On normalization and type checking for tree transducers

Tree transducers are an expressive formalism for reasoning about tree structured data. Practical applications range from XSLT-like document transformations to translations of natural languages. Important problems for transducers are to decide whether two transducers are equivalent, to construct normal forms, give semantic characterizations, and type checking, i.e., to check whether the produced outputs satisfy given structural constraints. This thesis addresses these problems for important classes of tree transducers. Constructive solutions are provided and classes of transducers for which these algorithms run in polynomial time, are identified. Equivalence testing, normalization, and semantic characterization are often solved together by the use of a Myhill-Nerode theorem. This identifies necessary and sufficient semantic properties for transformations definable by a specific class of transducers. The theorem also implies that a unique normal form of those transducers exists. Moreover, it implies that, given a transducer, the normal transducer can be constructed. This immediately leads to the question: Are there classes of tree transducers for which a Myhill-Nerode theorem exists? We give an affirmative answer for the class of deterministic bottom-up tree transducers. A semantic characterization of transformations definable by these transducers is presented, and, moreover, it is evidenced that for every deterministic bottom-up tree transducer, a unique equivalent transducer can be constructed, which is minimal. The construction is based on a sequence of normalizing transformations, which, among others, guarantee that non-trivial output is produced as early as possible. For a deterministic bottom-up transducer where every state produces either none or infinitely many outputs, the minimal transducer can be constructed in polynomial time. One of the useful properties of tree walking transducers is decidability of type checking: Given a transducer and input and output types, it can be checked statically whether each document adhering to the input type is necessarily transformed by the transducer into documents adhering to the output type. Here, a “type” means a regular set of trees specified by a finite-state tree automaton. Usually, type checking of tree transducers is extremely expensive; already for simple top-down tree transducers it is known to be EXPTIME-complete. Are there expressive classes of tree transducers for which type checking can be performed in polynomial time? Most of the previous approaches are based on inverse type inference. In contrast, the approach presented here uses forward