This might be relevant (apologies if not, I'm somewhat unsophisticated
and perhaps this is all old news, or not interesting.) :)
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, openeing, 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'.[4]
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[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
--
http://twitter.com/SimonForman
My blog: http://firequery.blogspot.com/
Also my blog: http://calroc.blogspot.com/
"The history of mankind for the last four centuries is rather like that of
an imprisoned sleeper, stirring clumsily and uneasily while the prison that
restrains and shelters him catches fire, not waking but incorporating the
crackling and warmth of the fire with ancient and incongruous dreams, than
like that of a man consciously awake to danger and opportunity."
--H. P. Wells, "A Short History of the World"
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