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$[0,1]$内的逻辑系统。此类逻辑既可用于定义查询，也可用于定义概率分布。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，每个结果都以收敛律作为推论。这些结果以及若干早期结论，均可表述为关于推理框架相对渐近表达力的研究结果。推理框架$\mathbf{F}$是由二元组$(\mathbb{P}, L)$构成的类，其中$\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots)$，$\mathbb{P}_n$是定义在所有定义域为$\{1, \ldots, n\}$的$\sigma$-结构集合$\mathbf{W}_n$上的概率分布（$\sigma$为一阶签名），$L$是真值在单位区间$[0, 1]$内的逻辑系统。若对于任意$(\mathbb{P}, L) \in \mathbf{F}$，均存在$(\mathbb{P}', L') \in \mathbf{F}'$使得$\mathbb{P}$与$\mathbb{P}'$渐近全变差等价，且对于任意$\varphi(\bar{x}) \in L$，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$\mathbf{F}$具有同等表达力。该关系构成预序。若在此基础上$\mathbf{F}$也至少与$\mathbf{F}'$具有同等表达力，则称$\mathbf{F}$与$\mathbf{F}'$具有渐近等同的表达力。本文的第三项贡献在于：通过系统整合本文的新结果与若干已有结论，对机器学习与人工智能领域中具有重要意义的推理系统建立了预序关系。