I was investigating G. Spencer-Brown's
Laws of Form by implementing it
in Python, and I found a very interesting
interpretation in terms of the
SK combinator calculus.
You can represent the "marks" of LoF as data-structures in Python composed entirely of tuples. For example:
- A mark: ()
- A mark next to a mark: (), ()
- A mark within a mark: ((),)
- and so on...
It is known that the propositional calculus can be represented by the "arithmetic of the mark". The two rules suffice:
((),) == nothing
() == (), ()
There are
details, but essentially math and logic can be derived from the behaviour of "the mark".
After reading "
Every Bit Counts" I spent some time trying to come up with an EBC coding for the circle language (Laws of Form expressed as tuples, See Burnett-Stuart's "
The Markable Mark")
With a little skull-sweat I found the following encoding:
- For the empty state (no mark) write '0'.
- Start at the left, if you encounter a boundary (one "side" of a mark, or the left, opening, parenthesis) write a '1'.
- Repeat for the contents of the mark, then the neighbors.
_ = 0
() = 100
(), () = 10100
((),) = 11000
and so on...
I recognized these numbers as the patterns of
the language called 'Iota'.
Briefly, the SK combinators:
S = λx.λy.λz.xz(yz)
K = λx.λy.x
Or, in Python:
S = lambda x: lambda y: lambda z: x(z)(y(z))
K = lambda x: lambda y: x
can be used to define the combinator used to implement Iota:
i = λc.cSK
or,
i = lambda c: c(S)(K)
And the bitstrings are decoded like so: if you encounter '0' return i, otherwise decode two terms and apply the first to the second.
In other words, the empty space, or '0', corresponds to i:
_ = 0 = i
and the mark () corresponds to i applied to itself:
() = 100 = i(i)
which is an Identity function I.
The S and K combinators can be "recovered" by application of i to itself like so (this is Python code, note that I am able to use 'is' instead of the weaker '==' operator. The i combinator is actually recovering the very same lambda functions used to create it. Neat, eh?):
K is i(i(i(i))) is decode('1010100')
S is i(i(i(i(i)))) is decode('101010100')
Where decode is defined (in Python) as:
decode = lambda path: _decode(path)[0]
def _decode(path):
bit, path = path[0], path[1:]
if bit == '0':
return i, path
A, path = _decode(path)
B, path = _decode(path)
return A(B), path
(I should note that there is an interesting possibility of encoding the tuples two ways: contents before neighbors (depth-first) or neighbors before content (breadth-first). Here we look at the former.)
So, in "Laws of Form" K is ()()() and S is ()()()() and, amusingly, the identity function I is ().
The term '(())foo' applies the identity function to foo, which matches the behaviour of the (()) form in the Circle Arithmetic (()) == _ ("nothing".)
(())A = i(i)(A) = I(A) = A
I just discovered this (that the Laws of Form have a direct mapping to the combinator calculus by means of λc.cSK) and I haven't found anyone else mentioning yet (although
this article might, I haven't worked my way all the way through it yet.)
There are many interesting avenues to explore from here, and I personally am just beginning, but this seems like something worth reporting.
