Random expansions of trees with bounded height

Abstract

We consider a sequence $\mathbf{T} = (\mathcal{T}_n : n \in \mathbb{N}^+)$ of trees $\mathcal{T}_n$ where, for some $\Delta \in \mathbb{N}^+$ every $\mathcal{T}_n$ has height at most $\Delta$ and as $n \to \infty$ the minimal number of children of a nonleaf tends to infinity. We can view every tree as a (first-order) $\tau$-structure where $\tau$ is a signature with one binary relation symbol. For a fixed (arbitrary) finite and relational signature $\sigma \supseteq \tau$ we consider the set $\mathbf{W}_n$ of expansions of $\mathcal{T}_n$ to $\sigma$ and a probability distribution $\mathbb{P}_n$ on $\mathbf{W}_n$ which is determined by a (parametrized/lifted) Probabilistic Graphical Model (PGM) $\mathbb{G}$ which can use the information given by $\mathcal{T}_n$. The kind of PGM that we consider uses formulas of a many-valued logic that we call $PLA^*$ with truth values in the unit interval $[0, 1]$. We also use $PLA^*$ to express queries, or events, on $\mathbf{W}_n$. With this setup we prove that, under some assumptions on $\mathbf{T}$, $\mathbb{G}$, and a (possibly quite complex) formula $\varphi(x_1, \ldots, x_k)$ of $PLA^*$, as $n \to \infty$, if $a_1, \ldots, a_k$ are vertices of the tree $\mathcal{T}_n$ then the value of $\varphi(a_1, \ldots, a_k)$ will, with high probability, be almost the same as the value of $\psi(a_1, \ldots, a_k)$, where $\psi(x_1, \ldots, x_k)$ is a ``simple'' formula the value of which can always be computed quickly (without reference to $n$), and $\psi$ itself can be found by using only the information that defines $\mathbf{T}$, $\mathbb{G}$ and $\varphi$. A corollary of this, subject to the same conditions, is a probabilistic convergence law for $PLA^*$-formulas.