We consider logics with truth values in the unit interval $[0,1]$. Such logics are used to define queries and to define probability distributions. In this context the notion of almost sure equivalence of formulas is generalized to the notion of asymptotic equivalence. We prove two new results about the asymptotic equivalence of formulas where each result has a convergence law as a corollary. These results as well as several older results can be formulated as results about the relative asymptotic expressivity of inference frameworks. An inference framework $\mathbf{F}$ is a class of pairs $(\mathbb{P}, L)$, where $\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots)$, $\mathbb{P}_n$ are probability distributions on the set $\mathbf{W}_n$ of all $\sigma$-structures with domain $\{1, \ldots, n\}$ (where $\sigma$ is a first-order signature) and $L$ is a logic with truth values in the unit interval $[0, 1]$. An inference framework $\mathbf{F}'$ is asymptotically at least as expressive as an inference framework $\mathbf{F}$ if for every $(\mathbb{P}, L) \in \mathbf{F}$ there is $(\mathbb{P}', L') \in \mathbf{F}'$ such that $\mathbb{P}$ is asymptotically total variation equivalent to $\mathbb{P}'$ and for every $\varphi(\bar{x}) \in L$ there is $\varphi'(\bar{x}) \in L'$ such that $\varphi'(\bar{x})$ is asymptotically equivalent to $\varphi(\bar{x})$ with respect to $\mathbb{P}$. This relation is a preorder. If, in addition, $\mathbf{F}$ is at least as expressive as $\mathbf{F}'$ then we say that $\mathbf{F}$ and $\mathbf{F}'$ are asymptotically equally expressive. Our third contribution is to systematize the new results of this paper and several previous results in order to get a preorder on a number of inference systems that are of relevance in the context of machine learning and artificial intelligence.

翻译：我们考虑真值在单位区间$[0,1]$内的逻辑。此类逻辑被用于定义查询及概率分布。在此背景下，公式的几乎必然等价概念被推广为渐近等价概念。我们证明了关于公式渐近等价的两个新结果，其中每个结果都以收敛律作为推论。这些结果以及若干较早的研究成果，均可表述为关于推理框架相对渐近表达能力的结果。一个推理框架$\mathbf{F}$是一类对$(\mathbb{P}, L)$的集合，其中$\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots)$，$\mathbb{P}_n$是定义在所有具有论域$\{1, \ldots, n\}$的$\sigma$-结构（其中$\sigma$为一阶签名）的集合$\mathbf{W}_n$上的概率分布，而$L$是一个真值在单位区间$[0, 1]$内的逻辑。若对于每个$(\mathbb{P}, L) \in \mathbf{F}$，均存在$(\mathbb{P}', L') \in \mathbf{F}'$使得$\mathbb{P}$与$\mathbb{P}'$渐近全变差等价，且对于每个$\varphi(\bar{x}) \in L$，均存在$\varphi'(\bar{x}) \in L'$使得$\varphi'(\bar{x})$相对于$\mathbb{P}$与$\varphi(\bar{x})$渐近等价，则称推理框架$\mathbf{F}'$在渐近意义上至少与推理框架$\mathbf{F}$具有同等表达能力。此关系构成一个预序。若在此基础上，$\mathbf{F}$也至少与$\mathbf{F}'$具有同等表达能力，则称$\mathbf{F}$与$\mathbf{F}'$是渐近同等表达的。我们的第三项贡献是，将本文的新结果与若干先前结果系统化，从而在机器学习和人工智能背景下相关的多个推理系统上建立一个预序关系。