but I don't understand why it puts a blank line each line,

and, worst why it cuts the end of the posts. So

I resend the cutted line. Apology for your mail box.

COMBINATORS VI (sequel):

Question: is there a systematic method such

that giving any behavior like

Xxyztuv = x(yx)(uvut) (or what you want)

can generate systematically a corresponding

SK or BWCK -combinator?

The answer is yes. I let you meditate the question.

(This point will be made clear when we will meet

the terrible little cousins of the combinators

which are the *lambda _expression_*, (and which

in our context will just abbreviate complex

combinators), but I propose to progress

and make sure that the

SK combinator are Turing Universal.

I am not yet decided when I really should

introduce you to the paradoxical combinators,

which are rather crazy. Smullyan call

them wise birds, but I guess

it is an euphemism!

Mmhhh... Showing turing-universality

through the use of some paradoxical

combinator is easy (once we have defined

the numbers), but there is a risk you take

bad habits! Actually we don't need the

paradoxical combinators to prove the turing

universality of the SK forest, mmmmh...

Well actually I will be rather buzy so I give

you the definition of a paradoxical combinator

and I let you search one.

First show that for any combinator A there

is a combinator B such that AB = B.

B is called a fixed point of A. (like the center

of a wheel C is a fixed point of the rotations of

the wheel: RC = C). It is a bit amazing that

all combinators have a fixed point and that is

what I propose you try to show. Here are hints for

different arguments. 1) Show how to find a fixed

point of A (Arbitrary combinator) using just B, M

and A. (Mx = xx I recall). 2) The same using just the

Lark L (Lxy = x(yy) I recall).

Now, a paradoxical combinator Y is just a

combinator which applied on that A will give the

fixed point of A; that is YA will give a B such that

AB = B, that is A(YA) gives YA, or more generally

Y is a combinator satisfying Yx = x(Yx).

Bruno

============================

COMBINATORS I is

__http://www.escribe.com/science/theory/m5913.html__

COMBINATORS II is

__http://www.escribe.com/science/theory/m5942.html__

COMBINATORS III is

__http://www.escribe.com/science/theory/m5946.html__

COMBINATORS IV is

__http://www.escribe.com/science/theory/m5947.html__

COMBINATORS V is

__http://www.escribe.com/science/theory/m5948.html__

Resume:

Kxy = x

Sxyz = xz(yz)