Random expansions of trees with bounded height

2024-10-15

arXiv.org

AI Summary

This work investigates the asymptotic probabilistic behavior of complex logical queries over sequences of random trees with bounded height. To address the computational intractability of evaluating intricate logical formulas as tree size grows, we propose the many-valued logic framework PLA*, which—uniquely—embeds probabilistic semantics into tree-structure expansions and constructs a parametric probabilistic graphical model. We prove that, under the regime where tree height is bounded and branching factor diverges to infinity, the value of any PLA* formula converges in probability to that of its efficiently computable “simplified formula.” This yields a rigorous probabilistic convergence law. Our contribution establishes the *asymptotic reducibility* theory for logical queries on tree structures, providing both a novel paradigm and scalable algorithmic foundations for probabilistic inference over large-scale hierarchical data.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Inference
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $n o infty$ the minimal number of children of a nonleaf tends to infinity. We can view every tree as a (first-order) $ au$-structure where $ au$ is a signature with one binary relation symbol. For a fixed (arbitrary) finite and relational signature $sigma supseteq au$ 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 $n o infty$, 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.

Problem

• Study random expansions of bounded-height trees using probabilistic graphical models.
• Analyze convergence of complex logic formulas to simpler, computable forms.
• Establish probabilistic convergence laws for many-valued logic formulas on trees.

Innovation

• Uses Probabilistic Graphical Model (PGM)
• Employs many-valued logic PLA*
• Simplifies complex formulas probabilistically