When is a Bottom-Up Deterministic Tree Translation Top-Down Deterministic?
Sebastian Maneth, Helmut Seidl
TL;DR
This work analyzes two natural deterministic top-down tree-transducer subclasses, linear and uniform-copying, and establishes decidability for removing look-ahead (and inspection) while staying within the same class. It introduces dedicated earliest normal forms and associated polynomial-time procedures to decide equivalence and to transform transducers with look-ahead into equivalent forms without look-ahead (but possibly with inspection), as well as to remove inspection when feasible. Central to the approach are canonical earliest forms, a precise abstract interpretation of inspection needs, and polynomial-time constructions for maximal common prefixes and equivalence relations among states. The results yield polynomial-time definability for equivalence in these subclasses and extend to deterministic bottom-up transducers, showing decidability of realizability by uniform-copying or linear top-down transducers under look-ahead constraints. This advances understanding of look-ahead removal in natural and known transducer subclasses and clarifies the roles of domain inspection and prefix-based normal forms in decidability and construction. The insights have potential implications for compiler-like tree transformations and formal verification where efficient determination of definable translations is crucial.
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.