Reflection in Probabilistic Logic

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Paul Christiano has devised a new fundamental approach to the "Löb Problem" wherein Löb's Theorem seems to pose an obstacle to AIs building successor AIs, or adopting successor versions of their own code, that trust the same amount of mathematics as the original.  (I am currently writing up a more thorough description of the question this preliminary technical report is working on answering.  For now the main online description is in a quick Summit talk I gave.  See also Benja Fallenstein's description of the problem in the course of presenting a different angle of attack.  Roughly the problem is that mathematical systems can only prove the soundness of, aka 'trust', weaker mathematical systems.  If you try to write out an exact description of how AIs would build their successors or successor versions of their code in the most obvious way, it looks like the mathematical strength of the proof system would tend to be stepped down each time, which is undesirable.)

Paul Christiano's approach is inspired by the idea that whereof one cannot prove or disprove, thereof one must assign probabilities: and that although no mathematical system can contain its own truth predicate, a mathematical system might be able to contain a reflectively consistent probability predicate.  In particular, it looks like we can have:

∀a, b: (a < P(φ) < b)          ⇒  P(a < P('φ') < b) = 1
∀a, b: P(a ≤ P('φ') ≤ b) > 0  ⇒  a ≤ P(φ) ≤ b

Suppose I present you with the human and probabilistic version of a Gödel sentence, the Whitely sentence "You assign this statement a probability less than 30%."  If you disbelieve this statement, it is true.  If you believe it, it is false.  If you assign 30% probability to it, it is false.  If you assign 29% probability to it, it is true.

Paul's approach resolves this problem by restricting your belief about your own probability assignment to within epsilon of 30% for any epsilon.  So Paul's approach replies, "Well, I assign almost exactly 30% probability to that statement - maybe a little more, maybe a little less - in fact I think there's about a 30% chance that I'm a tiny bit under 0.3 probability and a 70% chance that I'm a tiny bit over 0.3 probability."  A standard fixed-point theorem then implies that a consistent assignment like this should exist.  If asked if the probability is over 0.2999 or under 0.30001 you will reply with a definite yes.

We haven't yet worked out a walkthrough showing if/how this solves the Löb obstacle to self-modification, and the probabilistic theory itself is nonconstructive (we've shown that something like this should exist, but not how to compute it).  Even so, a possible fundamental triumph over Tarski's theorem on the undefinability of truth and a number of standard Gödelian limitations is important news as math qua math, though work here is still in very preliminary stages.  There are even whispers of unrestricted comprehension in a probabilistic version of set theory with ∀φ: ∃S: P(x ∈ S) = P(φ(x)), though this part is not in the preliminary report and is at even earlier stages and could easily not work out at all.

It seems important to remark on how this result was developed:  Paul Christiano showed up with the idea (of consistent probabilistic reflection via a fixed-point theorem) to a week-long "math squad" (aka MIRI Workshop) with Marcello Herreshoff, Mihaly Barasz, and myself; then we all spent the next week proving that version after version of Paul's idea couldn't work or wouldn't yield self-modifying AI; until finally, a day after the workshop was supposed to end, it produced something that looked like it might work.  If we hadn't been trying to solve this problem (with hope stemming from how it seemed like the sort of thing a reflective rational agent ought to be able to do somehow), this would be just another batch of impossibility results in the math literature.  I remark on this because it may help demonstrate that Friendly AI is a productive approach to math qua math, which may aid some mathematician in becoming interested.

I further note that this does not mean the Löbian obstacle is resolved and no further work is required.  Before we can conclude that we need a computably specified version of the theory plus a walkthrough for a self-modifying agent using it.

See also the blog post on the MIRI site (and subscribe to MIRI's newsletter here to keep abreast of research updates).

This LW post is the preferred place for feedback on the paper.

EDIT:  But see discussion on a Google+ post by John Baez here.  Also see here for how to display math LaTeX in comments.

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