We show that every prenex universal syntactic first-order safety property can be compiled into a universal invariant of a first-order transition system using quantifier-free substitutions only. We apply this insight to prove that every such safety property is decidable for first-order transition systems with stratified guarded updates only.

It is well known that for a given bottom-up tree automaton it can be decided whether or not an equivalent deterministic top-down tree automaton exists. Recently it was claimed that such a decision can be carried out in polynomial time (Leupold and Maneth, FCT'2021); but their procedure and corresponding property is wrong. Here we address this mistake and present a correct property which allows to determine in polynomial time whether or not a given tree language can be recognized by a deterministic top-down tree automaton. Furthermore, our new property is stated for arbitrary deterministic bottom-up tree automata, and not only for minimal such automata (as before).

We consider a notion of origin for deterministic macro tree transducers with look-ahead which records for each output node, the corresponding input node for which a rule-application generated that output node. With respect to this natural notion, we show that “origin equivalence” is decidable — whenever the transducers are weakly self-nesting. The latter means that whenever two nested calls on the same input node occur, then there must be at least one other node (a terminal output node or a call on another input node) in between these nested calls. Besides origin equivalence we are also able to decide “origin injectivity” for such transducers.

MSO logic on unranked trees has been identified as a convenient theoretical framework for reasoning about expressiveness and implementations of practical XML query languages. As a corresponding theoretical foundation of XML transformation languages, the "transformation language" TL is proposed. This language is based on the "document transformation language" DTL of Maneth and Neven which incorporates full MSO pattern matching, arbitrary navigation in the input tree using also MSO patterns, and named procedures. The new language generalizes DTL by additionally allowing procedures to accumulate intermediate results in parameters. It is proved that TL -- and thus in particular DTL - despite their expressiveness still allow for effective inverse type inference. This result is obtained by means of a translation of TL programs into compositions of top-down finite state tree transductions with parameters, also called (stay) macro tree transducers.

A query against a database behind a site like Napster may search, e.g., for all users who have downloaded more jazz titles than pop music titles. In order to express such queries, we extend classical monadic second-order logic by Presburger predicates which pose numerical restrictions on the children (content) of an element node and provide a precise automata-theoretic characterization. While the existential fragment of the resulting logic is decidable, it turns out that satisfiability of the full logic is undecidable. Decidable satisfiability and a querying algorithm even with linear data complexity can be obtained if numerical constraints are only applied to those contents of elements where ordering is irrelevant. Finally, it is sketched how these techniques can be extended also to answer questions like, e.g., whether the total price of the jazz music downloaded so far exceeds a user's budget.