Re: Belief Statements

[Hal Ruhl]

Hi All:
As I indicated in my last post I now see choice as an essential part of my 
(2).  But what do I mean by choice and how does choice operate on the 
dynamic?

Speculation:
What is my idea of choice?  In my (2) choice is the ability of a kernel 
currently having physical reality to select in part which kernel(s) next 
have physical reality and this selection is not a constant while a kernel 
has physical reality.  This is not necessarily the same as "free will".

I think the first thing to notice is that by my definition of kernel the 
boundary establishing potential of a given kernel need not be associated 
with a fully fixed boundary.  A boundary could for example have something 
like oil canning dents in it.  The issue re the kernel is whether it is a 
particular boundary not a question of whether or not it is completely 
static.   Such flexing or partially indeterminate features of a boundary 
partially determine the next kernel given physical reality.  That is the 
current condition of the current kernel when the external dynamic moves 
physical reality partially determines which kernel(s) receive it.

Thus the external dynamic would be inconsistent with its history which is a 
requirement of (2).

Is this "free will"?  "Free will" seems to require that one oil canning 
dent influence the state sequence of another oil canning dent in the same 
kernel while that kernel has physical reality.  This seems an unnecessary 
step up in complexity for the total dynamic but it is not forbidden by the 
model since within a kernel the actual oil canning of a dent and its causes 
are not necessarily fixed.  Surely this applies to large regions of the 
boundary containing many oil canning dents.

What is a SAS in this venue?  Taken over a large enough region of the 
boundary the mechanism described above could account for "self aware" in 
what is actually an overall "timeless" moment.

I will try to put this all in a post to the "An All/Nothing multiverse 
model" thread.

Hal Ruhl

COMBINATORS II (solution of exercises)

[Bruno Marchal]

COMBINATORS I  was
http://www.escribe.com/science/theory/m5913.html
I recall all you need to know:
Kxy = x
Sxyz = xz(yz)
That's all!(Well you are supposed to remember also that
abc is an abbreviation of ((ab)c), and a(bc) is an abbreviation for (a(bc)).
I recall the exercices taken from "My First Everything Theory" Primary 
school Year 2127 :)
Solution are below.
Evaluate:

(SS)KKK =
KKK(SS) = ?
(KK)(KK)(KK) = ?
(KKK)(KKK)(KKK) = ?
Evaluate:
K
KK
KKK

K
KK
KKK

K
KK
A little more advanced exercices: is there a molecule, let us called it I, 
having
the following dynamic: (X refers to any molecule).

IX = XI = ?
(Note I will use in this context the words molecules, birds, combinators, 
programs
as synonymous).

SOLUTIONS:

(SS)KKK = SK(KK)K = KK(KKK) = K
KKK(SS) = K(SS)
(KK)(KK)(KK) = KK(KK)(KK) = K(KK)
(KKK)(KKK)(KKK) = KKK = K
Note that the passage (KK)(KK)(KK) = KK(KK)(KK) comes just
from a use of the parentheses abbreviation rule which help to see
the match with the dynamic of K : Kxy = x, and indeed KK(KK),
when occuring at a beginning, matches Kxy with x = K  and y = (KK) = KK.
K=  K
KK=  KK
KKK=  K
   =  KK
K=  K
KK   =  KK
KKK=  K
   =  KK
K=  K
KK   =  KK
ok? (this was easy! if you have not succeed it means you
are imagining difficulties).
The next exercise is slightly less easy,
we are to program some identity operatort.
Ix = xI = ?
We must find a program (that is a combinator, that is a
combination of K and S) which applied on any X gives that X.
We want for example that
I(KK) = (KK)
I(SSS) = SSS etc.
So we want that for all x   Ix = x.
But only Kxy is able to give x
so x = Kx? and we want Kx? matching the rule for S (we have only this one),
it is easy because whatever ? represents, Kx? gives x. So we can take
? = (K x) or (S x) or etc.
This gives x = Kx(Kx)   (or x = Kx(Sx) )
so that the rule S can be applied so that
x = Kx(Kx) = SKKx (or x = Kx(Sx) = SKS)
Thus SKKx = x, and so a solution is
 I = SKK
It is our first program!
Another one is I = SKS (actually SK would work).
Let us verify. i.e. let us test SKK and SKS on KK:
  SKK(KK) = K(KK)(K(KK)) = KK
  SKS(KK) = K(KK)(S(KK)) = KK
more general verification:
SKKx = Kx(Kx) = x
Any problem?  You see that programming is really inverse-executing.
===
New programming exercises:
Find combinators M, B, W, L, T such that
Mx = xx   (Hint: use your "subroutine" I, as a "macro" for SKK)
Bxyz = x(yz)
Wxy = xyy
Lxy = x(yy)
Txy = yx
Bruno
http://iridia.ulb.ac.be/~marchal/