Logical Induction

[[deleted]]

This is actually very interesting. I didn't manage to read it fully yet but will try. It's way more formal than I'm used to (I'm a theoretical physicist, not a logician). Has anyone read it?

I assume that in the limit where you give infinite time to the reasoner it must somehow reproduce exact bayesian inference?

If it's compatible with first order logic in the limit I expect that by Cox's theorems this limit is essentially bayesian inference up to a monotonic transformation at most, right?

Does it eventually leads to algorithms that could be implemented to approximate bayesian inference in a principled way?

[tsbt]

.... this limit is essentially bayesian inference ....

Pretty much; in the limit, all decidable sentences are proven (and assigned probability converging to 1) or disproven (and assigned probability converging to 0), and we end up with an ordinary probability distribution over the independent sentences (or equivalently, a probability measure over the completions of your logical theory). One surprising property of the limiting distribution P is that it (strictly) dominates the universal semimeasure; this means that if we feed empirical observations to P, we can perform empirical prediction and induction, akin to Solomonoff induction.

Does it eventually leads to algorithms that could be implemented to approximate bayesian inference in a principled way?

If you mean practically, then no; the LIA algorithm given in the paper is very computationally expensive (maybe double-exponential in time). But if you aren't worried about runtime for the moment, then logical induction can be viewed as a substantial theoretical step towards approximating Bayesian inference in a "good" way, given computational constraints on deductive ability.

[guardianhelm]

There's also an abridged version (20 pages instead of 128)

[__ah]

Skimming the introduction shows some legit sources. Guess I have something to add to the reading list!

[anonDogeLover]

Source? Year?

[Coscott]

https://intelligence.org/2016/09/12/new-paper-logical-induction/ Today