Thursday, May 17, 2018

Diamond Bracket Forms and How to Count to Two; 4 of 10

From "Cybernetics & Human Knowing", Vol. 24 (2017), No. 3-4, pp. 161-188
3. Phase Order and Fixedpoints

Recall “phase order”:
x < y        iff         x min y  =  x        iff         x max y  =  y

<        <
I        <        J
<        <
This structure is a lattice; and it is preserved by any diamond-logic function:
a < b     implies    F(a)  <  F(b)

We can extend < to form vectors:
x   =    ( x1, x2, x3, ... , xn )
x  <  y   if and only if   ( xi < yi ) for all i

Given any diamond-logic function f(x), define
a left seed for f is any vector a such that     f(a) < a;
a right seed for f is any vector a such that    a < f(a).
a fixedpoint for f is any vector a such that    a  = f(a).
A fixedpoint is both a left seed and a right seed.

Right seeds generate fixedpoints, thus:
If a is a right seed for f, then  a < f(a). Since f is diamond-logic, it preserves order; so f(a) < f2(a); and f 2(a) < f 3(a); and so on:
a  <  f(a) < f 2(a) <  f 3(a) < f 4(a) _ ...
If f has n dimensions, then each component can move at most two steps right before stopping, so the form vector must reach its right bound within 2n steps.
Therefore f 2n(a) is a fixedpoint for f:   f(f 2n(a))  =   f 2n(a)
This is the leftmost fixedpoint right of a.

Right seeds grow rightwards towards fixedpoints.  Similarly, left seeds grow leftwards towards fixedpoints.
All fixedpoints are both left and right seeds - of themselves.

The Self-Reference Theorem:
Any self-referential diamond-logic system has a fixedpoint:
F(x) = x

Proof: Recall that all diamond-logic functions preserve order.
i is the leftmost set of values, hence this holds:
i     <    F(i)
Therefore, i is a right seed for F:
i   <  F(i)  <  F2(i)  <  F3(i)  < ... F2n(i)  =   F(F2n(i))
i generates the “leftmost” fixedpoint.  QED.

Similarly, j generates the “rightmost” fixedpoint:
F(F2n(j))      =       F2n(j)
All other fixedpoints lie between the two extremes:
x = F(x)       implies  that      F2n(i)  <   x   <   F2n(j)

I call this process “productio ex absurdo”; literally, production from the absurd; in contrast to “reduction to the absurd”, boolean logic’s refutation method. Diamond logic begins where boolean logic ends.

To see productio ex absurdo in action, consider this system:
A  =   false
B  =   if B then A
C  =   A or not A
D  =   A but C
E  =   C but A
F  =   D or not D
G  =   E or not E
H  =   F and G

If we iterate this system from default value i, we get:
(A,B,C,D,E,F,G,H) =   (i,i,i,i,i,i,i,i)
è  (f,i,i,i,i,i,i,i)
è  (f,i,t,f,i,i,i,i)
è  (f,i,t,j,i,t,i,i)
è  (f,i,t,j,i,j,i,i)
è  (f,i,t,j,i,j,i,f)
=   fixedpoint.

If we had started from default value j, we would have gotten the rightmost fixedpoint (f,J,t,j,i,j,i,f). These are the only two fixedpoints.

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