John Tromp wrote:
I am reading the book "The lambda calculus: Its syntax and Semantics" in the
chapter about Godel Numbering but I am confused in some points.
We know for Church Numerals, we have Cn = \fx.f^n(x) for some n>=0,
i.e. C0= \fx.x and C
1 = \fx.fx.
>From the above definition, I could guess the purpose of this kind of
encoding is trying to encode numeral via terms.
How about the Godel Numbering? From definition we know people say #M is the
godel number of M and we also have [M] = C#M to enjoy the second fixed point
theorem : for all F there exists X s.t. F[X] = X.
What the mapping function # is standing for? How could I use it? What the #M
will be? How to make use of the Godel Numbering?
Thank you very much!
My Wikipedia page on Binary Lambda Calculus.
http://en.wikipedia.org/wiki/Binary_lambda_calculus
shows a simple encoding of lc terms as bitstrings, which in
turn are in 1-1 correspondence with the natural numbers.
It also shows how to represent bitstrings as lambda terms,
and gives an interpreter that extracts any closed term M
from its encoding #M.
It also give several examples of how all that is applicable
to program size complexity.
Also don't forget Jot[1]
[1] http://barker.linguistics.fas.nyu.edu/Stuff/Iota/
--
Live well,
~wren
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