Author: Curtis Yarvin (curtis.yarvin@gmail.com)
Date: 3/8/2008
Version: 0.13

1. Manifest

    This file defines one Turing-complete function, "nock."

    nock is in the public domain.  So far as I know, it is
    neither patentable nor patented.  Use it at your own risk.
 
2. Data

    Both the domain and range of nock are "nouns."

    A "noun" is either an "atom" or a "cell."  An "atom" is an unsigned 
    integer of any size.  A "cell" is an ordered pair of any two nouns,
    the "head" and "tail."

3. Pseudocode

    nock is defined in a pattern-matching pseudocode.

    Match precedence is top-down.  Operators are prefix.  Parens
    denote cells, and group right: (a b c) is (a (b c)).  

4. Definition

4.1 Transformations

      *(a 0 b c)   => *(*(a b) c)
      *(a 0 b)     => /(b a)
      *(a 1 b)     => (b)
      *(a 2 b)     => **(a b)
      *(a 3 b)     => &*(a b)
      *(a 4 b)     => ^*(a b)
      *(a 5 b)     => =*(a b)
      *(a 6 b c d) => *(a 2 (0 1) 2 (1 c d) (1 0) 2 (1 2 3) (1 0) 4 4 b)
      *(a b c)     => (*(a b) *(a c))
      *(a)         => *(a)

4.2 Operators

4.2.1 Goto (*)

      *(a)             -> nock(a)

4.2.2 Deep (&)
    
      &(a b)           -> 0
      &(a)             -> 1
 
4.2.3 Bump (^)

      ^(a b)           -> ^(a b)
      ^(a)             -> a + 1

4.2.4 Same (=)

      =(a a)           -> 0
      =(a b)           -> 1
      =(a)             -> =(a)

4.2.5 Snip (/)

      /(1 a)           -> a
      /(2 a b)         -> a
      /(3 a b)         -> b
      /((a + a) b)     -> /(2 /(a b))
      /((a + a + 1) b) -> /(3 /(a b))
      /(a)             -> /(a)