Logical Induction

Abstract

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).

Temporal Inference with Finite Factored Sets

“…Upon discovering logical induction, one of the first things we considered was the possibility of inferring logical causality using our probabilities on logical sentences (Garrabrant et al 2016). We considered doing this using the Pearlian paradigm, but it now seems like that approach was doomed to fail, because we had many deterministic relationships between our variables.…”

Elementary Logic in Linear Space

“…We use the phrase "reasoning process" informally to suggest that the process is not necessarily deductive. 8 Nevertheless, deduction is an essential reasoning process and so we turn our attention towards this kind of reasoning next.…”

“…Logical uncertainty Logical uncertainty and the problem of logical omniscience is a problem of interest in philosophy (e.g., see [3,11,30,35]) and there have been many solutions proposed for addressing it. One approach proposes assigning probabilities to sentences such that logically equivalent statements are not necessarily assigned the same probability (e.g., see [8] and [15]). 14 Another approach syntactically models an agent's knowledge (e.g., see [6] and [7]).…”

“…. , ϕ n is encoded as the one sentence system ϕ 1 ∧ • • • ∧ ϕ n 8. For example, Huang[15] shows how a local Bayesian update process can be used to update the assignment of plausibility to logical possibilities.…”

Scalable agent alignment via reward modeling: a research direction

“…These formal tools could be necessary for formal verification of agents designing upgraded versions of themselves. Yet while there has been some of progress on this research agenda (Barasz et al, 2014;Garrabrant et al, 2016;Everitt, 2018), some questions turned out to be quite difficult. But even if we had formal solutions to the problems put forth by Soares & Fallenstein, there would still persist a gap to transfer these solutions to align agents in practice.…”