Asymptotic Logical Uncertainty and the Benford Test

Garrabrant, Scott; Benson-Tilsen, Tsvi; Bhaskar, Siddharth; Demski, Abram; Garrabrant, Joanna; Koleszarik, George; Lloyd, Evan

doi:10.1007/978-3-319-41649-6_20

Cited by 4 publications

(6 citation statements)

References 26 publications

(16 reference statements)

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“…However, we have no reason to expect that an inductively coherent M would have this property. Garrabrant et al (2016) study computable distributions that can do this; it is not yet clear how to reconcile our framework with theirs.…”

Section: Discussionmentioning

confidence: 99%

Inductive Coherence

Garrabrant,

Fallenstein,

Demski

et al. 2016

Preprint

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While probability theory is normally applied to external environments, there has been some recent interest in probabilistic modeling of the outputs of computations that are too expensive to run. Since mathematical logic is a powerful tool for reasoning about computer programs, we consider this problem from the perspective of integrating probability and logic. Recent work on assigning probabilities to mathematical statements has used the concept of coherent distributions, which satisfy logical constraints such as the probability of a sentence and its negation summing to one. Although there are algorithms which converge to a coherent probability distribution in the limit, this yields only weak guarantees about finite approximations of these distributions. In our setting, this is a significant limitation: Coherent distributions assign probability one to all statements provable in a specific logical theory, such as Peano Arithmetic, which can prove what the output of any terminating computation is; thus, a coherent distribution must assign probability one to the output of any terminating computation.To model uncertainty about computations, we propose to work with approximations to coherent distributions. We introduce inductive coherence, a strengthening of coherence that provides appropriate constraints on finite approximations, and propose an algorithm which satisfies this criterion.

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Section: Discussionmentioning

confidence: 99%

Inductive Coherence

Garrabrant,

Fallenstein,

Demski

et al. 2016

Preprint

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“…If a reasoner thought the 10 100 th digit of π was almost surely a 9, but had no reason for believing this this, we would be suspicious of their reasoning methods. Desideratum 4 is difficult to state formally; for two attempts, refer to Garrabrant, Benson-Tilsen, et al (2016) and Garrabrant, Soares, and Taylor (2016).…”

Section: Desiderata For Reasoning Under Logical Uncertaintymentioning

confidence: 99%

Logical Induction

Garrabrant,

Benson-Tilsen,

Critch

et al. 2016

Preprint

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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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“…Such algorithms are not entirely realistic, but one way to interpret them is as real-time efficient (since we assume polynomial time) algorithms that require inefficient precomputation (at least this interpretation is valid when the advice strings are computable). The strength of the concept of an "F(Γ)-optimal polynomialtime estimator" depends ambiguously on the size of Γ A , since on the one hand larger Γ A allows for a greater choice of candidate optimal polynomial-time estimators, on the other hand the estimator is required to be optimal in a larger class 5 . Sometimes it is possible to get the best of both worlds by having an estimator which uses few or no advice but is optimal in a class of estimators which use much advice (see e.g.…”

Section: Overviewmentioning

confidence: 99%

“…Several authors starting from Gaifman studied the idea of assigning probabilities to sentences in formal logic [1][2][3][4][5]. Systems of formal logic such as Peano Arithmetic are very expressive, so such an assignment would have much broader applicability than most of the examples we are concerned about in the present work.…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Polynomial-Time Estimators: A Bayesian Notion of Approximation Algorithm

Kosoy¹,

Appel²

2016

Preprint

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The concept of an "approximation algorithm" is usually only applied to optimization problems, since in optimization problems the performance of the algorithm on any given input is a continuous parameter. We introduce a new concept of approximation applicable to decision problems and functions, inspired by Bayesian probability. From the perspective of a Bayesian reasoner with limited computational resources, the answer to a problem that cannot be solved exactly is uncertain and therefore should be described by a random variable. It thus should make sense to talk about the expected value of this random variable, an idea we formalize in the language of average-case complexity theory by introducing the concept of "optimal polynomialtime estimators." We prove some existence theorems and completeness results, and show that optimal polynomial-time estimators exhibit many parallels with "classical" probability theory.

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Asymptotic Logical Uncertainty and the Benford Test

Abstract: We give an algorithm AL,T which assigns probabilities to logical sentences. For any simple infinite sequence {φs n } of sentences whose truthvalues appear indistinguishable from a biased coin that outputs "true" with probability p, we have limn→∞ AL,T (sn) = p.

Cited by 4 publications

References 26 publications

Inductive Coherence

Inductive Coherence

Logical Induction

Optimal Polynomial-Time Estimators: A Bayesian Notion of Approximation Algorithm

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