empty) (list (P #\0) L L) o)
                  (make-m-cfg (list 'any) (list R R) f)))])
         b))

With the output:

.
ǝǝ0.0
ǝǝ0.0
ǝǝ0.0
ǝǝ0.0..
ǝǝ0.0.1
ǝǝ0.0.1
ǝǝ0.0.1
ǝǝ0.0.1
ǝǝ0.0.1
ǝǝ0.0.1
ǝǝ0.0.1..
ǝǝ0.0.1.0
ǝǝ0.0.1x0
ǝǝ0.0.1x0
ǝǝ0.0.1x0
ǝǝ0.0.1x0
ǝǝ0.0.1x0..
ǝǝ0.0.1x0.1
ǝǝ0.0.1x0.1
ǝǝ0.0.1.0.1
ǝǝ0.0.1.0.1
ǝǝ0.0.1.0.1..
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1
ǝǝ0.0.1.0.1.1..
ǝǝ0.0.1.0.1.1.0
ǝǝ0.0.1.0.1.1x0
ǝǝ0.0.1.0.1x1x0
ǝǝ0.0.1.0.1x1x0
ǝǝ0.0.1.0.1x1x0
ǝǝ0.0.1.0.1x1x0
ǝǝ0.0.1.0.1x1x0
ǝǝ0.0.1.0.1x1x0..
ǝǝ0.0.1.0.1x1x0.1
ǝǝ0.0.1.0.1x1x0.1
ǝǝ0.0.1.0.1x1.0.1
ǝǝ0.0.1.0.1x1.0.1
ǝǝ0.0.1.0.1x1.0.1..
ǝǝ0.0.1.0.1x1.0.1.1
ǝǝ0.0.1.0.1x1.0.1.1
ǝǝ0.0.1.0.1x1.0.1.1
ǝǝ0.0.1.0.1x1.0.1.1
ǝǝ0.0.1.0.1.1.0.1.1
ǝǝ0.0.1.0.1.1.0.1.1
ǝǝ0.0.1.0.1.1.0.1.1
ǝǝ0.0.1.0.1.1.0.1.1
ǝǝ0.0.1.0.1.1.0.1.1..
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1..
ǝǝ0.0.1.0.1.1.0.1.1.1.0
ǝǝ0.0.1.0.1.1.0.1.1.1x0
ǝǝ0.0.1.0.1.1.0.1.1x1x0
ǝǝ0.0.1.0.1.1.0.1x1x1x0
ǝǝ0.0.1.0.1.1.0.1x1x1x0
ǝǝ0.0.1.0.1.1.0.1x1x1x0
ǝǝ0.0.1.0.1.1.0.1x1x1x0
ǝǝ0.0.1.0.1.1.0.1x1x1x0
ǝǝ0.0.1.0.1.1.0.1x1x1x0
ǝǝ0.0.1.0.1.1.0.1x1x1x0..
ǝǝ0.0.1.0.1.1.0.1x1x1x0.1
ǝǝ0.0.1.0.1.1.0.1x1x1x0.1
ǝǝ0.0.1.0.1.1.0.1x1x1.0.1
ǝǝ0.0.1.0.1.1.0.1x1x1.0.1
ǝǝ0.0.1.0.1.1.0.1x1x1.0.1..
ǝǝ0.0.1.0.1.1.0.1x1x1.0.1.1
ǝǝ0.0.1.0.1.1.0.1x1x1.0.1.1
ǝǝ0.0.1.0.1.1.0.1x1x1.0.1.1
ǝǝ0.0.1.0.1.1.0.1x1x1.0.1.1
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1..
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1x1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1..
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1
ǝǝ0.0.1.0.1.1.0.1.1.1.0.1.1.1.1

Here we can see the scratch-pad being used. We can also see the beginnings of a pattern of execution; notice how the machine returns to the beginning of the tape after completing each set of 1s.

Opus Magnum

Next up was my main challenge. One of the major contributions of Turing’s paper was to display a machine which could emulate any other machine you wanted. In essence, you don’t actually need lots of different machines. You can just build one, and it can do anything any of the other machines can do! This is the principle behind modern general-purpose computers.

This is a very big, and complex machine. Some of the intricacies not involved in the other machines are:

• m-functions; that is, configurations which accept parameters—luckily, this was surprisingly easy to implement using Scheme; I simply wrap the delayed code inside another lambda
• m-configurations which are parametrized over all symbols on the machine—this is solved by just mapping a lambda over the list of symbols
• variadic m-functions—solved by the magic of case-lambda
• poorly-scanned journal document containing Fraktur and Greek letters

But not only was this by far the biggest and most complex machine supplied by Turing, I had read texts mentioning unspecified bugs in the program.

… and sure enough, I ran into a ‘bug’. There were some configurations used in the machine which weren’t defined in the paper! At first I thought this was due to the low resolution of the PDF, but even enhancing the image didn’t help.

After much supplication and burnt offerings to the God of the Internet, I managed to find that:

Someone else did the work already

Yay!

More specifically, I found a paper entitled Understanding Turing’s Universal Machine — Personal Style in Program Description (which is unfortunately not available to the public), a marvelous paper that not only explains the errors made in detail, but also provides a nice, corrected version of Turing’s exposition of his machine.

After painstakingly re-checking all the states again, I arrived at this:

; need this to generate a couple of cfgs
(define u-symbols (list
		   #\A #\C #\D #\0 #\1
		   #\u #\v #\w #\x #\y #\z
		   #\; #\L #\R #\N
		   #\∷ #\:
		   ))
 
; yo dawg
(define u (letrec
        (
         [f (lambda (C B a) (delay (list
                (make-m-cfg (list #\ǝ) (list L) (f1 C B a))
                (make-m-cfg (list 'any) (list L) (f C B a)))))]
         [f1 (lambda (C B a) (delay (list
                (make-m-cfg (list a) (list) C)
                (make-m-cfg (list 'empty) (list R) (f2 C B a))
                (make-m-cfg (list 'any) (list R) (f1 C B a)))))]
         [f2 (lambda (C B a) (delay (list
                (make-m-cfg (list a) (list) C)
                (make-m-cfg (list 'empty) (list R) B)
                (make-m-cfg (list 'any) (list R) (f1 C B a)))))]
         [fdash (lambda (C B a) (delay (list
                (make-m-cfg (list 'any) (list) (f (l C) B a)))))]
         [fdashdash (lambda (C B a) (delay (list
                (make-m-cfg (list 'any) (list) (f (r C) B a)))))]
         [r (lambda (C) (delay (list
                (make-m-cfg (list 'any) (list R) C))))]
         [l (lambda (C) (delay (list
                (make-m-cfg (list 'any) (list L) C))))]
         [q (case-lambda
              [(C) (delay (list
                (make-m-cfg (list 'empty) (list R) (q1 C))
                (make-m-cfg (list 'any) (list R) (q C))))]
         [(C a) (delay (list
                (make-m-cfg (list 'any) (list) (q (q1 C a)))))])]
         [q1 (case-lambda
               [(C) (delay (list
                (make-m-cfg (list 'empty) (list) C)
                (make-m-cfg (list 'any) (list R) (q C))))]
               [(C a) (delay (list
                (make-m-cfg (list a) (list) C)
                (make-m-cfg (list 'any) (list L) (q1 C a))))])]
         [pe (lambda (C b) (delay (list
                (make-m-cfg (list 'any) (list) (f (pe1 C b) c #\ǝ)))))]
         [pe1 (lambda (C b) (delay (list
                (make-m-cfg (list 'empty) (list (P b)) C)
                (make-m-cfg (list 'any) (list R R) (pe1 C b)))))]
         [pe2 (lambda (C a b) (delay (list
                (make-m-cfg (list 'any) (list) (pe (pe C b) a)))))]
         [c (lambda (C B a) (delay (list
                (make-m-cfg (list 'any) (list) (fdash (c1 C) B a)))))]
         [c1 (lambda (C) (delay (map
                    (lambda (b) (make-m-cfg (list b) (list) (pe C b)))
                      u-symbols)))]
         [ce (case-lambda
               [(C B a) (delay (list
                (make-m-cfg (list 'any) (list) (c (e C B a) B a))))]
               [(B a) (delay (list
                (make-m-cfg (list 'any) (list) (ce (ce B a) B a))))])]
         [ce2 (lambda (B a b) (delay (list
                (make-m-cfg (list 'any) (list) (ce (ce B b) a)))))]
         [ce3 (lambda (B a b g) (delay (list
                (make-m-cfg (list 'any) (list) (ce (ce2 B b g) a)))))]
         [ce5 (lambda (B a b g d e) (delay (list
                (make-m-cfg (list 'any) (list) (ce3 (ce2 B d e) a b g)))))] ; added
         [cp (lambda (C U F a b) (delay (list
                (make-m-cfg (list 'any) (list) (fdash (cp1 C U b) (f U F b) a)))))]
         [cp1 (lambda (C U b) (delay (map
                   (lambda (g) (make-m-cfg (list g) (list) (fdash (cp2 C U g) U b)))
                     u-symbols)))]
         [cp2 (lambda (C U g) (delay (list
                (make-m-cfg (list g) (list) C)
                (make-m-cfg (list 'any) (list) U))))]
         [cpe (case-lambda 
                [(C U F a b) (delay (list
                 (make-m-cfg (list 'any) (list) (cp (e (e C C b) C a) U F a b))))]
                [(U F a b) (delay (list
                 (make-m-cfg (list 'any) (list) (cpe (cpe U F a b) U F a b))))])]
         [e (case-lambda
               [(C) (delay (list
                (make-m-cfg (list #\ǝ) (list R) (e1 C))
                (make-m-cfg (list 'any) (list L) (e C))))]
               [(B a) (delay (list 
                (make-m-cfg (list 'any) (list) (e (e B a) B a))))]
               [(C B a) (delay (list 
                (make-m-cfg (list 'any) (list) (f (e1 C B a) B a))))])]
         [e1 (case-lambda
               [(C) (delay (list
                (make-m-cfg (list 'empty) (list) C) 
                (make-m-cfg (list 'any) (list R E R) (e1 C))))]
               [(C B a) (delay (list
                (make-m-cfg (list 'any) (list E) C)))])]
         [con (lambda (C a) (delay (list
                (make-m-cfg (list #\A) (list L (P a) R) (con1 C a))
                (make-m-cfg (list 'any) (list R R) (con C a)))))]
         [con1 (lambda (C a) (delay (list
                (make-m-cfg (list #\A) (list R (P a) R) (con1 C a))
                (make-m-cfg (list #\D) (list R (P a) R) (con2 C a)))))]
         [con2 (lambda (C a) (delay (list
                (make-m-cfg (list #\C) (list R (P a) R) (con2 C a))
                (make-m-cfg (list 'any) (list R R) C))))]
         [b (delay (list
                (make-m-cfg (list 'any) (list) (f b1 b1 #\∷))))]
         [b1 (delay (list
                (make-m-cfg (list 'any) (list R R (P #\:) R R (P #\D) R R (P #\A) R R (P #\D)) anf)))] ; added "R R PD"
         [anf (delay (list
                (make-m-cfg (list 'any) (list) (q anf1 #\:))))] ; corrected from "(g ..."
         [anf1 (delay (list
                (make-m-cfg (list 'any) (list) (con fom #\y))))]
         [fom (lambda () (list
                (make-m-cfg (list #\;) (list R (P #\z) L) (con fmp #\x))
                (make-m-cfg (list #\z) (list L L) fom)
                (make-m-cfg (list #\ǝ) (list H) fom)
                (make-m-cfg (list 'any) (list L) fom)))]
         [fmp (delay (list
                (make-m-cfg (list 'any) (list) (cpe (e (e anf #\x) #\y) sim #\x #\y))))] ; corrected
         [sim (delay (list
                (make-m-cfg (list 'any) (list) (fdash sim1 sim1 #\z))))]
         [sim1 (delay (list
                (make-m-cfg (list 'any) (list) (con sim2 'empty))))]
         [sim2 (delay (list
                (make-m-cfg (list #\A) (list) sim3)
                (make-m-cfg (list 'any) (list L (P #\u) R R R) sim2)))] ; corrected from "R ..."
         [sim3 (delay (list
                (make-m-cfg (list #\A) (list L (P #\y) R R R) sim3)
                (make-m-cfg (list 'any) (list L (P #\y)) (e mf #\z))))]
         [mf (delay (list
                (make-m-cfg (list 'any) (list) (q mf1 #\:))))] ; corrected from "(g mf ..."
         [mf1 (delay (list
                (make-m-cfg (list #\A) (list L L L L) mf2)
                (make-m-cfg (list 'any) (list R R) mf1)))]
         [mf2 (delay (list
                (make-m-cfg (list #\C) (list R (P #\x) L L L) mf2)
                (make-m-cfg (list #\:) (list) mf4)
                (make-m-cfg (list #\D) (list R (P #\x) L L L) mf3)))]
         [mf3 (delay (list
                (make-m-cfg (list #\:) (list) mf4)
                (make-m-cfg (list 'any) (list R (P #\v) L L L) mf3)))]
         [mf4 (delay (list
                (make-m-cfg (list 'any) (list) (con (l (l mf5)) 'empty))))]
         [mf5 (delay (list
                (make-m-cfg (list 'empty) (list (P #\:)) sh)
                (make-m-cfg (list 'any) (list R (P #\w) R) mf5)))]
         [sh (delay (list
                (make-m-cfg (list 'any) (list) (f sh1 inst #\u))))]
         [sh1 (delay (list
                (make-m-cfg (list 'any) (list L L L) sh2)))]
         [sh2 (delay (list
                (make-m-cfg (list #\D) (list R R R R) sh3) ; corrected from "sh2"
                (make-m-cfg (list 'any) (list) inst)))]
         [sh3 (delay (list
                (make-m-cfg (list #\C) (list R R) sh4)
                (make-m-cfg (list 'any) (list) inst)))]
         [sh4 (delay (list
                (make-m-cfg (list #\C) (list R R) sh5)
                (make-m-cfg (list 'any) (list) (pe2 inst #\0 #\:))))]
         [sh5 (delay (list
                (make-m-cfg (list #\C) (list) inst)
                (make-m-cfg (list 'any) (list) (pe2 inst #\1 #\:))))]
         [inst (delay (list                                 ; note that inst1 is forced here!
                                                            ; this is because it is a zero-arity varargs
                (make-m-cfg (list 'any) (list) (q (l (inst1)) #\u))))] ; corrected from "(g ..." 
         [inst1 (case-lambda
                 [() (delay (map
                       (lambda (a) (make-m-cfg (list a) (list R E) (inst1 a)))
                       u-symbols))]
                 [(x) (case x
                      [(#\L) (delay (list (make-m-cfg (list 'any) (list) (ce5 ov #\v #\y #\x #\u #\w))))]
                      [(#\R) (delay (list (make-m-cfg (list 'any) (list) (ce5 ov #\v #\x #\u #\y #\w))))]
                      [(#\N) (delay (list (make-m-cfg (list 'any) (list) (ce5 ov #\v #\x #\y #\u #\w))))])])]
         [ov (delay (list
               (make-m-cfg (list 'any) (list) (q (r (r ov1)) #\A))))] ; changed from original
         [ov1 (delay (list
               (make-m-cfg (list #\D) (list) (e anf))
               (make-m-cfg (list 'empty) (list (P #\D)) (e anf))))]
         ) b)) ;; start in state 'b'

I don’t think I can blame Turing much for the errors in the paper. It seems as though some arose through printing typos, and attempting to debug this baroque machine by hand, on paper, would have been a difficult task. (I don’t think he even had Visual Studio!)

A machine in a machine

Of course, the final test of this is to see whether this machine can actually emulate another like it is supposed to. I defined a short procedure to set up a machine on a tape according to Turing’s ingenious encoding.

(define (setup-tape tape inits) 
  (let ([t (move-right (print (move-right (print tape #\ǝ)) #\ǝ))]
  	[inits (append inits (list #\∷))])
   (go-far-left (fold-left (lambda (tape init)
  	       (move-right (move-right (print tape init)))) 
	     t inits))))

A quick explanation of Turing’s encoding

The idea is to first simplify, then encode the states. Turing noted that many of the m-configurations’ operations could be considered redundant:

• as mentioned above, instead of having “erase” we can simply print the ‘blank’ symbol
• instead of not modifying the symbol, we simply print the symbol already present

There are then only three types of operation each state needs to perform:

1. print something and go left
2. print something and go right
3. print something and stay put

States which have a sequence of operations can be split into a series of states, each of which transfers control to the next one.

Now, if we encode all the symbols and configurations as numbers, we can write them out. Turing chose to encode them via this scheme:

• configurations’ numbers are the symbol ‘D’ followed by n ‘A’s, where n is the number of the state
• symbols’ numbers are similar, with ‘D’ followed by n ‘C’s. (Turing set ‘blank’ to always be symbol 0, represented as simply “D”)

So to encode each configuration, we write down its number, the symbol it accepts, the symbol it outputs, which direction to move, and which state to go to next. We prefix each configuration with ‘;’. (When we input it into the machine we also sandwich the whole thing between ‘ǝǝ’ and ‘∷’.)

Here is the example which Turing gives in his paper. I have formatted it to make it easier to read. Can you tell what it does?

(define example (setup-tape empty-tape (list 
			     #\; #\D #\A             #\D #\D #\C     #\R #\D #\A #\A
			     #\; #\D #\A #\A         #\D #\D         #\R #\D #\A #\A #\A
			     #\; #\D #\A #\A #\A     #\D #\D #\C #\C #\R #\D #\A #\A #\A #\A
			     #\; #\D #\A #\A #\A #\A #\D #\D         #\R #\D #\A)))

If we translate the symbols to get numbers we have the following:

; 1 0 0 R 2
; 2 0 . R 3
; 3 0 1 R 4
; 4 0 . R 1

This machine prints alternately 0.1.0.1.. In fact, when I first typed it up, I left off an ‘A’ on the 3rd state and couldn’t figure out why the machine was printing 0.11111...!

Just to show it works

Here is some output of the universal machine running the ’0101′ machine. I have only included a snippet as the machine takes a while to get to this stage. You’ll also notice the output format is different from the other machines; this one outputs some state information and the output of the emulated machine (in this case, 0 and 1), separated by colons. So far, after a minute or so, the machine has output 010

.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C
.ǝǝ;.D.A.D.DuCuR.DyAyAy;.D.A.A.D.D.R.D.A.A.A.;.D.A.A.A.D.D.C.C.R.D.A.A.A.A.;.D.A.A.A.A.D.D.R.D.A.∷.:.D.A.D.:.0.:.D.C.D.A.A.D.:.D.C.D.D.A.A.A.D.:.1.:.D.C.D.D.C.C.D.A.A.A.A.D.:.D.CvDvDvCvCvDxD.A.D.:.0.:.D.C

And that’s enough for today! Feel free to post corrections, additions, your own Turing machines, and so on

If we want to restrict ourselves to assigned, non-control, non-private Unicode characters, then by my reckoning that gives us 129,775 available characters.

wget http://unicode.org/Public/UNIDATA/UnicodeData.txt
awk -F ';' UnicodeData.txt -f countUnichars.awk | bc

countUnichars.awk source:

BEGIN { print "ibase=16" } # set bc to hex mode
 
$2 ~ /Private/ { # skip any lines with "private" in the description
    getline;
}
 
n { # if n is set, then print the range for bc to calculate
    printf("(%s-%s+1)+",$1,n);
    n="";
}
 
$2 ~ /First>/ { # set n if the start of a range
    n=$1;
    getline;
}
 
$3 !~ "C." { # otherwise count anything that isn't some kind of a control character
    i++;
}
 
END { # print out the count of everything else
    printf("%X\n",i)
}

This means we can store exactly 2377 bits (297 bytes) per message (this is ), so if we use a 16-colour palette we can store about 594 pixels (), which can almost reproduce the Mona Lisa thumbnail in the contest page.

s = cmds.sphere()
r = cmds.sphere(s, query=True, radius=True)
cmds.sphere(s, edit=True, radius=10)

Nasty.

So, here is a wrapper to make things nicer:

import maya.cmds as cmds
 
class Wrapper(object):
	def __init__(self, other):
		self.__dict__['me'] = other()
 
	def __getattr__(self, thing):
		i = 0 if cmds.attributeQuery(thing, node=self.__dict__['me'][0], exists=True) else 1
		return cmds.getAttr(self.__dict__['me'][i] + "." + thing)
 
	def __setattr__(self, thing, value):
		i = 0 if cmds.attributeQuery(thing, node=self.__dict__['me'][0], exists=True) else 1
		cmds.setAttr(self.__dict__['me'][i] + "." + name, value)

The above example is now:

s = Wrapper(cmds.sphere)
r = s.radius
s.radius = 10

David R. MacIver has recently written this blog post about cleaning up a set of tags. This blog post, on the other hand, is about a nice old Unix tool called ‘awk’.

Awk is one of those programs that is often overlooked. It is really a small domain-specific language for processing text. In some ways it resembles sed, but it is more powerful, and it especially excels at processing line- and field-structured input.

Processing cite-u-like’s data

First of all, before we begin I want to say this this isn’t a criticism of David’s work. Using a general-purpose language like Ruby to process data comes with several benefits. This post is to explain the benefits of awk and what it excels at doing.

Now, cite-u-like’s tag data comes in a pipe-separated format, so we have input like this:

42|61baaeba8de136d9c1aa9c18ec3860e8|2004-11-04 02:25:05.373798+00|ecoli
42|61baaeba8de136d9c1aa9c18ec3860e8|2004-11-04 02:25:05.373798+00|metabolism
42|61baaeba8de136d9c1aa9c18ec3860e8|2004-11-04 02:25:05.373798+00|barabasi
42|61baaeba8de136d9c1aa9c18ec3860e8|2004-11-04 02:25:05.373798+00|networks
43|61baaeba8de136d9c1aa9c18ec3860e8|2004-11-04 02:25:51.839281+00|control
43|61baaeba8de136d9c1aa9c18ec3860e8|2004-11-04 02:25:51.839281+00|engineering
43|61baaeba8de136d9c1aa9c18ec3860e8|2004-11-04 02:25:51.839281+00|robustness
44|61baaeba8de136d9c1aa9c18ec3860e8|2004-11-04 02:26:33.156319+00|networks
44|61baaeba8de136d9c1aa9c18ec3860e8|2004-11-04 02:26:33.156319+00|strogatz
44|61baaeba8de136d9c1aa9c18ec3860e8|2004-11-04 02:26:33.156319+00|survey

From now on, whenever you see x-separated data, I want you to scream ‘USE AWK!’

Awk scripts look something like this:

pattern { expression }

pattern is used to match against a record, and if successful, the action in the braces (expression) will be carried out. But what is a record? Awk allows you to define what a line is using the variable RS (short for ‘record separator’). By default it is set to \n (so that each line is a record), which is what we want here.

Within the expression you can refer to fields of the current record, using the syntax $n. $0 refers to the whole record, while $1,$2… refer to individual fields.

Awk also allows you to define what the field separator will be via the variable FS.

So how do we set these variables? Awk has two special patterns BEGIN and END, which run before and after everything else. In this case, we want the fields to be separated by |, so we use the pattern:

BEGIN { FS = "|" }

This is rather long-winded, so mawk (an implementation of awk) also allows you to set FS via a command-line option -F.

For this application, we want the last field of the record (we could hard-code it as $4, but we’re exploring awk!). Awk provides the number of fields in the record as the variable NF (number of fields), so we want to access this field. We do so using the record syntax $ and the variable NF.

So to put all this together, what we want is the command:

awk -F "|" '{ print $NF }'

We set the field separator to “|”, and write an awk expression to print the last field of each record. Since we left the pattern empty, this expression is evaluated on every record.

Awk vs. Ruby

So what good is learning all this, anyway? There are a couple of reasons:

• awk is standard on *nix operating systems. In order to use David’s code I had to install Ruby; with awk you can generally count on it being there.
• awk is fast (I should say at least in the interpreter ‘mawk’ which is the standard for Ubuntu). On my machine the awk version completes in under a third of the time that it takes for David’s Ruby version to complete. An interesting thing was that the awk version didn’t even max out the CPU, indicating that it is IO-bound and would go faster if I had faster disks (I’m currently on a laptop).
• Awk is ideal for record- and field-based input, as I hope this post will show you

Filtering data with awk

After the above we use sort and uniq the same as David does to get the results in the following form:

 212595 bibtex-import
 157136 no-tag
  27926 elegans
  27887 celegans
  27825 c_elegans
  27795 nematode
  27738 wormbase
  27736 caenorhabditis_elegans
  18933 review
  15280 all-articles

David uses the following to filter out lines with no alphabetical content:

ruby -ne 'puts $_ if !($_ =~ /^[^a-z]+$/)'

We can use awk’s patterns to do the same thing:

$0 ~ /[a-zA-Z]/

Here we use the ~ (match) operator to write a pattern that matches only the records with an alphabetical character in them. (Remember that $0 refers to the entire record.) Notice also that we can leave off the expression after the pattern, because it defaults to { print }, which is exactly what we want.

In this case, awk really shines. On my machine it outperforms the Ruby version by a factor of 8–9.

Programming with awk

The third task that David does is to consolidate all tags which are differentiated only by hyphens or underscores. That is, ‘a-tag’, ‘atag’, and ‘a_tag’ should all be considered the same. We choose which one to put into the output by whichever one is used the most times (and then we normalize the tag by replacing ‘-’ with ‘_’ so there are only underscores in the output).

Here is David’s code to do the job:

tag_counts = {}
STDIN.lines.each{|l| c, t = l.split; tag_counts[t.strip] = c.to_i }
duplicates = Hash.new{|h, k| h[k] = []}
tag_counts.keys.each{|k| duplicates[k.gsub(/-|_/, "")] << k }
duplicates.values.each{|vs| vs.sort!{|x, y| tag_counts[y] <=> tag_counts[x]} }
 
new_tag_counts = {}
duplicates.values.each{|vs| new_tag_counts[vs[0].gsub(/(_|-)+/, "_")] = vs.map{|v| tag_counts[v]}.inject(0, &:+)}
puts new_tag_counts.to_a.sort{|x, y| y[1] <=> x[1]}.map{|t, c| " #{c} #{t}" }

I’m not going to explain it here, because that’s not the point of the post

Here’s my awk script:

{ tag_counts[$2] = $1 }
END {
	for (tag in tag_counts)
	{
		normtag=tag
		gsub(/-|_/,"",normtag)
 
		count=tag_counts[tag]
		sum[normtag]+=count
 
		if (count > max[normtag])
		{
			names[normtag]=tag
			max[normtag]=count
		}
	}
 
	for (tag in names)
	{
		finaltag=names[tag]
		gsub(/(-|_)+/,"_",finaltag)
		print " " sum[tag] " " finaltag
	}
}

That’s right, you can use awk to do some ordinary programming tasks! Arrays are used using the usual syntax, and awk even has a foreach-style loop for looping over them. I’ll walk through the rest of the script slowly.

First we apply an expression to each record, creating an array of tags and their counts:

{ tag_counts[$2] = $1 }

Then, once everything has finished (the END pattern), we process this array. For each tag, we do the following:

1. Normalize the tag (the gsub function overwrites the variable, so we have to make a copy):

   normtag=tag
   gsub(/-|_/,"",normtag)

   (Notice the similarity between this and Ruby’s equivalent normtag.gsub!(/-|_/, "")!)

2. Get the count for that tag and add it to the count for the normalized version:

   count=tag_counts[tag]
   sum[normtag]+=count

   Like in PHP and Perl, if a value is not present in an array it is automatically added with a default value.

3. Next we check to see if the current tag is the commonest version of the normalized tag, and if so we save its name and count in two other arrays:

   if (count > max[normtag])
   {
   	names[normtag]=tag
   	max[normtag]=count
   }

   Notice again the usefulness of a default value for nonexistent keys: count > max[normtag] will be true if max[normtag] doesn’t exist.

Now we have all we need to print out the answer. For each tag we normalize it to the final version (remembering to make a copy):

finaltag=names[tag]
gsub(/(-|_)+/,"_",finaltag)

Then we print out the line (concatenation is done by simply juxtaposing variables or strings):

print " " sum[tag] " " finaltag

If you’ve been watching closely you’ll notice there is a small difference between the awk and the Ruby scripts; Ruby sorts before outputting, while the awk version will come out in a non-defined order. This is fine! We can use the standard *nix tool ‘sort’ to sort the lines;

awk -f consolidate_tags.awk < tags | sort -nr > fixed_tags

(Note that this fits in with the *nix philosophy of ‘do one thing well’ and reusing small components.)

Again, the awk version outperforms the Ruby by a factor of 3–4.

Reading external commands and files

Unfortunately there doesn’t seem to be a command-line stemming program (a quick Perl script would suffice but it isn’t what we’re here for!), so I’ll skip that stage (here’s one of the aforementioned weaknesses of a non-general-purpose language). Instead we’ll go straight to implementing stopwords.

Here’s David’s Ruby code again:

require "set"
 
stopwords = Set[*
  IO.read("smart.txt").lines.reject{|x| x =~ /^#/}.map(&:strip)
]
 
STDIN.lines.each do |l|
  c, t = l.split
  puts l unless stopwords.include? t.strip
end

And here’s my equivalent in awk:

BEGIN {
    while (getline < "smart.txt")
    { stopwords[$0] = 1 }
}
!($2 in stopwords)

Here I use the ‘getline’ function which does as its name suggests. We make an array of all the stopwords, with 1 as a placeholder value. The pattern is then short and simple: Print every record where the tag isn’t in the stopwords (again, we can leave off the expression to print the whole record).

Note: There is a discrepancy here: David claims this eliminated 46 tags, while I get a value of 368 for both my awk code and his Ruby code.

Again, the Ruby takes about 8 times as long to execute.

Conclusion

Here’s a couple of points:

• Record- or field-oriented data? Think awk.

• Don’t discount it just because it’s venerable. It is very well-suited to its task.

• pattern { expression } syntax is extremely flexible.

• Awk’s regular expressions are fast.

Nowadays everybody wanna talk
    like they got something to say
But nothing comes out
    when they move their lips
Just a bunch of gibberish
And motherf—kers act
    like they forgot about Awk

f x y z = ...

… and you want:

f x y z w = ...

First of all you should check the contexts where f is used. For example, if f is quite often used as an argument to map (fmap, <$>):

elsewhere xs = map (f a b) xs

If you naïvely add w to the end of the argument list you’ll end up with this instead:

elsewhere xs = map (\x-> f a b x c) xs

On the other hand, if you take a look at the usage of f you can decide that because f is often used to map over things, you can insert the additional parameter before the last, and end up with nicer code:

elsewhere xs = map (f a b c) xs

You may even discover that the existing f is used in a lot of lambdas that could be avoided by rearranging its arguments!

The brief was to write both an additive and subtractive synthesizer, in addition to an effects processor. I generalized this a little bit so that you can combine random parts together, so the additive/subtractive synthesizers aren’t really separate any more.

Basically my code allows you to write ‘scores’ like this:

square wave:square:
glassharp:adsr:square wave,0.1,0.2,0.6,0.6
glassharpDelay:delay:glassharp,0.4
glassharpDelayed:gain:glassharpDelay,0.5
instrument:add:glassharp,glassharpDelayed
!instrument c e8 g c4 e8 g c’4 e’ g’ c e8 g c4 e8 g c’4 e’ g’

The first 5 lines are ‘instrument lines’ and the last one is a ‘score line’.

Instrument lines

The instrument lines are somewhat SSA-like in that you can only do one thing on one line. Each line has the structure [instrument name]:[source or filter]:[parameters]. The provided sources include sine, square, sawtooth, triangle, and additive (this latter has parameters which are the coefficients for additive synthesis).

An example of using additive synthesis (here used to approximate a square wave more and more closely):

sq1:additive:1
sq2:additive:1,0,0.333
sq3:additive:1,0,0.333,0,0.2
sq4:additive:1,0,0.333,0,0.2,0,0.14285714
!sq1 c d e f g r !sq2 c d e f g r !sq3 c d e f g r !sq4 c d e f g r

Filters include (parameters in brackets): delay (source instrument, seconds), gain (source instrument, amount), adsr (source instrument, attack time, decay time, sustain gain, release time), lowpass (instrument, frequency, order), highpass (instrument, frequency, order), add (instrument, instrument), subtract (instrument, instrument).

Score lines

Score lines specify the notes for the instruments to play. Notes are specified like this: [note name][optional sharp or flat][optional octave shift][optional length]. For example, to specify a C-major arpeggio using crotchets, over two octaves, you would use:

c4 e g c' e' g'

The note length ‘sticks’ for all following notes in the same line, and defaults to 4 (a crotchet) at the start of each line. Note length is specified as , so c8 is half as long as c4.

Octave shift is indicated using commas or apostrophes:

c,,, c,, c, c c' c'' c'''

Sharps and flats are indicated using ‘s’ and ‘b’, respectively:

c cs d ds e f gb g ab a as b c'

To change instrument in the middle of the score line, use the syntax ‘!instrument_name’. (This operation takes no time in the score.)

Multiple score lines will be played simultaneously. An example:

g2 a g  g
e2 f f  e
c2 c b, c

Rests are specified using the ‘note name’ ‘r’.

Caveats

• The code isn’t written for widespread consumption (it is very get-the-job-done).
• It is possible to send the program into an infinite loop using a filter that refers to itself. (e.g. infinite:delay:infinite,0.5)
• Currently you cannot control filter parameters via the score lines; this means you cannot (e.g.) create sound from white noise using a series of filters, as you cannot change the pitch.
• Wavetables aren’t implemented, but should be easy to add.
• The tempo is currently fixed at 120 BPM.
• Gain for the whole audio file will be reduced automatically to prevent clipping.
• The file format itself is not very forgiving. It is best to stick closely to the examples shown above! (In particular, white space at the end of a score line will break things with obscure messages.)

Code & example

You can listen to an example rendered by my code here. The code can be downloaded (including the score for that example) here.

Comments on implementation

The various files with ‘handle’ in their name are there to support Matlab’s form of closures (this is what I got from the documentation); if you don’t do it this way then all the references to ‘synth_options{n}’ end up being to the wrong options as Matlab’s anonymous functions are in some ways call-by-name.

The file ‘lookup’ is used instead of interfacing directly to java.util.Hashtable because you can’t (AFAIK) marshal function_handles across to Java and back.

Due to minor demand, the code is also available: BigNum source.

Please note that I haven’t actually touched the code since it was first written. I’m sure there’s some things that don’t work properly. If you’re doing anything big with this you probably want to write some ‘destructive’ update functions for adding, etc. At the moment every time you add, subtract, etc, a completely new BigInt is returned. To make it faster you’d want to just update the number in-place.

Original content…

I’ve created a simple wrapper for GMP. At the moment only BigInts are implemented, but I expect to have BigRationals and BigFloats coming along soon enough. (Edit: this never happened.)

Usage

I think this is very close to as-you’d-expect. The only minor thing that might come as a surprise is that BigInts aren’t value types; since they need destructors, they can’t be. The rest is fairly straightforward:

static void Main(string[] args) {
    BigInt n;
    for (ulong i = 100; i <= 999; i++) {
        n = BigInt.Factorial(i);
        Console.WriteLine(n);
    }
    Console.ReadLine();
}

This code executes with sub 5-second times, which I’m pretty pleased with. I’m not sure what kind of performance decrease you get through using GMP under managed code.

Caveats

• Consider this alpha software
• The included GMP DLL is compiled with none of the magical assembly they provide. If you want to, you can compile it yourself with all this enabled, and you should be able to just use it as a drop-in replacement.

Download

Available here: BigNum library. Documentation is available.

public static IEnumerable<T> Map<F,T>(Func<F,T> func, IEnumerable<F> source)
{
    return
        from it in source
        select func(it);
}

Actually, it’s trivial to implement without the new syntax (although I’ve left in Func because it’s nicer than defining a separate delegate):

public static IEnumerable<T> Map<F,T>(Func<F,T> func, IEnumerable<F> from)
{
    foreach (F it in from)
        yield return func(it);
}

But why stop at map? We can also define a fold:

public static T FoldL<F, T>(Func<T, F, T> func, IEnumerable<F> source, T init)
{
    T last = init;
    foreach (F it in source)
        last = func(last, it);
    return last;
}

Or the non-empty-list variant. This is made easier through the query operator Skip:

public static T FoldL1<T>(Func<T, T, T> func, IEnumerable<T> source)
{
    T last = source.First();
    foreach (T it in source.Skip(1))
        last = func(last, it);
    return last;
}

Finally, a simple compose operator for sticking two functions together, again using the nicer syntax to make this simple:

public static Func<X,Z> Compose<X,Y,Z>(this Func<Y,Z> second, Func<X,Y> first) {
    return x => second(first(x));
}

Example

A simple example including functionality from each part above (I put everything into an FFC utility class):

int[] numbers = { 1, 2, 3, 4, 5 };
Func<int, int> addOne = n => n + 1;
 
foreach (var k in FFC.Map(addOne, numbers))
    Console.Write(k + " ");
Console.ReadLine();
 
Func<int,int> addTwo = addOne.Compose(addOne);
 
foreach (var k in FFC.Map(addTwo, numbers))
    Console.Write(k + " ");
Console.ReadLine();
 
var sum = FFC.FoldL((x, y) => x + y, numbers, 0);
Console.Write(sum + " ");
Console.ReadLine();

A first attempt

The first thing I thought of was to implement them using an IEnumerable with yield statements, so I began with a construction borrowed from Haskell: the iterate function.

static class FakeLazyList
{
    public static IEnumerable<T> Iterate<T>(Func<T, T> f, T x)
    {
        while (true)
        {
            yield return x;
            x = f(x);
        }
    }
}

While this works (try FakeLazyList.Iterate(i => i + 1, 0)), it is obviously a poor solution; it is not flexible as it requires a seperate function to be written for every incarnation.

Lazy data in general

My next approach was to begin with creating a generic type for lazy data. All it has to do is act as a wrapper for a piece of data. However, since in C# arguments must be evaluated before they are used, the data must in turn be wrapped inside a delegate so that the delegate’s execution can be postponed. This resulted in:

public class Lazy<T> 
{
    public Lazy(Func<T> del)
    {
        Del = del;
    }
    private Func<T> Del;
    private T PValue;
    private bool HasValue = false;
    public static implicit operator T(Lazy<T> f)
    {
        if (!f.HasValue)
        {
            f.PValue = f.Del.Invoke();
            f.HasValue = true;
        }
        return f.PValue;
    }
}

As you can see, execution of the data-bearing delegate is delayed until the lazy data is forced to become normal data.

Implementing the list

For the list implementation itself, I chose to go with the traditional cons list structure, so that each list item has an object of data, and a pointer to the next item in the list. In order to make the list lazy, the next item of the list should be generated only on-access, so it is wrapped in the above Lazy<T> class.

public class LazyList<T> : IEnumerable<T>, IEnumerable
{
    private T _First;
    private Lazy<LazyList<T>> _Rest;
 
    public T First { get { return _First; } }
    public LazyList<T> Rest { get { return _Rest; } }
 
    public LazyList(T first, Lazy<LazyList<T>> rest)
    {
        _First = first; _Rest = rest;
    }

Next comes some boilerplate so that the class can be used as an enumerable structure:

    public IEnumerator<T> GetEnumerator() {
        return new Enumerator(this);
    }
    IEnumerator IEnumerable.GetEnumerator()
    {
        return new Enumerator(this);
    }
 
    private class Enumerator : IEnumerator<T>, IEnumerator
    {
        private LazyList<T> curr = null;
        private LazyList<T> next;
        public Enumerator(LazyList<T> parent)
        {
            next = parent;
        }
        /* must implement both interfaces */
        object IEnumerator.Current { get { return this.Current; } }
        public T Current {
                get {
                    if (curr == null)
                        throw new InvalidOperationException();
                    else
                        return curr._First;
                }
        }
 
        public bool MoveNext() { curr = next; next = next._Rest; return curr != null; }
        public void Reset() { throw new NotSupportedException(); }
        public void Dispose() { return; }
    }

Last of all comes a couple of utility functions for constructing lists, both of which I took from Haskell. The first, iterate, generates a list of items by taking a function f and a datum x and generating the list like this [x,f(x),f(f(x)),f(f(f(x))),...].

    public static LazyList<X> Iterate<X>(Func<X, X> f, X x)
    {
        return new LazyList<X>(x, new Lazy<LazyList<X>>(delegate { return Iterate(f, f(x)); }));
    }

The next takes a datum and generates the infinite list of just that datum:

    public static LazyList<X> Repeat<X>(X x)
    {
        return new LazyList<X>(x, new Lazy<LazyList<X>>(delegate { return Repeat(x); }));
    }
}

As you can see, the usage of the LazyList constructor isn’t the most elegant. I had hoped that I would be able to use an implicit operator on the Lazy class, such as:

public static implicit operator Lazy<T>(Func<T> f)
{
    return new Lazy<T>(f);
}

… and then use it like this …

new LazyList<X>(x, () => Repeat(x));

Unfortunately, this doesn’t work, giving error CS1660. Apparently the C# compiler cannot use the fact that an implicit cast has been defined. Still, this implementation works well.

Usage

The usage of the LazyList in normal code is very simple, and as shown in this example, you can also use the new C# 3.0-defined list operators Take, Drop, Where etc:

IEnumerable<int> ys;
 
// Lazy list of all positive integers
ys = LazyList<int>.Iterate(i => i + 1, 0).Take(20);
foreach (var y in ys) Console.WriteLine(y);
 
//Lazy list of [89, 89, 89, 89, ...]
ys = LazyList<int>.Repeat(89).Take(20);
foreach (var y in ys) Console.WriteLine(y);
 
Console.ReadLine();

A Comparison

Firstly, I’ll show a short comparison between some Java code and some code from a language that doesn’t have NullPointerExceptions, but does have something that allows you to accomplish anything you might want to do with null pointers.

This Java code:

AnObject thing = someMethod();

is equivalent to the following Haskell:

let thing = Maybe AnObject

That is, every Java object reference is possibly null. What this means is that (using the type notation mentioned in earlier posts) every Java reference has the type:

(Every object reference for an object is either to ‘null’ (AKA ‘unit’ or ‘singleton’, or the object itself.)

… and when you’re dealing with so many objects—Java is, after all, an object-oriented language, so (almost) everything is an object—this simple little “+1″ gets lost easily.

Why so bad?

This wouldn’t seem to be such a big deal, but when contrasted against Java’s penchant for making things (sometimes painfully) explicit—think public static void main (string argh!)—a small, implicit item like this is easily overlooked. Conversely, one might say that the verbosity of Java increases the cognitive load of understanding its code; thus helping the chances of a small mistake like this sneaking through.

Either way, I think I can say that this is objectively A Bad Thing. C# has gone some of the way to a ‘correct’ solution in its introduction of possibly-null primitives, which are explicitly marked with a ? (such as int?), but unfortunately this hasn’t been ‘back-ported’ to the rest of the language. If one were able to force between possibly-null and definitely-not-null references in C#—perhaps by the analogous object? vs. object, this would help to reduce the number of errors ‘hidden’ by the syntax.