"Laws of Form" are a notation for the SK calculus, demo in Python.

forman.simon at gmail.com forman.simon at gmail.com
Sun Mar 31 20:55:29 CEST 2013

I was investigating G. Spencer-Brown's Laws of Form[1] by implementing it
in Python.  You can represent the "marks" of LoF as datastructures 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"[2] 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"[3])

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

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


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[5] by means of λc.cSK) and I haven't found anyone
else mentioning yet (although [6] 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.

Warm regards,
~Simon Peter Forman

[1] http://en.wikipedia.org/wiki/Laws_of_Form
[2] research.microsoft.com/en-us/people/dimitris/every-bit-counts.pdf
[3] http://www.markability.net/
[4] http://semarch.linguistics.fas.nyu.edu/barker/Iota/
[5] http://en.wikipedia.org/wiki/SKI_combinator_calculus have 
[6] http://memristors.memristics.com/Combinatory%20Logic%20and%20LoF/Combinatory%20Logic%20and%20the%20Laws%20of%20Form.html

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