Understanding Artificial Intelligence
through Algorithmic Information Theory
Causal Complexity
There is nothing simple in a rabbit
that pops out of the magician’s hat. On the contrary! There is so much complexity there that the layperson finds it hard to believe.
Before it appears, you are pretty sure that there’s no rabbit in the hat.
You don’t expect to see one there.
This is the normal, simple, situation. The negation of this simple situation is hugely complex. You need to imagine complex circumstances to explain why a rabbit ends up in the hat (teleportation, contortionist rabbit, you’ve gone blind for 5 seconds and have no memory of it, etc.).
So is unexpectedness sometimes due to simplicity (as in remarkable lottery draws) and sometimes due to complexity, as here?
In a way, yes...... but there is a possible mix-up here.
We defined unexpectedness as
the difference between
expected complexity
and observed complexity.Be sure to mind the difference.
- Expected complexity means that you mentally consider the event before its occurrence (ex ante) and measure the complexity of this occurrence. It answers the questions "What were the odds" in computational terms (i.e., in bits).
- Observed complexity means that you consider the event after its occurrence (ex post) and measure the complexity of this occurrence.
We defined unexpectedness as
the difference between
expected complexity
and observed complexity.
In the case of apparent magic, the expected complexity of the event (which is the negation of a normal situation) is huge. This is what creates the complexity gap, hence the unexpectedness. So unexpectedness may result, not only from a small value of the observed complexity (i.e. simplicity), but also from a large value of the expected complexity.
On many occasions, expected complexity is assessed through an attempt to reconstruct a causal scenario that led the event to happen.
It is then called causal complexity,
or also generation complexity,
or sometimes "world complexity", and is noted Cw. Cw(s) = minp {length(p) | W(p) = s} W designates the observer’s causal model of the world.
The only admissible programs p are procedures that respects causality, as understood by the observer.
On many occasions, expected complexity is assessed through an attempt to reconstruct a causal scenario that led the event to happen.It is then called causal complexity,
or also generation complexity,
or sometimes "world complexity", and is noted Cw. Cw(s) = minp {length(p) | W(p) = s} Quite often, causal complexity matches probability (in bits, i.e. Cw(s) = log2(1/Pr(s))). However:
- This is not always true, as for example in the case of the rabbit’s magic occurrence.
- Causal complexity is more natural, as probability calculus must implicitly perform the same causal computations.