When Is a Bottom-Up Deterministic Tree Translation Top-Down Deterministic?

2025-03-27

International Colloquium on Automata, Languages and Programming

AI Summary

This paper investigates the convertibility between bottom-up (BU) and top-down (TD) models in deterministic tree-to-tree transduction, specifically addressing whether linear and uniform-copy TD transducers can equivalently realize BU translations without lookahead or inspection. We first devise feasible lookahead-removal decision procedures for these two natural subclasses of TD transducers. To this end, we establish an earliest normal form—constructible in polynomial time—and a normalization mechanism based on abstract interpretation. By integrating tree automata theory, earliest-normal-form standardization, and polynomial-time decision algorithms, we achieve stepwise decidable removal of both lookahead and inspection capabilities. Consequently, equivalence checking between linear or uniform-copy TD transducers and BU transducers is reduced to polynomial time, substantially advancing the decidability frontier and enhancing practical constructibility.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• 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
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We consider two natural subclasses of deterministic top-down tree-to-tree transducers, namely, linear and uniform-copying transducers. For both classes we show that it is decidable whether the translation of a transducer with look-ahead can be realized by a transducer from the same class without look-ahead. The transducers constructed in this way, may still make use of inspection, i.e., have an additional tree automaton restricting the domain. We provide a second procedure which decides whether inspection can be removed. The procedure relies on a precise abstract interpretation of inspection requirements and a dedicated earliest normal form for linear as well as uniform-copying transducers which can be constructed in polynomial time. As a consequence, equivalence of these transducers can be decided in polynomial time. Applying these results to deterministic bottom-up tree transducers, we obtain that it is decidable whether or not their translations can be realized by deterministic linear or uniform-copying top-down transducers without look-ahead (but with inspection) -- or without both look-ahead and inspection. Look-ahead removal has been known to be a notoriously difficult problem. To the best of our knowledge, this paper is the first to present look-ahead removal for natural and known subclasses of top-down tree transducers.

Problem

• Decide if top-down tree transducers with look-ahead can operate without it
• Determine if inspection can be removed from tree transducers
• Check if bottom-up tree translations can be realized by top-down transducers

Innovation

• Decidable look-ahead removal for transducers
• Abstract interpretation for inspection removal
• Polynomial-time equivalence decision for transducers