Abstract
Almost all formal theories of intelligence suffer from the problem of logical omniscience, the assumption that an agent already knows all consequences of its beliefs. Logical uncertainty codifies uncertainty about the consequences of existing beliefs. This implies a departure from beliefs governed by standard probability theory. Here, we study the asymptotic properties of beliefs on quickly computable sequences of logical sentences. Motivated by an example we call the Benford test, we provide an approach which identifies when such subsequences are indistinguishable from random, and learns their probabilities.
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Notes
- 1.
We will use the term “probability” to refer to degrees of belief generally, whether or not the probability axioms are obeyed.
- 2.
The test presumes that the frequencies of the first digits in the sequence \(3\uparrow ^n 3\) satisfy Benford’s law. Though this seems likely, the conjecture is not too important; any sufficiently fast-growing sequence satisfying Benford’s law could serve as an example.
- 3.
We tailored this definition of irreducible pattern to our needs. The theory of algorithmic randomness may offer alternatives. However, algorithmic randomness generally considers all computable tests and focuses on the case where \(p=\frac{1}{2}\) [22, 23]. We believe that any reasonable definition inspired by algorithmic randomness would imply Definition 2.
- 4.
See pre-print version [24] for the full proof.
- 5.
See pre-print version [24] for the full proof.
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Garrabrant, S. et al. (2016). Asymptotic Logical Uncertainty and the Benford Test. In: Steunebrink, B., Wang, P., Goertzel, B. (eds) Artificial General Intelligence. AGI 2016. Lecture Notes in Computer Science(), vol 9782. Springer, Cham. https://doi.org/10.1007/978-3-319-41649-6_20
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