A Markov logic network (MLN) $\mathbb{M}$ determines a probability distribution $\mathbb{P}_n^\mathbb{M}$ on the set $\mathbf{W}_n$ of structures, or ``possible worlds'', with domain $\{1, \ldots, n\}$. We study the properties of such distributions as $n$ tends to infinity. We show that with mild assumptions on an MLN $\mathbb{M}$ with one soft constraint with an arbitrary positive weight the distribution $\mathbb{P}_n^\mathbb{M}$ will behave quite differently from the uniform distribution $\mathbb{P}_n^{uni}$ on $\mathbf{W}_n$ for all sufficiently large $n$. For a language with only one relation symbol $R$ which has arity 1 we give an almost complete characterization of the possible asymptotic behaviours of $\mathbb{P}_n^\mathbb{M}$ as $n \to \infty$, where $\mathbb{M}$ may be any MLN for this language. The asymptotic behaviour depends on the soft constraints and weights of the MLN. This characterization is used to show that if the language under consideration contains at least one relation symbol of arity 1 then the following holds: (a) There is an MLN $\mathbb{M}$ such that for every lifted Bayesian network (LBN) $\mathbb{G}$ there are infinitely many $n$ such that $\mathbb{M}$ and $\mathbb{G}$ determine different distributions on $\mathbf{W}_n$. (b) There is an LBN $\mathbb{G}$ such that for every MLN $\mathbb{M}$ there are infinitely many $n$ such that $\mathbb{G}$ and $\mathbb{M}$ determine different distributions on $\mathbf{W}_n$. We also show that, in the limit, the weight dimension and the domain size dimension may behave completely differently.

翻译：一个马尔可夫逻辑网络（MLN）$\mathbb{M}$在结构集（或“可能世界”）$\mathbf{W}_n$上确定一个概率分布$\mathbb{P}_n^\mathbb{M}$，其中域为$\{1, \ldots, n\}$。我们研究当$n$趋于无穷时此类分布的性质。我们证明，在温和假设下，对于一个具有一个任意正权重软约束的MLN $\mathbb{M}$，对于所有足够大的$n$，分布$\mathbb{P}_n^\mathbb{M}$的行为将与$\mathbf{W}_n$上的均匀分布$\mathbb{P}_n^{uni}$截然不同。对于仅包含一个一元关系符号$R$的语言，我们给出了当$n \to \infty$时$\mathbb{P}_n^\mathbb{M}$可能渐近行为的几乎完整刻画，其中$\mathbb{M}$可以是该语言的任意MLN。这种渐近行为取决于MLN的软约束和权重。该刻画被用于证明：若考虑的语言至少包含一个一元关系符号，则以下结论成立：(a)存在一个MLN $\mathbb{M}$，使得对于每个提升贝叶斯网络（LBN）$\mathbb{G}$，存在无穷多个$n$使得$\mathbb{M}$和$\mathbb{G}$在$\mathbf{W}_n$上确定不同的分布。(b)存在一个LBN $\mathbb{G}$，使得对于每个MLN $\mathbb{M}$，存在无穷多个$n$使得$\mathbb{G}$和$\mathbb{M}$在$\mathbf{W}_n$上确定不同的分布。我们还表明，在极限情况下，权重维度和域规模维度可能表现出完全不同的行为。