Domain size asymptotics for Markov logic networks 文章

arXiv:2509.04192v2 Announce Type: replace Abstract: 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.