I found this fascinating. Some interesting results about AI. How to get
around the assumption of perfect inference.
https://www.youtube.com/watch?v=UOddW4cXS5Y
/Logical Induction//
//
//Scott Garrabrant, Tsvi Benson-Tilsen, Andrew Critch, Nate Soares,
Jessica Taylor//
//(Submitted on 12 Sep 2016 (v1), last revised 2 Oct 2016 (this version,
v3))/
/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)./
Subjects: Artificial Intelligence (cs.AI); Logic in Computer Science
(cs.LO); Logic (math.LO); Probability (math.PR)
Cite as: arXiv:1609.03543 [cs.AI]
(or arXiv:1609.03543v3 [cs.AI] for this version)
Brent
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