逻辑归纳
Logical Induction
摘要：
我们提出了一个可计算的算法，分配概率的每一个逻辑语句在一个给定的形式语言，并细化这些概率随着时间的推移。例如，如果语言是皮亚诺算术，它会为所有算术语句分配概率，包括关于孪生素数猜想的声明、长时间运行计算的输出以及它自己的概率。我们表明，我们的算法，我们称之为逻辑电感器的一个实例，满足了一些直观的desiderata，包括：（1）它学会预测模式的真和假的逻辑语句，往往早在有资源来评估的语句，只要模式可以写下来在多项式时间;（2）它学会使用适当的统计摘要来预测真值呈现伪随机的语句序列;（3）它学会对自己当前的信念有准确的信念，以避免自我参照的标准悖论。例如，如果给定的计算机程序仅产生一定范围内的输出，则逻辑感应器以及时的方式学习该事实;并且如果$π$的十进制扩展中的后期数字难以预测，则逻辑感应器学习将$\approx 10\%$概率分配给“$π$的$n$第位是7”，用于大$n$。逻辑归纳器也学会相信他们的未来信念，而不是他们的当前信念，并且他们的信念在极限中是一致的（无论何时$ϕ\implies ψ$，$\mathbb{P}_\infty(ϕ) \le \mathbb{P}_\infty(ψ)$，等等）;逻辑归纳器严格支配极限中的泛半测度。这些性质和许多其他性质都遵循一个单一的逻辑归纳标准，这是由一系列股票交易类比的动机。粗略地说，每个逻辑句子$ϕ$都与每股价值1美元的股票相关联，如果[.]
英文摘要：
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 $\approx 10\%$ probability to "the $n$th 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 $ϕ\implies ψ$, $\mathbb{P}_\infty(ϕ) \le \mathbb{P}_\infty(ψ)$, 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 [...]
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
