and understand how it resolves paradoxical self reference, but I’m not sure what the inductor will do in situations where multiple stable solutions exist.
Let f:[0,1]→[0,1]
If f is continuous then it must have a fixed point. Even if it has finitely many discontinuities, it must have an “almost fixed” point. An x such that ∀ϵ>0:infy∈(x−ϵ,x)f(y)≤x≤supy∈(x,x+ϵ)f(y)
However some f have multiple such points.
f(x)={0x<121x≥12
Has “almost fixed” points at 0, 12 and 1.
A similar continuous f is
f(x)=⎧⎪
⎪
⎪⎨⎪
⎪
⎪⎩0x≤133x−113≤x≤231x≥23
With
f(x)=x
Having every point fixed.
Consider ϕn="f(En(ϕn))"
These functions make ϕn the logical inductor version of “this statement is true”. Multiple values can be consistently applied to this logically uncertain variable. None of the possible values allow a money pump, so the technique of showing that some behaviour would make the market exploitable that is used repeatedly in the paper don’t work here.
Is the value of En(ϕn) uniquely defined or does it depend on the implementation details of the logical inductor? Does it tend to a limit as n→∞ ? Is there a sense in which
f(x)={0.830.82≤x≤0.840.1else
causes En(ϕn) has a stronger attractor to 0.1 than it does to 0.83?
Can En(ϕn) be 0.6 where
f(x)={0.6x=0.60.1else
because the smallest variation would force it to be 0.1?
[Question] Logical inductors in multistable situations.
I was reading about logical induction at
https://intelligence.org/files/LogicalInduction.pdf
and understand how it resolves paradoxical self reference, but I’m not sure what the inductor will do in situations where multiple stable solutions exist.
Let f:[0,1]→[0,1]
If f is continuous then it must have a fixed point. Even if it has finitely many discontinuities, it must have an “almost fixed” point. An x such that ∀ϵ>0:infy∈(x−ϵ,x)f(y)≤x≤supy∈(x,x+ϵ)f(y)
However some f have multiple such points.
Has “almost fixed” points at 0, 12 and 1.
A similar continuous f is
With
Having every point fixed.
Consider ϕn="f(En(ϕn))"
These functions make ϕn the logical inductor version of “this statement is true”. Multiple values can be consistently applied to this logically uncertain variable. None of the possible values allow a money pump, so the technique of showing that some behaviour would make the market exploitable that is used repeatedly in the paper don’t work here.
Is the value of En(ϕn) uniquely defined or does it depend on the implementation details of the logical inductor? Does it tend to a limit as n→∞ ? Is there a sense in which
causes En(ϕn) has a stronger attractor to 0.1 than it does to 0.83?
Can En(ϕn) be 0.6 where
because the smallest variation would force it to be 0.1?