Logical Induction

09/12/2016
by   Scott Garrabrant, et al.
0

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 ϕψ, 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 [...]

READ FULL TEXT

page 1

page 2

page 3

page 4

research
10/12/2015

Asymptotic Logical Uncertainty and The Benford Test

We give an algorithm A which assigns probabilities to logical sentences....
research
04/24/2019

On Learning to Prove

In this paper, we consider the problem of learning a (first-order) theor...
research
01/25/2023

Towards a Unification of Logic and Information Theory

We examine the problem of efficient transmission of logical statements f...
research
04/27/2010

Ontology-based inference for causal explanation

We define an inference system to capture explanations based on causal st...
research
09/14/2017

Integrating Context of Statements within Description Logics

We address the problem of providing contextual information about a logic...
research
10/28/2012

Illustrating a neural model of logic computations: The case of Sherlock Holmes' old maxim

Natural languages can express some logical propositions that humans are ...

Please sign up or login with your details

Forgot password? Click here to reset