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Fixed markdown.

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commit be83f787b9642239d56010b7e37ead13936f96a0 1 parent 964127c
@grig authored
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48 combinators1-5.md
@@ -1,11 +1,11 @@
-# See http://github.com/raganwald/homoiconic/tree/master/2008-11-12/combinator_chemistry.md
+See http://github.com/raganwald/homoiconic/tree/master/2008-11-12/combinator_chemistry.md
-** Intro
+# Intro
sxyz = xz(yz)
kxy = x
-** Exercises
+# Exercises
(ss)kkk = sk(kk)k = kk((kk)k) = kkk = k
kkk(ss) = k(ss)
@@ -35,42 +35,42 @@ A little more advanced exerciaes: is there a molecule, let us called it I, havin
Ix = skkx = kx(kx) = x
-** New programming exercises:
+## New programming exercises:
Find combinators M, B, W, L, T such that
Mx = xx
-*** Solution
+### Solution
xx = Ix(Ix) = SIIx => M = SII
-** B
+## B
Bxyz = x(yz)
-*** Solution
+### Solution
x(yz) = S?yz, ?z = x. Kxz = x => x(yz) = S(Kx)yz = ((KS)x)(Kx)yz = S(KS)Kxyz
B = S(KS)K
-** W
+## W
Wxy = xyy
-*** Solution
+### Solution
xyy = (xy)y =
= (xy)(?xy) = (xy)(KIxy) = Sx(KIx)y = Sx((KI)x)y
= SS(KI)xy
-*** Check:
+### Check:
SS(KI)xy = Sx(KIx)y = xy(KIxy) = xy(Iy) = xyy
W = SS(KI)
-** L
+## L
Lxy = x(yy)
-*** Solution
+### Solution
Lxy = x(yy) = (Kxy)(yy) = S(Kx)yy = (KSx)(Kx)yy = S(KS)Kxyy =
S(KS)K(Wxy)
@@ -88,27 +88,27 @@ Find combinators M, B, W, L, T such that
L = S(S(KS)K)(KM)
-*** Check
+### Check
S(S(KS)K)(KM)xy = S(KS)Kx(KMx)y = S(KS)KxMy = (KSx)(Kx)My = S(Kx)My =
(Kxy)(My) = x(My) = x(yy)
-*** Another solution
+### Another solution
x(yy) = Bxyy = (Bx)yy = W(Bx)y = BWBxy
-*** check
+### check
BWBxy = W(Bx)y -- dynamics of B
= (Bx)yy -- dynamics of W
= Bxyy
= x(yy)
-** T
+## T
Txy = yx
-*** Solution
+### Solution
yx = y(Kxy) = (Iy)(Kxy) = SI(Kx)y = (K(SI)x)(Kx)y = S(K(SI))Kxy
@@ -116,15 +116,15 @@ Find combinators M, B, W, L, T such that
Txy = S(K(SI))Kxy = (K(SI)x)(Kx)y = SI(Kx)y = Iy(Kxy) = yx
-*** Another
+### Another
yx = y(Kxy) = SI(Kx)y = B(SI)Kxy
-** C
+## C
Cxyz = xzy
-*** Solution 1
+### Solution 1
xzy = xz(Kyz) = Sx(Ky)z = K(Sx)y(Ky)z = S(K(Sx))Kyz = S((KKx)(Sx))Kyz
= S(S(KK)Sx)Kyz = ((KS)x)((S(KK)S)x)Kyz
@@ -134,23 +134,23 @@ Find combinators M, B, W, L, T such that
C = S(S(KS)(S(KK)S))(KK)
-*** Solution 2
+### Solution 2
xzy = xz(Kyz) = Sx(Ky)z = B(Sx)Kyz = BBSxKyz = BBSx(KKx)yz
= S(BBS)(KK)xyz
C = S(BBS)(KK)
-** INFINITY
+## INFINITY
INFINITY dynamically transform into INFINITY itself.
-*** Solution
+### Solution
MM = MM, thus,
INFINITY = MM
-** S
+## S
We have seen how to program the blue bird B, the cardinal C and the
Warbler W with the kestrel K and the starling S. Could you define the
View
1  combinators1.md
@@ -1 +0,0 @@
-Find file: ~/Projects/combinators/[exercises.txt]
View
20 combinators6.md
@@ -1,4 +1,4 @@
-* Question: is there a systematic method such that giving any behavior like
+Question: is there a systematic method such that giving any behavior like
Xxyztuv = x(yx)(uvut) (or what you want)
@@ -12,11 +12,11 @@ Well actually I will be rather busy so I give you the definition of a paradoxica
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).
-** Problem: fixed point of A
+## Problem: fixed point of A
Show that for any combinator A there is a combinator B such that AB = B
-*** Solution
+### Solution
Ax = x, x=?
@@ -51,17 +51,17 @@ let X' = SX'', then
X' = BSM
X = M(BSM)
-** Check:
+### Check:
M(BSM)yz = BSM(BSM)yz = S(M(BSM))yz
-Fixpoint of I:
+## Fixpoint of I:
Ix = x
any combinator is a fixpoint of I
-Fixpoint of K
+## Fixpoint of K
Kx = x
Kxy = xy = x, that is, x is such that for all y
@@ -83,7 +83,7 @@ Search X as M(X')
X = M(BKM)
-** Fixpoint of B
+## Fixpoint of B
BX = X
@@ -95,17 +95,17 @@ Xyz = M(BBM)yz = BBM(BBM)yz = B(M(BBM))yz = BXyz
X = M(BBM) = BBM(BBM) = B(M(BBM)) = BX
-** Fixpoint of M
+## Fixpoint of M
X = M(BMM)
Xy = M(BMM)y = BMM(BMM)y = M(M(BMM))y
-** Fixpoint of A
+## Fixpoint of A
X = M(BAM) = BAM(BAM) = A(M(BAM)) = AX
-* Y
+# Y
YA = M(BAM)
View
20 combinators8.md
@@ -10,14 +10,14 @@ been replaced by a variable a). This is a traditional (non recursive)
exercise. Then YB gives the solution of the recursive equation. (Y is
the traditional name for a paradoxical combinator). Exercise: why?
-** Solution
+## Solution
B(YB) = YB
B(YB)xyz = xx(YB)(YByy)z = YBxyz
let YB = A, then
Axyz = xxA(Ayy)z
-* exercises
+# exercises
Find an infinite eliminator E, that is a bird which eliminates all its
variables: Ex = E, Exy = E, etc.
@@ -31,9 +31,9 @@ Etc. I mean: solve the following equations (little letters like x, y z
are put for any combinator, A is put for the precise combinator we are
ask searching for):
-Ax = A
+# Ax = A
-** solution
+## solution
Let's find B such that
@@ -41,31 +41,31 @@ Let's find B such that
B = K
A = YB = YK = SLLK
-* Ax = xA
+# Ax = xA
Bax = xa
B = T = B(SI)K
A = YB = YT = SLLT
-* Axy = Ayx
+# Axy = Ayx
Baxy = ayx
B = C
A = YB = YC
-* Ax = AAx
+# Ax = AAx
Bax = aax
aax = (aa)x = Max, so B = M, A = YB = YM
-* A = AA
+# A = AA
Ba = aa
B = M => A = YM
-* Ax = AA
+# Ax = AA
Xax = aa
@@ -73,7 +73,7 @@ Let's find B such that
X = BKM
A = YX = YBKM
-* Ax = x(Ax)
+# Ax = x(Ax)
Bax = x(ax)
x(ax) = Ix(ax) = SIax

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