Type-Checking for Pattern-Based Tree Transformations

2026-10-08

AI Summary

This study addresses the long-standing limitation regarding the decidability of type checking in pattern tree transformations, which hinders ensuring that such transformations preserve regular properties. To overcome this challenge, the authors propose a finitely representable model for pattern tree transformations that integrates pattern matching with tree automata theory. By reducing the type checking problem to the emptiness decision problem for alternating tree automata, this work breaks through the traditional undecidability barrier associated with equivalence while retaining strong expressive power. The authors rigorously prove the decidability of type checking under this model and successfully construct a corresponding decision procedure. This procedure effectively verifies that the proposed transformations preserve regular properties, offering both theoretical advancement and practical utility for typed tree transformation systems.

Abstract

We introduce and study pattern-based tree transformations. As an illustrating example, consider a source pattern $(x \cdot y) + (x \cdot z)$ and a target pattern $x \cdot (y + z)$ as a pair. This source pattern matches any expression $e$ of the form $(e_1 \cdot e_2) + (e_1 \cdot e_3)$ (by substituting $x$ with $e_1$, $y$ with $e_2$, and $z$ with $e_3$) and the pair transforms it into the expression $e_1 \cdot (e_2 + e_3)$ as dictated by the target pattern. Note that in this example, the set of expressions that match the source pattern is not a regular tree language. We propose a model of tree transformations given by a finite representation of a (possibly infinite) set of such (source pattern, target pattern) pairs. The expressive power of this model comes at the cost of undecidability of checking equivalence. Nevertheless, we show that the type-checking problem is decidable for our model of pattern-based tree transformations. The type-checking problem asks whether applying a given transformation to trees having a given regular property (type) preserves the property. Our decision procedure is by a reduction to the emptiness problem of alternating tree automata.

Problem

• pattern-based tree transformations
• type-checking
• regular tree language
• decidability

Innovation

• pattern-based tree transformations
• type-checking
• alternating tree automata
• decidability
• tree automata