Pattern-based tree transformations type-checking made decidable
Type-Checking for Pattern-Based Tree Transformations
Formal Languages and Automata TheoryComputation and Language
Summary
The paper deals with a way to change tree-structured data by matching and rewriting patterns, like turning parts of a math expression into a simpler form. The authors look at complicated patterns that can match many trees, even infinitely many, which makes checking if two transformations are the same impossible. However, they found that you can still decide if a transformation always keeps certain tree properties intact, which helps ensure safe changes to data structures. They do this by converting the problem into one related to special tree automata, a kind of computational model.
What this means in practice
- •For program analysis teams: Verify that code transformations preserve important structural invariants when manipulating abstract syntax trees for optimization or refactoring.
- •For compiler developers: Ensure correctness when applying complex pattern-based tree rewrites during compilation stages for safe and predictable code generation.
A theory result. No direct application yet.
Authors
C. Aiswarya, Sahil Mhaskar, M. Praveen
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.