I love Mencius to death, but let's face it - he's a strange guy. And his blog is even stranger. Frankly, I'd stay off it.
So here's a copy of the original UR post - the Nock spec, plus some mysterious non-clarification:
1 Context This spec defines one function, Nock. 2 Structures A noun is an atom or a cell. An atom is any unsigned integer. A cell is an ordered pair of any two nouns. 3 Pseudocode Brackets enclose cells. [a b c] is [a [b c]]. *a is Nock(a). Reductions match top-down. 4 Reductions ?[a b] => 0 ?a => 1 ^[a b] => ^[a b] ^a => (a + 1) =[a a] => 0 =[a b] => 1 =a => =a /[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 *[a 0 b] => /[b a] *[a 1 b] => b *[a 2 b c d] => *[a 3 [0 1] 3 [1 c d] [1 0] 3 [1 2 3] [1 0] 5 5 b] *[a 3 b] => **[a b] *[a 4 b] => ?*[a b] *[a 5 b] => ^*[a b] *[a 6 b] => =*[a b] *[a [b c] d] => [*[a b c] *[a d]] *a => *a
For context and comparison, see this or this or this. Here at UR, axioms are axioms - we leave no "niceties such as arithmetic" to the programmer's imagination. Just sayin'.
To grok Nock, construct a formula
(s - 1)for any atomic nonzero subject
s. The equivalent formula for
(s + 1)is
[5 0 1]. The first 16 people who mail me a correct answer may (or may not) win a prize, which may (or may not) be valuable. (Please include your interpreter - in any language, it should fit on a page. Please do not post the answer here or anywhere, and let me know if you see it posted. This is not a difficult problem.)
As you'll quickly see if you try this exercise, raw Nock is not a usable programming language. But nor is it an esoteric language or automaton, like SK combinators. Rather, Nock is a tool for defining higher-level languages - comparable to the lambda calculus, but meant as foundational system software rather than foundational metamathematics.
To define a language with Nock, construct two nouns,
r, such that
*[s *[p r]]is a useful functional language. In this description,
pis the function source;
qis your language definition, as source;
ris your language definition, as data;
sis the input data. You will find this a tractable and entertaining problem.