研究带权重的关系网络在大域下的逻辑概率极限，发现缩放权重后出现确定性规律。
Random coloured digraphs defined by a Markov logic network
- 通过缩放权重使每条逻辑命题概率趋近0或1
- 未缩放权重时出现7种不同渐近行为，可能有相变现象
- 结果对大规模推理有帮助，尤其适合关注渐近性质的研究者
马尔可夫逻辑网络（MLN）是一种用于统计关系人工智能的概率关系模型，定义在任意有限领域D上的可能世界概率分布。本文考虑一个语言体系，包含属性P(x)和关系R(x,y)，并假设所有P(x)与R(x,y)的布尔组合均为软约束（附带非负实数权重）。设领域大小为n，我们证明：当所有权重按1/n缩放时，对任意一阶逻辑句子φ，其成立概率随n→∞趋于0或1——即满足一阶逻辑的0-1律；且该极限与权重无关。若采用标准语义（不缩放权重），极限行为更复杂，依赖于权重，出现7种定性不同的情况：部分满足0-1律，部分不满足但可能存在收敛律。权重影响可能引发从一种情形到另一种的“突变式相变”。收敛律的存在对大规模领域上的推理具有积极意义。
原文摘要 · Abstract (English)
A Markov Logic Network (MLN) is a probabilistic relational model used in Statistical Relational Artificial Intelligence for defining a probability distribution on the set of possible worlds with domain $D$ for an arbitrary finite domain $D$. An MLN consists of soft constraints with associated weights which are nonnegative real numbers. In this study we consider a language speaking about a property $P(x)$ and a relation $R(x, y)$. We consider an MLN for which every Boolean combination of $P(x)$ and $R(x, y)$ is a soft constraint (with associated weight). Let $n$ denote the size (cardinality) of the domain. We show that, for every choice of weights, if the weights are scaled by $1/n$ then, for every first-order sentence $φ$, the probability that $φ$ holds tends to either 0 or 1 as $n \to \infty$; that is, a 0-1 law for first-order logic holds. Morover, the limit probability does {\em not} depend on the weights. If we instead use the standard semantics of MLNs, in the case of which the weights are {\em not} scaled, then the limit behaviour is more complicated and {\em depends} on the weights. With unscaled weights we get 7 qualitatively different cases which depend on the weights. In some cases we have a 0-1 law for first-order logic, in some cases not, but we may still have a convergence law. The influence of the weights on the asymptotic probability of a first-order sentence may be in the form of a sudden ``phase transition'' from one of the 7 cases to another. The presence of a convergence law has positive implications for inference on large domains.
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