Random expansions of trees with bounded height

Abstract

We consider a sequence $T=(T_n:n\in N^{+})$ of trees $T_n$ where, for some $\Delta \in N^{+}$ every $T_n$ has height at most Δ 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) τ-structure where τ is a signature with one binary relation symbol. For a fixed (arbitrary) finite and relational signature $\sigma \supseteq \tau$ we consider the set $W_n$ of expansions of $T_n$ to σ and a probability distribution $P_n$ on $W_n$ which is determined by a (parametrized/lifted) Probabilistic Graphical Model (PGM) $G$ which can use the information given by $T_n$.

The kind of PGM that we consider uses formulas of a many-valued logic that we call $PL{A}^{⁎}$ with truth values in the unit interval $[0,1]$. We also use $PL{A}^{⁎}$ to express queries, or events, on $W_n$. With this setup we prove that, under some assumptions on T, $G$, and a (possibly quite complex) formula $\phi (x_1,\dots ,x_k)$ of $PL{A}^{⁎}$, as $n\to \infty$, if $a_1,\dots ,a_k$ are vertices of the tree $T_n$ then the value of $\phi (a_1,\dots ,a_k)$ will, with high probability, be almost the same as the value of $\psi (a_1,\dots ,a_k)$, where $\psi (x_1,\dots ,x_k)$ is a “simple” formula the value of which can always be computed quickly (without reference to n), and ψ itself can be found by using only the information that defines T, $G$ and φ. A corollary of this, subject to the same conditions, is a probabilistic convergence law for $PL{A}^{⁎}$-formulas.