Consider this iterative system:
A = 1/A + B
B = 1/B + C
C = 1/C + 1/A
If we use the logistic interpretation: 0 = True, Infinity = False, + = 'and', 1/x = 'not', then this translates into this paradox system:
A = B and not A
B = C and not B
C = not A and not C
I have experimented with this system, and find that whatever initial values I input for ABC, it quickly converges to a period-2 oscillation. But not the _same_ oscillation for different initial values!
My question is: what does the set of period-2 wobbles look like in 3d space? Can you do this: Program a computer to take a large number of random points, iterate all to a wobble, plot all the resulting point pairs, and display it, as seen along the three axes, as gifs. Or perhaps some stereograms?
Iterated twice, the system is:
A = (A {+} b) + (b + C)
b = (b + C) {+} (C {+} A)
C = (C {+} A) + (A {+} b)
where b = 1/B, and {+} is the reciprocal addition operator 1/((1/x)+(1/y)). This translates by logistic into this:
A = (A or b) and b and C
b = (b and C) or C or A
C = (C or A) and (A or b)
This converges to fixedpoints, but again not the same fixedpoint for different initial values. And this too I would like to see some gifs of. Can you do this?
The reciprocal dual of the systems noted above are:
A = 1/A {+} B = B / (AB+1)
B = 1/B {+} C = C / (BC+1)
C = 1/C {+} 1/A = 1 / (C+A)
A: “If I’m right, then B is right.”
B: “If I’m right, then C is right.”
C: “If I’m right, then A is wrong.”
This too converges to a period 2 wobble, but different ones for different initial conditions. This too I'd like to see a gif of.
Its second iterate is:
A = (A + b) {+} (b {+} C)
b = (b {+} C) + (C + A)
C = (C + A) {+} (A + b)
- where b = 1/B. It translates to:
A = (A and b) or b or C
b = (b or C) and C and A
C = (C and A) or (A and b)
No comments:
Post a Comment