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Logical Induction


Scott Garrabrant,, Machine Intelligence Institute, Logical Induction, here.

We present a computable algorithm that assigns probabilities to every logical
statement in a given formal language, and refines those probabilities over time.
For instance, if the language is Peano arithmetic, it assigns probabilities to
all arithmetical statements, including claims about the twin prime conjecture,
the outputs of long-running computations, and its own probabilities. We show
that our algorithm, an instance of what we call a logical inductor, satisfies a
number of intuitive desiderata, including: (1) it learns to predict patterns
of truth and falsehood in logical statements, often long before having the
resources to evaluate the statements, so long as the patterns can be written
down in polynomial time; (2) it learns to use appropriate statistical summaries
to predict sequences of statements whose truth values appear pseudorandom;
and (3) it learns to have accurate beliefs about its own current beliefs, in a
manner that avoids the standard paradoxes of self-reference. For example, if
a given computer program only ever produces outputs in a certain range, a
logical inductor learns this fact in a timely manner; and if late digits in the
decimal expansion of π are difficult to predict, then a logical inductor learns
to assign ≈ 10% probability to “the nth digit of π is a 7” for large n. Logical
inductors also learn to trust their future beliefs more than their current beliefs,
and their beliefs are coherent in the limit (whenever φ → ψ, P∞(φ) ≤ P∞(ψ),
and so on); and logical inductors strictly dominate the universal semimeasure
in the limit.
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. This criterion
bears strong resemblance to the “no Dutch book” criteria that support both
expected utility theory (von Neumann and Morgenstern 1944) and Bayesian
probability theory (Ramsey 1931; de Finetti 1937)



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