MIRI is releasing a paper introducing a new model of deductively limited reasoning: “Logical induction,” authored by Scott Garrabrant, Tsvi Benson-Tilsen, Andrew Critch, myself, and Jessica Taylor. Readers may wish to start with the abridged version.
Consider a setting where a reasoner is observing a deductive process (such as a community of mathematicians and computer programmers) and waiting for proofs of various logical claims (such as the abc conjecture, or “this computer program has a bug in it”), while making guesses about which claims will turn out to be true. Roughly speaking, our paper presents a computable (though inefficient) algorithm that outpaces deduction, assigning high subjective probabilities to provable conjectures and low probabilities to disprovable conjectures long before the proofs can be produced.
This algorithm has a large number of nice theoretical properties. Still speaking roughly, the algorithm learns to assign probabilities to sentences in ways that respect any logical or statistical pattern that can be described in polynomial time. Additionally, it learns to reason well about its own beliefs and trust its future beliefs while avoiding paradox. Quoting from the abstract:
These properties and many others all follow from a single logical induction criterion, which is motivated by a series of stock trading analogies. Roughly speaking, each logical sentence φ is associated with a stock that is worth $1 per share if φ is true and nothing otherwise, and we interpret the belief-state of a logically uncertain reasoner as a set of market prices, where ℙn(φ)=50% means that on day n, shares of φ may be bought or sold from the reasoner for 50¢. The logical induction criterion says (very roughly) that there should not be any polynomial-time computable trading strategy with finite risk tolerance that earns unbounded profits in that market over time.