# Asymptotic Conditional Probabilities: The Unary Case

@article{Grove1996AsymptoticCP,
title={Asymptotic Conditional Probabilities: The Unary Case},
author={Adam J. Grove and Joseph Y. Halpern and Daphne Koller},
journal={SIAM J. Comput.},
year={1996},
volume={25},
pages={1-51}
}
• Published 1 February 1996
• Mathematics, Computer Science
• SIAM J. Comput.
Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences $\phi$ and $\theta$, we consider the structures with domain $\{1,\ldots, N\}$ that satisfy $\theta$, and compute the fraction of them in which $\phi$ is true. We then consider what happens to this fraction as $N$ gets large. This extends the work on 0-1 laws that considers the limiting probability of…

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## References

SHOWING 1-10 OF 57 REFERENCES

### Asymptotic conditional probabilities: The non-unary case

• Mathematics
Journal of Symbolic Logic
• 1996
This work considers the problem of computing asymptotic conditional probabilities for first-order sentences, and shows the complexity of three problems with respect to this limit: deciding whether it is well defined, whether it exists, and whether it lies in some nontrivial interval to be highly undecidable.

### Asymptomatic conditional probabilities for first-order logic

• Computer Science, Mathematics
STOC '92
• 1992
It is shown that in this general case, almost all the questions one might want to ask (such as deciding whether the asymptotic probability exists) are highly undecidable, and that the situation with unary predicates only is much better.

### Random worlds and maximum entropy

• Computer Science
[1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science
• 1992
It is shown that when the vocabulary underlying phi and theta uses constants and unary predicates only, one can in many cases use a maximum entropy computation to compute the degree of belief.

### Concerning measures in first order calculi

~0. Introduction. The idea of treating probability as a real valued function defined on sentences is an old one (see ['6] and [7], where other references can be found). Carnap's at tempt to set up a

### Probabilities on finite models

It is shown that μ n (σ) always converges to 0 or 1 as n → ∞, and that the rate of convergence is geometrically fast, and the spectrum of a sentence σ is defined to be the set of cardinalities of finite models of σ.

### From Statistics to Beliefs

• Computer Science
AAAI
• 1992
An intelligent agent uses known facts, including statistical knowledge, to assign degrees of belief to assertions it is uncertain about, and three principled techniques for doing this are investigated, all of which are applications of the principle of indifference.

### Laws in Logic and Combinatorics

This is a survey of logical results concerning random structures. A class of relational structures on which a (finitely additive) probability measure has been defined has a 0–1 law for a particular