Deciding Linear Height and Linear Size-to-Height Increase for Macro Tree Transducers

TL;DR

This paper introduces a new normal form for macro tree transducers that helps determine whether their output growth is linear in height or size-to-height ratio, improving analysis of their output complexity.

Contribution

It proposes a depth proper normal form for macro tree transducers, enabling decision procedures for linear height and size-to-height increase in their translations.

Findings

Normal form guarantees parameters appear at arbitrary depths

Iterative construction achieves depth-normal form

Decides linear height and size-to-height increase

Abstract

We present a novel normal form for (total deterministic) macro tree transducers (mtts), called depth proper normal form. If an mtt is in this normal form, then it is guaranteed that each parameter of each state of the mtt appears at arbitrary depth in the output trees of that state. Intuitively, if some parameter only appears at certain bounded depths in the output trees of a state, then this parameter can be removed by in-lining the corresponding output paths at each call site of that state. We use regular look-ahead in order to determine which of the paths should be in-lined. As a consequence of changing the lookahead, a parameter that was previously appearing at unbounded depths, may be appearing at bounded depths for some new look-ahead; for this reason, our construction has be iterated in order to obtain an mtt in depth-normal form. Using the normal form, we can decide whether the translation of an mtt has linear height increase or has linear size-to-height increase.