The $\epsilon$-logic (which is called $\epsilon$E-logic in this paper) of
Kuyper and Terwijn is a variant of first order logic with the same syntax, in
which the models are equipped with probability measures and in which the
$\forall x$ quantifier is interpreted as "there exists a set $A$ of measure
$\ge 1 - \epsilon$ such that for each $x \in A$, ...." Previously, Kuyper and
Terwijn proved that the general satisfiability and validity problems for this
logic are, i) for rational $\epsilon \in (0, 1)$, respectively
$\Sigma^1_1$-complete and $\Pi^1_1$-hard, and ii) for $\epsilon = 0$,
respectively decidable and $\Sigma^0_1$-complete. The adjective "general" here
means "uniformly over all languages."
We extend these results in the scenario of finite models. In particular, we
show that the problems of satisfiability by and validity over finite models in
$\epsilon$E-logic are, i) for rational $\epsilon \in (0, 1)$, respectively
$\Sigma^0_1$- and $\Pi^0_1$-complete, and ii) for $\epsilon = 0$, respectively
decidable and $\Pi^0_1$-complete. Although partial results toward the countable
case are also achieved, the computability of $\epsilon$E-logic over countable
models still remains largely unsolved. In addition, most of the results, of
this paper and of Kuyper and Terwijn, do not apply to individual languages with
a finite number of unary predicates. Reducing this requirement continues to be
a major point of research.
On the positive side, we derive the decidability of the corresponding
problems for monadic relational languages --- equality- and function-free
languages with finitely many unary and zero other predicates. This result holds
for all three of the unrestricted, the countable, and the finite model cases.
Applications in computational learning theory, weighted graphs, and neural
networks are discussed in the context of these decidability and undecidability
results.

本文研究的是 epsilonE-logic 中关于有限模型的可满足性和有效性问题，特别是对于有理的 epsilon，结果为 Sigma^0_1 和 Pi^0_1 完备性，而对于 epsilon=0，结果为可判定，但对于可数模型的 εE 逻辑的可计算性仍然存在研究点；此外，还讨论了在计算学习理论，加权图和神经网络中的应用。

有限及可枚举模型中概率逻辑的有效性和可满足性的可计算性

Computabilities of Validity and Satisfiability in Probability Logics  over Finite and Countable Models

This note introduces the method of cross-conformal prediction, which is a
hybrid of the methods of inductive conformal prediction and cross-validation,
and studies its validity and predictive efficiency empirically.

介绍了交叉一致性预测方法，它是归纳一致性预测和交叉验证方法的混合，经验上研究了它的有效性和预测效率。